6620:
6068:
6615:{\displaystyle \operatorname {per} {\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}^{(\alpha ,\beta )}={\det }^{p-1}(A_{11})\operatorname {per} {\begin{pmatrix}A_{11}^{-1}&A_{11}^{-1}A_{12}\\A_{21}A_{11}^{-1}&A_{22}-A_{21}A_{11}^{-1}A_{12}\end{pmatrix}}^{((p-1){\vec {1}}_{n}-\beta ,(p-1){\vec {1}}_{n}-\alpha )}\left(\prod _{i=1}^{n}\alpha _{1,i}!\right)\left(\prod _{j=1}^{n}\beta _{1,j}!\right)(-1)^{n_{1}+\sum _{i=1}^{n}\alpha _{1,i}}}
2397:
5219:
4593:
7219:, to represent the permanent of positive-semidefinite matrices as the expected value of a specific random variable. The latter is then approximated by its sample mean. This algorithm, for a certain set of positive-semidefinite matrices, approximates their permanent in polynomial time up to an additive error, which is more reliable than that of the standard classical polynomial-time algorithm by Gurvits.
4693:-th columns are identical too.) The closure of that operator defined as the limit of its sequential application together with the transpose transformation (utilized each time the operator leaves the matrix intact) is also an operator mapping, when applied to classes of matrices, one class to another. While the compression operator maps the class of 1-semi-unitary matrices to itself and the classes of
5831:
2074:
4886:
4324:
5492:
2777:
5468:
denotes the n-vector all whose entries equal unity. This identity is an exact analog of the classical formula expressing a matrix's minor through a minor of its inverse and hence demonstrates (once more) a kind of duality between the determinant and the permanent as relative immanants. (Actually its
4697:
and 2-semi-unitary ones, the compression-closure of the 1-semi-unitary class (as well as the class of matrices received from unitary ones through replacing one row by an arbitrary row vector — the permanent of such a matrix is, via the
Laplace expansion, the sum of the permanents of 1-semi-unitary
2392:{\displaystyle {\begin{aligned}\operatorname {per} (A)&=(-1)^{n}\sum _{{J_{1}},\ldots ,{J_{p-1}}}\det(A_{J_{1}})\dotsm \det(A_{J_{p-1}})\\\det(A)&=(-1)^{n}\sum _{{J_{1}},\ldots ,{J_{p-1}}}\operatorname {per} (A_{J_{1}})\dotsm \operatorname {per} (A_{J_{p-1}})\end{aligned}}}
5214:{\displaystyle \operatorname {per} (A^{(\alpha ,\beta )})=\det ^{p-1}(A)\operatorname {per} (A^{-1})^{((p-1){\vec {1}}_{n}-\beta ,(p-1){\vec {1}}_{n}-\alpha )}\left(\prod _{i=1}^{n}\alpha _{i}!\right)\left(\prod _{j=1}^{n}\beta _{j}!\right)(-1)^{n+\sum _{i=1}^{n}\alpha _{i}}}
4588:{\displaystyle \operatorname {per} {\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}=\operatorname {det} ^{2}(A_{11})\operatorname {per} {\begin{pmatrix}A_{11}^{-1}&A_{11}^{-1}A_{12}\\A_{21}A_{11}^{-1}&A_{22}-A_{21}A_{11}^{-1}A_{12}\end{pmatrix}}}
3063:
752:
3182:
4645:+ 1)-matrix correspondingly, hence having introduced a compression operator (analogical to the Gaussian modification applied for calculating the determinant) that "preserves" the permanent in characteristic 3. (Analogically, it would be worth noting that the
5826:{\textstyle \operatorname {haf} ^{2}(A^{(\alpha ,\alpha )})=\det ^{p-1}(A)\operatorname {haf} ^{2}(A^{-1})^{((p-1){\vec {1}}_{n}-\alpha ,(p-1){\vec {1}}_{n}-\alpha )}\left(\prod _{i=1}^{n}\alpha _{i}!\right)^{2}(-1)^{n(p-1)/2+n+\sum _{i=1}^{n}\alpha _{i}}}
1882:
2529:
1326:
had asked the question of when it is possible to change the signs of some of the entries of a 01 matrix A so that the determinant of the new matrix is the permanent of A. Not all 01 matrices are "convertible" in this manner; in fact it is known
3732:
237:
3904:
266:σ while in this formula each product is unsigned. The formula may be directly translated into an algorithm that naively expands the formula, summing over all permutations and within the sum multiplying out each matrix entry. This requires
2887:
1495:
has a perfect matching, there are an odd number of edges directed along C (and thus an odd number with the opposite orientation). It was also shown that these graphs are exactly those that do not contain a subgraph homeomorphic to
49:
The permanent is defined similarly to the determinant, as a sum of products of sets of matrix entries that lie in distinct rows and columns. However, where the determinant weights each of these products with a ±1 sign based on the
1240:
3508:
576:
3350:
2919:
898:). The methods to find the formula are quite different, being related to the combinatorics of the Muir algebra, and to finite difference theory respectively. Another way, connected with invariant theory is via the
6061:
4875:
1105:
4086:
4698:
matrices and, accordingly, polynomial-time computable) is yet unknown and tensely related to the general problem of the permanent's computational complexity in characteristic 3 and the chief question of
4212:
587:
2079:
7026:
1393:
7205:
7172:
3070:
4713:
Besides, the problem of finding and classifying any possible analogs of the permanent-preserving compressions existing in characteristic 3 for other prime characteristics was formulated (
2457:
1592:
1751:
2781:
with the sum taken over the same set of indexes. Moreover, in characteristic zero a similar convolution sum expression involving both the permanent and the determinant yields the
7093:
6787:
3988:
3247:
5432:(if some row's or column's multiplicity equals zero it would mean that the row or column was removed, and thus this notion is a generalization of the notion of submatrix), and
7128:
807:
7174:, but this is considered an infeasible class in general. It is NP-hard to approximate permanents of PSD matrices within a subexponential factor, and it is conjectured to be
5466:
2495:
2059:
4706:), if such a compression-closure is the set of all square matrices over a field of characteristic 3 or, at least, contains a matrix class the permanent's computation on is
6965:
1493:
6709:
1664:
852:
416:
7607:
6736:
4226:
polynomial in characteristic 2: while computing it on the unitary matrices is polynomial-time feasible, the problem is #2-P-complete for the k-semi-unitary ones for any
3609:
2001:
5332:
1419:
135:
3768:
452:
362:
6642:
5977:
5948:
4771:
3824:
1690:
1527:
1141:
6662:
5915:
5888:
5861:
4791:
4313:
4286:
4259:
3564:
3382:
5418:
5379:
2914:
1912:
1619:
479:
389:
318:
7055:
6859:
2788:
7370:
Chakhmakhchyan, Levon; Cerf, Nicolas; Garcia-Patron, Raul (2017), "A quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices",
6939:
6919:
6899:
6879:
6756:
6682:
5487:
5352:
5289:
5269:
5249:
4743:
4156:
4129:
4109:
4008:
3950:
3930:
3817:
3788:
3604:
3528:
3418:
3267:
3209:
2522:
2021:
1972:
1952:
1932:
1735:
1467:
1439:
1349:
872:
2772:{\displaystyle \operatorname {haf} ^{2}(A)=(-1)^{n}\sum _{{J_{1}},\ldots ,{J_{p-1}}}\det(A_{J_{1}})\dotsm \det(A_{J_{p-1}})(-1)^{|J_{1}|+\dots +|J_{(p-1)/2}|}}
910:). The formula generalizes to infinitely many others, as found by all these authors, although it is not clear if they are any faster than the basic one. See (
6805:
from the set of all perfect matchings in a given bipartite graph: in other words, a fully polynomial almost uniform sampler (FPAUS). This can be done using a
1146:
8019:
3423:
487:
7574:
Jerrum, M.; Sinclair, A.; Vigoda, E. (2001), "A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries",
3280:
7928:
4234:) and it was proven that in characteristic 3 there is a simple formula relating the permanents of a square matrix and its partial inverse (for
262:. This formula differs from the corresponding formula for the determinant only in that, in the determinant, each product is multiplied by the
7914:
7869:
7844:
7718:
7439:
923:
7861:
5982:
4796:
7593:
7249:
6810:
1286:
computes the number of perfect matchings in polynomial time by changing the signs of a carefully chosen subset of the entries in the
290:
7666:
7524:
4016:
886:
Another formula that appears to be as fast as Ryser's (or perhaps even twice as fast) is to be found in the two Ph.D. theses; see (
7727:
Little, C. H. C. (1974), "An extension of
Kasteleyn's method of enumerating the 1-factors of planar graphs", in Holton, D. (ed.),
7575:
4161:
3058:{\displaystyle \operatorname {ham} (A)=\sum _{J\subseteq \{2,\dots ,n\}}\det(A_{J})\operatorname {per} (A_{\bar {J}})(-1)^{|J|}.}
747:{\displaystyle \operatorname {perm} (A)=(-1)^{n}\sum _{S\subseteq \{1,\dots ,n\}}(-1)^{|S|}\prod _{i=1}^{n}\sum _{j\in S}a_{ij}.}
100:
The development of both exact and approximate algorithms for computing the permanent of a matrix is an active area of research.
8039:
7738:
7499:, Ph.D. Thesis, Department of Statistics, Loyola College, Madras, India, vol. T073, Indian Statistical Institute, Calcutta
66:
4613:− 1 rows expressible as linear combinations of another (disjoint) subset of k rows to the computation of the permanent of an (
4649:
polynomial in characteristic 2 does possess its invariant matrix compressions as well, taking into account the fact that ham(
7807:
6802:
70:
4710:(like the class of 2-semi-unitary matrices) then the permanent is computable in polynomial time in this characteristic.
1307:
6970:
4214:, the permanent of a 1-semi-unitary matrix is computable in polynomial time over fields of characteristic 3, while for
85:. This puts the computation of the permanent in a class of problems believed to be even more difficult to compute than
3177:{\displaystyle \operatorname {ham} (A)=\sum _{J\subseteq \{2,\dots ,n\}}\det(A_{J})\operatorname {det} (A_{\bar {J}})}
7546:
3392:-matrix with the entry of indexes 1,1 replaced by 0. Moreover, it may, in turn, be further generalized for a unitary
2065:
1354:
7649:
Kogan, Grigoriy (1996), "Computing permanents over fields of characteristic 3: Where and why it becomes difficult",
7177:
7144:
1445:
who showed that such matrices are precisely those that are the biadjacency matrix of bipartite graphs that have a
7886:
7460:
7330:
Meiburg, Alexander (2023), "Inapproximability of
Positive Semidefinite Permanents and Quantum State Tomography",
6828:
of the permanent, by using the FPAUS in combination with a well-known reduction from sampling to counting due to
7134:
6825:
6818:
6806:
2406:
1877:{\displaystyle \operatorname {per} (A)=(-1)^{n}\sum _{J\subseteq \{1,\dots ,n\}}\det(A_{J})\det(A_{\bar {J}}),}
1311:
1543:
8024:
6771:
1275:
263:
51:
35:
1306:) is the number of perfect matchings. This technique can be generalized to graphs that contain no subgraph
8029:
7469:
7211:
are imposed, there are more efficient algorithms known. One randomized algorithm is based on the model of
7060:
1738:
17:
7491:
7138:
8034:
6798:
3955:
3214:
1295:
899:
7698:
7389:
7208:
7098:
6775:
1446:
760:
62:
39:
5435:
3727:{\textstyle \operatorname {ham_{K}} (A)=\sum _{\sigma \in H_{n}(K)}\prod _{i=1}^{n}a_{i,\sigma (i)}}
2462:
2026:
7474:
5836:
And, as an even wider generalization for the partial inverse case in a prime characteristic p, for
4699:
6944:
1472:
232:{\displaystyle \operatorname {perm} (A)=\sum _{\sigma \in S_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}.}
7998:
7980:
7945:
7824:
7688:
7672:
7599:
7563:
7405:
7379:
7341:
6687:
1624:
1537:
1271:
812:
394:
6714:
3899:{\displaystyle \operatorname {per} (A^{-1})\operatorname {det} ^{2}(A)=\operatorname {per} (A);}
1977:
5298:
1398:
7975:
7910:
7865:
7840:
7714:
7662:
7589:
7435:
5918:
4316:
3797:
3737:
421:
331:
6627:
5953:
5924:
4756:
2882:{\textstyle \operatorname {ham} (A)=\sum _{\sigma \in H_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}}
1669:
1499:
1113:
917:
The simplest known formula of this type (when the characteristic of the field is not two) is
7990:
7937:
7900:
7892:
7816:
7767:
7747:
7654:
7638:
7581:
7555:
7533:
7479:
7427:
7397:
7351:
6647:
5893:
5866:
5839:
4776:
4707:
4646:
4291:
4264:
4237:
4223:
4219:
3537:
3355:
3185:
2782:
1263:
903:
78:
74:
7314:
5396:
5357:
2892:
1890:
1597:
457:
367:
296:
7758:
Marcus, M.; Minc, H. (1961), "On the relation between the determinant and the permanent",
7031:
6835:
1693:
1267:
243:
86:
58:
7789:
7422:
Gurvits, Leonid (2005), "On the complexity of mixed discriminants and related problems",
6824:
It is possible to approximately count the number of perfect matchings in a graph via the
1323:
7702:
7393:
6786:
is the value of the permanent and ε > 0 is arbitrary. In other words, there exists a
65:, it is generally believed that the permanent cannot be computed in polynomial time. In
7923:
7891:, Lecture Notes in Computer Science, vol. 318, Springer-Verlag, pp. 233–242,
7878:
7626:
7622:
7458:
Allender, Eric; Gore, Vivec (1994), "A uniform circuit lower bound for the permanent",
7216:
7212:
6924:
6904:
6884:
6864:
6741:
6667:
5472:
5337:
5274:
5254:
5234:
4728:
4694:
4141:
4114:
4094:
3993:
3935:
3915:
3910:
3802:
3773:
3589:
3513:
3403:
3277:-matrix), because each minor of such a matrix coincides with its algebraic complement:
3252:
3194:
3189:
2507:
2006:
1957:
1937:
1917:
1720:
1452:
1424:
1334:
1235:{\displaystyle \delta =(\delta _{1}=1,\delta _{2},\dots ,\delta _{n})\in \{\pm 1\}^{n}}
857:
31:
8013:
7941:
7751:
7642:
7629:(1986), "Random generation of combinatorial structures from a uniform distribution",
1283:
7676:
7567:
7409:
7133:
Another class of matrices for which the permanent is of particular interest, is the
3793:
This formula also implies the following identities over fields of characteristic 3:
7949:
7603:
7332:
6814:
1287:
1279:
8002:
6797:
The most difficult step in the computation is the construction of an algorithm to
7853:
7618:
1303:
1299:
282:
255:
43:
7611:
7401:
7356:
4597:
and it allows to polynomial-time reduce the computation of the permanent of an
3770:
is the set of n-permutations whose each cycle contains at least one element of
7736:
Little, C. H. C. (1975), "A characterization of convertible (0, 1)-matrices",
7731:, Lecture Notes in Mathematics, vol. 403, Springer-Verlag, pp. 63–72
7559:
7537:
7483:
3503:{\displaystyle \operatorname {ham_{K}} (U)=\operatorname {det} ^{2}(U+I_{/K})}
571:{\displaystyle \operatorname {perm} (A)=\sum _{k=0}^{n-1}(-1)^{k}\Sigma _{k}.}
89:. It is known that computing the permanent is impossible for logspace-uniform
42:
is a problem that is thought to be more difficult than the computation of the
7896:
7772:
7658:
1710:. First, there is one using symbolic calculations with partial derivatives.
875:
7585:
4111:
is the matrix whose entries are the cubes of the corresponding entries of
3345:{\displaystyle \operatorname {ham} (U)=\operatorname {det} ^{2}(U+I_{/1})}
7881:(1988), "NC algorithms for computing the number of perfect matchings in K
1291:
7431:
3184:
what therefore provides an opportunity to polynomial-time calculate the
7828:
7508:, Ph.D. Dissertation, vol. 223, California Institute of Technology
7426:, Lecture Notes in Computer Science, vol. 3618, pp. 447–458,
7837:
Symmetric
Functionals on Random Matrices and Random Matchings Problems
2504:
The former formula possesses an analog for the hafnian of a symmetric
7994:
7905:
7805:
Reich, Simeon (1971), "Another solution of an old problem of pólya",
1696:
to compute the permanent modulo any number that is not a power of 2.
7820:
7985:
7693:
7384:
7346:
7260:
6056:{\textstyle {\sum _{i=1}^{n}\alpha _{i}=\sum _{j=1}^{n}\beta _{j}}}
4870:{\textstyle {\sum _{i=1}^{n}\alpha _{i}=\sum _{j=1}^{n}\beta _{j}}}
2061:, while the determinant of the empty submatrix is defined to be 1.
1100:{\displaystyle \operatorname {perm} (A)={\frac {1}{2^{n-1}}}\left,}
7544:
Glynn, David G. (2013), "Permanent formulae from the
Veronesean",
7651:
Proceedings of 37th
Conference on Foundations of Computer Science
90:
46:
of a matrix despite the apparent similarity of the definitions.
7973:
Barvinok, A. (2017), "Approximating permanents and hafnians",
7888:
Proc. 1st
Scandinavian Workshop on Algorithm Theory (SWAT '88)
5291:, both vectors having all their entries from the set {0, ...,
1441:. The characterization of "convertible" matrices was given by
581:
It may be rewritten in terms of the matrix entries as follows
7709:
van Lint, Jacobus
Hendricus; Wilson, Richard Michale (2001),
7316:
Shtetl
Optimized: Introducing some British people to P vs. NP
7130:
can be approximated by applying the same method recursively.
4230:> 0). The latter result was essentially extended in 2017 (
4081:{\displaystyle \operatorname {per} ^{2}(U)=\det(U+V)\det(-U)}
1449:: an orientation of the edges such that for every even cycle
7522:
Glynn, David G. (2010), "The permanent of a square matrix",
6684:
generate the corresponding row/column multiplicity vectors
4207:{\displaystyle \operatorname {rank} (A^{\textsf {T}}A-I)=k}
7517:, Caltech-CS-TR-96-04, California Institute of Technology
4749:-vectors (having all their entries from the set {0, ...,
2064:
The expansion above can be generalized in an arbitrary
6973:
6210:
6084:
5985:
5495:
4799:
4441:
4339:
3612:
2791:
2399:
where in both formulas the sum is taken over all the (
1717:= 3 there is the following formula for an n×n-matrix
7926:(1979), "The complexity of computing the permanent",
7180:
7147:
7101:
7063:
7034:
6947:
6927:
6907:
6887:
6867:
6838:
6829:
6788:
fully polynomial-time randomized approximation scheme
6744:
6717:
6690:
6670:
6650:
6630:
6071:
5956:
5927:
5896:
5869:
5842:
5475:
5438:
5399:
5360:
5340:
5301:
5277:
5257:
5237:
4889:
4779:
4759:
4731:
4327:
4294:
4267:
4240:
4164:
4144:
4117:
4097:
4019:
3996:
3958:
3938:
3918:
3827:
3805:
3776:
3740:
3592:
3540:
3516:
3426:
3406:
3358:
3283:
3255:
3217:
3197:
3073:
2922:
2916:
is the set of n-permutations having only one cycle):
2895:
2532:
2510:
2465:
2409:
2077:
2029:
2009:
1980:
1960:
1940:
1920:
1893:
1754:
1723:
1672:
1627:
1600:
1546:
1502:
1475:
1455:
1427:
1401:
1357:
1337:
1149:
1116:
926:
860:
815:
763:
590:
490:
460:
424:
397:
370:
334:
299:
138:
7729:
Proc. 2nd
Australian Conf. Combinatorial Mathematics
6791:
1540:
2, the permanent is the same as the determinant, as
3067:In characteristic 2 the latter equality turns into
1278:as the number of perfect matchings in a graph. For
7685:Some facts on Permanents in Finite Characteristics
7515:A finite-difference sieve to compute the permanent
7506:Finite-difference Algorithms for Counting Problems
7199:
7166:
7122:
7087:
7049:
7021:{\textstyle \rho ={\frac {M(G)}{M(G\setminus e)}}}
7020:
6959:
6933:
6913:
6893:
6873:
6853:
6817:whose distribution is close to uniform, and whose
6758:'s partial inverse in the equality's right side).
6750:
6730:
6703:
6676:
6656:
6636:
6614:
6055:
5971:
5942:
5909:
5882:
5855:
5825:
5481:
5460:
5412:
5373:
5346:
5326:
5283:
5263:
5243:
5213:
4869:
4785:
4765:
4737:
4587:
4307:
4280:
4253:
4206:
4150:
4123:
4103:
4080:
4010:is the identity matrix of the corresponding size,
4002:
3982:
3944:
3924:
3898:
3811:
3782:
3762:
3726:
3598:
3558:
3522:
3502:
3412:
3376:
3344:
3261:
3241:
3203:
3176:
3057:
2908:
2881:
2771:
2516:
2489:
2451:
2391:
2053:
2015:
1995:
1966:
1946:
1926:
1906:
1876:
1729:
1684:
1658:
1613:
1586:
1521:
1487:
1461:
1433:
1413:
1387:
1343:
1234:
1135:
1099:
866:
846:
801:
746:
570:
473:
446:
410:
383:
356:
312:
231:
7424:Mathematical Foundations of Computer Science 2005
7235:
6624:where the common row/column multiplicity vectors
281:The best known general exact algorithm is due to
54:, the permanent weights them all with a +1 sign.
7835:Rempała, Grzegorz A.; Wesolowski, Jacek (2008),
6166:
5541:
4928:
4063:
4045:
3126:
2975:
2652:
2623:
2234:
2198:
2169:
1845:
1826:
1379:
242:The sum here extends over all elements σ of the
6941:and counting how many of them are matchings in
6770:are nonnegative, the permanent can be computed
6738:, s,t = 1,2, for its blocks (the same concerns
1388:{\displaystyle \operatorname {per} T(A)=\det A}
887:
7860:, The Carus Mathematical Monographs, Vol. 14,
7246:
7200:{\displaystyle {\textsf {BPP}}^{\textsf {NP}}}
7167:{\displaystyle {\textsf {BPP}}^{\textsf {NP}}}
4717:), while giving the following identity for an
7271:
4877:, valid in an arbitrary prime characteristic
4714:
4703:
4231:
1274:, and the permanent of any 0-1 matrix can be
81:for matrices in which all entries are 0 or 1
8:
7780:Nijenhuis, Albert; Wilf, Herbert S. (1978),
5469:own analogue for the hafnian of a symmetric
3121:
3103:
2970:
2952:
2484:
2466:
2048:
2030:
1821:
1803:
1223:
1213:
657:
639:
94:
7577:Proc. 33rd Symposium on Theory of Computing
1270:is counted by the permanent of the graph's
895:
6967:, one can obtain an estimate of the ratio
6861:denote the number of perfect matchings in
2501:− 1 subsets, some of them possibly empty.
2071:as the following pair of dual identities:
1328:
882:Balasubramanian–Bax–Franklin–Glynn formula
293:formula that can be given as follows: Let
18:Computation of the permananent of a matrix
7984:
7904:
7771:
7692:
7473:
7383:
7355:
7345:
7191:
7190:
7189:
7184:
7183:
7182:
7179:
7158:
7157:
7156:
7151:
7150:
7149:
7146:
7100:
7062:
7033:
6980:
6972:
6946:
6926:
6906:
6886:
6866:
6837:
6743:
6722:
6716:
6695:
6689:
6669:
6649:
6629:
6598:
6588:
6577:
6564:
6559:
6526:
6516:
6505:
6476:
6466:
6455:
6429:
6418:
6417:
6386:
6375:
6374:
6354:
6340:
6327:
6322:
6312:
6299:
6284:
6279:
6269:
6255:
6242:
6237:
6222:
6217:
6205:
6189:
6170:
6165:
6143:
6129:
6117:
6103:
6091:
6079:
6070:
6046:
6036:
6025:
6012:
6002:
5991:
5986:
5984:
5962:
5957:
5955:
5933:
5928:
5926:
5901:
5895:
5874:
5868:
5847:
5841:
5815:
5805:
5794:
5773:
5754:
5735:
5721:
5711:
5700:
5673:
5662:
5661:
5630:
5619:
5618:
5598:
5585:
5569:
5544:
5516:
5500:
5494:
5474:
5452:
5441:
5440:
5437:
5404:
5398:
5365:
5359:
5339:
5306:
5300:
5276:
5256:
5236:
5203:
5193:
5182:
5171:
5144:
5134:
5123:
5100:
5090:
5079:
5053:
5042:
5041:
5010:
4999:
4998:
4978:
4965:
4931:
4903:
4888:
4860:
4850:
4839:
4826:
4816:
4805:
4800:
4798:
4778:
4758:
4730:
4571:
4558:
4553:
4543:
4530:
4515:
4510:
4500:
4486:
4473:
4468:
4453:
4448:
4436:
4421:
4405:
4384:
4372:
4358:
4346:
4334:
4326:
4299:
4293:
4272:
4266:
4245:
4239:
4180:
4179:
4178:
4163:
4143:
4116:
4096:
4024:
4018:
3995:
3965:
3964:
3963:
3957:
3937:
3917:
3857:
3841:
3826:
3804:
3775:
3745:
3739:
3703:
3693:
3682:
3661:
3650:
3624:
3613:
3611:
3591:
3546:
3545:
3539:
3515:
3487:
3486:
3464:
3438:
3427:
3425:
3405:
3364:
3363:
3357:
3329:
3328:
3306:
3282:
3254:
3224:
3223:
3222:
3216:
3196:
3159:
3158:
3136:
3096:
3072:
3045:
3037:
3036:
3008:
3007:
2985:
2945:
2921:
2900:
2894:
2858:
2848:
2837:
2825:
2814:
2790:
2762:
2752:
2736:
2727:
2713:
2707:
2698:
2697:
2667:
2662:
2638:
2633:
2608:
2603:
2587:
2582:
2581:
2571:
2537:
2531:
2509:
2464:
2452:{\displaystyle {J_{1}},\ldots ,{J_{p-1}}}
2436:
2431:
2415:
2410:
2408:
2368:
2363:
2336:
2331:
2303:
2298:
2282:
2277:
2276:
2266:
2213:
2208:
2184:
2179:
2154:
2149:
2133:
2128:
2127:
2117:
2078:
2076:
2028:
2008:
1982:
1981:
1979:
1959:
1939:
1919:
1898:
1892:
1856:
1855:
1836:
1796:
1786:
1753:
1722:
1671:
1638:
1626:
1605:
1599:
1565:
1545:
1507:
1501:
1474:
1454:
1426:
1400:
1356:
1336:
1226:
1201:
1182:
1163:
1148:
1121:
1115:
1080:
1070:
1060:
1049:
1039:
1028:
1013:
1003:
992:
977:
954:
945:
925:
859:
826:
814:
790:
774:
762:
732:
716:
706:
695:
684:
676:
675:
632:
622:
589:
559:
549:
524:
513:
489:
465:
459:
435:
423:
402:
396:
375:
369:
345:
333:
304:
298:
205:
195:
184:
172:
161:
137:
57:While the determinant can be computed in
7286:
7141:, they can be computed within the class
1587:{\displaystyle (-1)\equiv 1{\pmod {2}}.}
7958:CRC Concise Encyclopedia of Mathematics
7493:Combinatorics and Diagonals of Matrices
7261:CRC Concise Encyclopedia of Mathematics
7227:
7111:
7076:
7006:
6951:
1697:
1479:
757:Ryser's formula can be evaluated using
82:
7282:
1442:
7301:
7297:
6830:Jerrum, Valiant & Vazirani (1986)
4138:) that, if we define a square matrix
4135:
1742:
1703:
911:
907:
286:
7:
7885:-free graphs and related problems",
7683:Knezevic, Anna; Cohen, Greg (2017),
7207:-hard If further constraints on the
7088:{\displaystyle \rho M(G\setminus e)}
6792:Jerrum, Sinclair & Vigoda (2001)
6778:polynomial time, up to an error of ε
3574:-matrix with the entries of indexes
1702:There are various formulae given by
73:states that computing permanents is
7862:Mathematical Association of America
6881:. Roughly, for any particular edge
1934:induced by the rows and columns of
1737:, involving the matrix's principal
1706:for the computation modulo a prime
1573:
1282:(regardless of bipartiteness), the
891:
4222:. (A parallel theory concerns the
3983:{\displaystyle U^{\textsf {T}}U=I}
3625:
3621:
3617:
3614:
3439:
3435:
3431:
3428:
3242:{\displaystyle U^{\textsf {T}}U=I}
556:
399:
364:be the product of the row-sums of
25:
7525:European Journal of Combinatorics
5334:denotes the matrix received from
289:). Ryser's method is based on an
7513:Bax, Eric; Franklin, J. (1996),
7215:and it uses the tools proper to
6921:, by sampling many matchings in
6063:, there holds also the identity
1110:where the outer sum is over all
8020:Computational complexity theory
7760:Illinois Journal of Mathematics
7739:Journal of Combinatorial Theory
7547:Designs, Codes and Cryptography
7236:Rempała & Wesolowski (2008)
7123:{\displaystyle M(G\setminus e)}
4685:-th rows are identical and its
4665:having three equal rows or, if
2459:that are partitions of the set
1594:It can also be computed modulo
1566:
802:{\displaystyle O(2^{n-1}n^{2})}
67:computational complexity theory
7960:, Chapman & Hall/CRC, 2002
7713:, Cambridge University Press,
7135:positive-semidefinite matrices
7117:
7105:
7082:
7070:
7044:
7038:
7012:
7000:
6992:
6986:
6848:
6842:
6556:
6546:
6441:
6423:
6413:
6401:
6380:
6370:
6358:
6355:
6195:
6182:
6156:
6144:
5770:
5758:
5751:
5741:
5685:
5667:
5657:
5645:
5624:
5614:
5602:
5599:
5595:
5578:
5562:
5556:
5534:
5529:
5517:
5509:
5461:{\displaystyle {\vec {1}}_{n}}
5446:
5319:
5307:
5168:
5158:
5065:
5047:
5037:
5025:
5004:
4994:
4982:
4979:
4975:
4958:
4949:
4943:
4921:
4916:
4904:
4896:
4427:
4414:
4195:
4171:
4075:
4066:
4060:
4048:
4039:
4033:
3890:
3884:
3872:
3866:
3850:
3834:
3757:
3751:
3719:
3713:
3673:
3667:
3640:
3634:
3497:
3473:
3454:
3448:
3339:
3315:
3296:
3290:
3171:
3164:
3151:
3142:
3129:
3086:
3080:
3046:
3038:
3033:
3023:
3020:
3013:
3000:
2991:
2978:
2935:
2929:
2874:
2868:
2804:
2798:
2763:
2749:
2737:
2728:
2714:
2699:
2694:
2684:
2681:
2655:
2646:
2626:
2568:
2558:
2552:
2546:
2490:{\displaystyle \{1,\dots ,n\}}
2382:
2356:
2344:
2324:
2263:
2253:
2243:
2237:
2227:
2201:
2192:
2172:
2114:
2104:
2094:
2088:
2054:{\displaystyle \{1,\dots ,n\}}
1987:
1868:
1861:
1848:
1842:
1829:
1783:
1773:
1767:
1761:
1653:
1631:
1577:
1567:
1556:
1547:
1373:
1367:
1331:) that there is no linear map
1207:
1156:
939:
933:
841:
819:
796:
767:
685:
677:
672:
662:
619:
609:
603:
597:
546:
536:
503:
497:
441:
428:
351:
338:
221:
215:
151:
145:
104:Definition and naive algorithm
1:
7808:American Mathematical Monthly
7942:10.1016/0304-3975(79)90044-6
7929:Theoretical Computer Science
7752:10.1016/0095-8956(75)90048-9
7643:10.1016/0304-3975(86)90174-X
7631:Theoretical Computer Science
7490:Balasubramanian, K. (1980),
7247:van Lint & Wilson (2001)
6960:{\displaystyle G\setminus e}
1488:{\displaystyle G\setminus C}
418:be the sum of the values of
7272:Nijenhuis & Wilf (1978)
6704:{\displaystyle \alpha _{s}}
4715:Knezevic & Cohen (2017)
4704:Knezevic & Cohen (2017)
4232:Knezevic & Cohen (2017)
4218:> 1 the problem becomes
3932:, that is, a square matrix
1659:{\displaystyle O(n^{4k-3})}
1533:Computation modulo a number
847:{\displaystyle O(2^{n-1}n)}
411:{\displaystyle \Sigma _{k}}
8056:
7402:10.1103/PhysRevA.96.022329
7357:10.1007/s00453-023-01169-1
6731:{\displaystyle \beta _{t}}
4669:> 2, a pair of indexes
1996:{\displaystyle {\bar {J}}}
1290:of the graph, so that the
809:arithmetic operations, or
258:of the numbers 1, 2, ...,
7858:Combinatorial Mathematics
7711:A Course in Combinatorics
7560:10.1007/s10623-012-9618-1
7538:10.1016/j.ejc.2010.01.010
7484:10.1137/s0097539792233907
7461:SIAM Journal on Computing
5327:{\displaystyle M^{(x,y)}}
4605:-matrix with a subset of
1414:{\displaystyle n\times n}
34:, the computation of the
27:Problem in linear algebra
7936:(2), Elsevier: 189–201,
7897:10.1007/3-540-19487-8_27
7782:Combinatorial Algorithms
7659:10.1109/SFCS.1996.548469
7312:See open problem (4) at
6807:Markov chain Monte Carlo
3763:{\displaystyle H_{n}(K)}
3530:is a subset of {1, ...,
1329:Marcus & Minc (1961)
1312:complete bipartite graph
447:{\displaystyle P(A_{k})}
357:{\displaystyle P(A_{k})}
95:Allender & Gore 1994
7839:, Springer, p. 4,
7792:(1913), "Aufgabe 424",
7137:. Using a technique of
6762:Approximate computation
6637:{\displaystyle \alpha }
5972:{\displaystyle {n_{1}}}
5943:{\displaystyle {n_{1}}}
4766:{\displaystyle \alpha }
4158:as k-semi-unitary when
2785:polynomial (defined as
1685:{\displaystyle k\geq 2}
1522:{\displaystyle K_{3,3}}
1276:interpreted in this way
1136:{\displaystyle 2^{n-1}}
896:Bax & Franklin 1996
854:by processing the sets
273:arithmetic operations.
264:sign of the permutation
8040:Computational problems
7773:10.1215/ijm/1255630882
7201:
7168:
7124:
7089:
7051:
7022:
6961:
6935:
6915:
6895:
6875:
6855:
6809:algorithm that uses a
6752:
6732:
6705:
6678:
6658:
6657:{\displaystyle \beta }
6638:
6616:
6593:
6521:
6471:
6057:
6041:
6007:
5973:
5944:
5911:
5910:{\displaystyle A_{11}}
5884:
5883:{\displaystyle A_{22}}
5857:
5856:{\displaystyle A_{11}}
5827:
5810:
5716:
5489:and an odd prime p is
5483:
5462:
5414:
5375:
5348:
5328:
5285:
5265:
5245:
5215:
5198:
5139:
5095:
4871:
4855:
4821:
4787:
4786:{\displaystyle \beta }
4767:
4739:
4702:: as it was shown in (
4589:
4309:
4308:{\displaystyle A_{11}}
4282:
4281:{\displaystyle A_{22}}
4255:
4254:{\displaystyle A_{11}}
4208:
4152:
4125:
4105:
4082:
4004:
3984:
3946:
3926:
3900:
3813:
3784:
3764:
3728:
3698:
3600:
3582:replaced by 0 for all
3560:
3559:{\displaystyle I_{/K}}
3524:
3504:
3414:
3378:
3377:{\displaystyle I_{/1}}
3346:
3263:
3243:
3205:
3178:
3059:
2910:
2883:
2853:
2773:
2518:
2491:
2453:
2393:
2055:
2017:
1997:
1968:
1948:
1928:
1908:
1878:
1731:
1686:
1660:
1615:
1588:
1523:
1489:
1463:
1435:
1415:
1389:
1345:
1236:
1137:
1101:
1065:
1044:
1008:
868:
848:
803:
748:
711:
572:
535:
475:
448:
412:
385:
358:
314:
233:
200:
7586:10.1145/380752.380877
7202:
7169:
7125:
7090:
7052:
7023:
6962:
6936:
6916:
6896:
6876:
6856:
6753:
6733:
6706:
6679:
6659:
6639:
6617:
6573:
6501:
6451:
6058:
6021:
5987:
5974:
5945:
5912:
5885:
5858:
5828:
5790:
5696:
5484:
5463:
5415:
5413:{\displaystyle y_{j}}
5376:
5374:{\displaystyle x_{i}}
5349:
5329:
5286:
5266:
5246:
5216:
5178:
5119:
5075:
4872:
4835:
4801:
4788:
4768:
4740:
4590:
4310:
4283:
4256:
4209:
4153:
4126:
4106:
4083:
4005:
3985:
3947:
3927:
3901:
3814:
3785:
3765:
3729:
3678:
3601:
3561:
3525:
3505:
3415:
3379:
3347:
3264:
3244:
3206:
3179:
3060:
2911:
2909:{\displaystyle H_{n}}
2884:
2833:
2774:
2519:
2492:
2454:
2394:
2056:
2018:
2003:is the complement of
1998:
1969:
1949:
1929:
1909:
1907:{\displaystyle A_{J}}
1879:
1732:
1687:
1661:
1616:
1614:{\displaystyle 2^{k}}
1589:
1524:
1490:
1464:
1436:
1416:
1390:
1346:
1296:skew-symmetric matrix
1237:
1138:
1102:
1045:
1024:
988:
900:polarization identity
869:
849:
804:
749:
691:
573:
509:
476:
474:{\displaystyle A_{k}}
449:
413:
386:
384:{\displaystyle A_{k}}
359:
315:
313:{\displaystyle A_{k}}
234:
180:
7653:, pp. 108–114,
7580:, pp. 712–721,
7178:
7145:
7099:
7061:
7050:{\displaystyle M(G)}
7032:
6971:
6945:
6925:
6905:
6885:
6865:
6854:{\displaystyle M(G)}
6836:
6813:to define and run a
6766:When the entries of
6742:
6715:
6688:
6668:
6648:
6628:
6069:
5983:
5954:
5925:
5894:
5867:
5840:
5493:
5473:
5436:
5397:
5358:
5338:
5299:
5275:
5255:
5235:
4887:
4797:
4777:
4757:
4729:
4325:
4292:
4265:
4238:
4162:
4142:
4115:
4095:
4017:
3994:
3956:
3936:
3916:
3825:
3803:
3774:
3738:
3610:
3590:
3538:
3514:
3424:
3404:
3356:
3281:
3253:
3215:
3195:
3071:
2920:
2893:
2789:
2530:
2508:
2463:
2407:
2075:
2027:
2007:
1978:
1958:
1938:
1918:
1914:is the submatrix of
1891:
1752:
1721:
1670:
1625:
1598:
1544:
1500:
1473:
1453:
1447:Pfaffian orientation
1425:
1399:
1355:
1335:
1147:
1114:
924:
888:Balasubramanian 1980
858:
813:
761:
588:
488:
458:
422:
395:
368:
332:
297:
136:
108:The permanent of an
71:a theorem of Valiant
63:Gaussian elimination
7854:Ryser, Herbert John
7703:2017arXiv171001783K
7432:10.1007/11549345_39
7394:2017PhRvA..96b2329C
7139:Stockmeyer counting
6335:
6292:
6250:
6230:
4566:
4523:
4481:
4461:
4134:It was also shown (
291:inclusion–exclusion
7924:Valiant, Leslie G.
7879:Vazirani, Vijay V.
7504:Bax, Eric (1998),
7197:
7164:
7120:
7085:
7047:
7018:
6957:
6931:
6911:
6891:
6871:
6851:
6748:
6728:
6701:
6674:
6654:
6634:
6612:
6348:
6318:
6275:
6233:
6213:
6137:
6053:
5969:
5940:
5907:
5880:
5853:
5823:
5479:
5458:
5410:
5371:
5344:
5324:
5281:
5261:
5241:
5211:
4867:
4783:
4763:
4735:
4585:
4579:
4549:
4506:
4464:
4444:
4392:
4305:
4278:
4251:
4204:
4148:
4121:
4101:
4078:
4000:
3980:
3942:
3922:
3896:
3809:
3780:
3760:
3724:
3677:
3596:
3556:
3520:
3500:
3410:
3374:
3342:
3259:
3239:
3201:
3188:polynomial of any
3174:
3125:
3055:
2974:
2906:
2879:
2832:
2769:
2622:
2514:
2487:
2449:
2389:
2387:
2317:
2168:
2051:
2013:
1993:
1964:
1944:
1924:
1904:
1874:
1825:
1727:
1682:
1656:
1611:
1584:
1519:
1485:
1459:
1431:
1411:
1385:
1341:
1272:biadjacency matrix
1232:
1133:
1097:
982:
864:
844:
799:
744:
727:
661:
568:
471:
454:over all possible
444:
408:
381:
354:
310:
229:
179:
7976:Discrete Analysis
7916:978-3-540-19487-3
7871:978-1-61444-014-7
7846:978-0-387-75145-0
7794:Arch. Math. Phys.
7720:978-0-521-00601-9
7441:978-3-540-28702-5
7340:(12): 3828–3854,
7193:
7186:
7160:
7153:
7016:
6934:{\displaystyle G}
6914:{\displaystyle G}
6894:{\displaystyle e}
6874:{\displaystyle G}
6826:self-reducibility
6751:{\displaystyle A}
6677:{\displaystyle A}
6426:
6383:
5670:
5627:
5555:
5482:{\displaystyle A}
5449:
5347:{\displaystyle M}
5284:{\displaystyle y}
5264:{\displaystyle x}
5244:{\displaystyle M}
5050:
5007:
4942:
4738:{\displaystyle A}
4647:Hamiltonian cycle
4224:Hamiltonian cycle
4182:
4151:{\displaystyle A}
4124:{\displaystyle U}
4104:{\displaystyle V}
4003:{\displaystyle I}
3967:
3945:{\displaystyle U}
3925:{\displaystyle U}
3812:{\displaystyle A}
3783:{\displaystyle K}
3646:
3599:{\displaystyle K}
3523:{\displaystyle K}
3413:{\displaystyle U}
3262:{\displaystyle I}
3226:
3204:{\displaystyle U}
3186:Hamiltonian cycle
3167:
3092:
3016:
2941:
2810:
2783:Hamiltonian cycle
2577:
2517:{\displaystyle A}
2272:
2123:
2016:{\displaystyle J}
1990:
1967:{\displaystyle J}
1947:{\displaystyle A}
1927:{\displaystyle A}
1864:
1792:
1730:{\displaystyle A}
1692:. However, it is
1462:{\displaystyle C}
1434:{\displaystyle A}
1344:{\displaystyle T}
1294:of the resulting
1264:perfect matchings
973:
966:
867:{\displaystyle S}
712:
628:
320:be obtained from
157:
52:parity of the set
16:(Redirected from
8047:
8005:
7995:10.19086/da.1244
7988:
7961:
7952:
7919:
7908:
7874:
7849:
7831:
7801:
7785:
7784:, Academic Press
7776:
7775:
7754:
7732:
7723:
7705:
7696:
7679:
7645:
7614:
7570:
7540:
7532:(7): 1887–1891,
7518:
7509:
7500:
7498:
7486:
7477:
7468:(5): 1026–1049,
7445:
7444:
7419:
7413:
7412:
7387:
7367:
7361:
7360:
7359:
7349:
7327:
7321:
7320:
7310:
7304:
7295:
7289:
7280:
7274:
7269:
7263:
7258:
7252:
7244:
7238:
7234:As of 2008, see
7232:
7206:
7204:
7203:
7198:
7196:
7195:
7194:
7188:
7187:
7173:
7171:
7170:
7165:
7163:
7162:
7161:
7155:
7154:
7129:
7127:
7126:
7121:
7094:
7092:
7091:
7086:
7056:
7054:
7053:
7048:
7027:
7025:
7024:
7019:
7017:
7015:
6995:
6981:
6966:
6964:
6963:
6958:
6940:
6938:
6937:
6932:
6920:
6918:
6917:
6912:
6900:
6898:
6897:
6892:
6880:
6878:
6877:
6872:
6860:
6858:
6857:
6852:
6757:
6755:
6754:
6749:
6737:
6735:
6734:
6729:
6727:
6726:
6710:
6708:
6707:
6702:
6700:
6699:
6683:
6681:
6680:
6675:
6663:
6661:
6660:
6655:
6643:
6641:
6640:
6635:
6621:
6619:
6618:
6613:
6611:
6610:
6609:
6608:
6592:
6587:
6569:
6568:
6545:
6541:
6537:
6536:
6520:
6515:
6495:
6491:
6487:
6486:
6470:
6465:
6445:
6444:
6434:
6433:
6428:
6427:
6419:
6391:
6390:
6385:
6384:
6376:
6353:
6352:
6345:
6344:
6334:
6326:
6317:
6316:
6304:
6303:
6291:
6283:
6274:
6273:
6260:
6259:
6249:
6241:
6229:
6221:
6194:
6193:
6181:
6180:
6169:
6160:
6159:
6142:
6141:
6134:
6133:
6122:
6121:
6108:
6107:
6096:
6095:
6062:
6060:
6059:
6054:
6052:
6051:
6050:
6040:
6035:
6017:
6016:
6006:
6001:
5978:
5976:
5975:
5970:
5968:
5967:
5966:
5949:
5947:
5946:
5941:
5939:
5938:
5937:
5916:
5914:
5913:
5908:
5906:
5905:
5889:
5887:
5886:
5881:
5879:
5878:
5862:
5860:
5859:
5854:
5852:
5851:
5832:
5830:
5829:
5824:
5822:
5821:
5820:
5819:
5809:
5804:
5777:
5740:
5739:
5734:
5730:
5726:
5725:
5715:
5710:
5689:
5688:
5678:
5677:
5672:
5671:
5663:
5635:
5634:
5629:
5628:
5620:
5593:
5592:
5574:
5573:
5554:
5540:
5533:
5532:
5505:
5504:
5488:
5486:
5485:
5480:
5467:
5465:
5464:
5459:
5457:
5456:
5451:
5450:
5442:
5419:
5417:
5416:
5411:
5409:
5408:
5380:
5378:
5377:
5372:
5370:
5369:
5353:
5351:
5350:
5345:
5333:
5331:
5330:
5325:
5323:
5322:
5290:
5288:
5287:
5282:
5271:and an m-vector
5270:
5268:
5267:
5262:
5250:
5248:
5247:
5242:
5220:
5218:
5217:
5212:
5210:
5209:
5208:
5207:
5197:
5192:
5157:
5153:
5149:
5148:
5138:
5133:
5113:
5109:
5105:
5104:
5094:
5089:
5069:
5068:
5058:
5057:
5052:
5051:
5043:
5015:
5014:
5009:
5008:
5000:
4973:
4972:
4941:
4927:
4920:
4919:
4876:
4874:
4873:
4868:
4866:
4865:
4864:
4854:
4849:
4831:
4830:
4820:
4815:
4792:
4790:
4789:
4784:
4772:
4770:
4769:
4764:
4744:
4742:
4741:
4736:
4594:
4592:
4591:
4586:
4584:
4583:
4576:
4575:
4565:
4557:
4548:
4547:
4535:
4534:
4522:
4514:
4505:
4504:
4491:
4490:
4480:
4472:
4460:
4452:
4426:
4425:
4410:
4409:
4397:
4396:
4389:
4388:
4377:
4376:
4363:
4362:
4351:
4350:
4314:
4312:
4311:
4306:
4304:
4303:
4287:
4285:
4284:
4279:
4277:
4276:
4260:
4258:
4257:
4252:
4250:
4249:
4217:
4213:
4211:
4210:
4205:
4185:
4184:
4183:
4157:
4155:
4154:
4149:
4130:
4128:
4127:
4122:
4110:
4108:
4107:
4102:
4087:
4085:
4084:
4079:
4029:
4028:
4009:
4007:
4006:
4001:
3989:
3987:
3986:
3981:
3970:
3969:
3968:
3951:
3949:
3948:
3943:
3931:
3929:
3928:
3923:
3905:
3903:
3902:
3897:
3862:
3861:
3849:
3848:
3818:
3816:
3815:
3810:
3789:
3787:
3786:
3781:
3769:
3767:
3766:
3761:
3750:
3749:
3733:
3731:
3730:
3725:
3723:
3722:
3697:
3692:
3676:
3666:
3665:
3630:
3629:
3628:
3606:, and we define
3605:
3603:
3602:
3597:
3566:is the identity
3565:
3563:
3562:
3557:
3555:
3554:
3550:
3529:
3527:
3526:
3521:
3509:
3507:
3506:
3501:
3496:
3495:
3491:
3469:
3468:
3444:
3443:
3442:
3419:
3417:
3416:
3411:
3384:is the identity
3383:
3381:
3380:
3375:
3373:
3372:
3368:
3351:
3349:
3348:
3343:
3338:
3337:
3333:
3311:
3310:
3269:is the identity
3268:
3266:
3265:
3260:
3248:
3246:
3245:
3240:
3229:
3228:
3227:
3211:(i.e. such that
3210:
3208:
3207:
3202:
3183:
3181:
3180:
3175:
3170:
3169:
3168:
3160:
3141:
3140:
3124:
3064:
3062:
3061:
3056:
3051:
3050:
3049:
3041:
3019:
3018:
3017:
3009:
2990:
2989:
2973:
2915:
2913:
2912:
2907:
2905:
2904:
2888:
2886:
2885:
2880:
2878:
2877:
2852:
2847:
2831:
2830:
2829:
2778:
2776:
2775:
2770:
2768:
2767:
2766:
2761:
2760:
2756:
2731:
2717:
2712:
2711:
2702:
2680:
2679:
2678:
2677:
2645:
2644:
2643:
2642:
2621:
2620:
2619:
2618:
2593:
2592:
2591:
2576:
2575:
2542:
2541:
2523:
2521:
2520:
2515:
2496:
2494:
2493:
2488:
2458:
2456:
2455:
2450:
2448:
2447:
2446:
2421:
2420:
2419:
2398:
2396:
2395:
2390:
2388:
2381:
2380:
2379:
2378:
2343:
2342:
2341:
2340:
2316:
2315:
2314:
2313:
2288:
2287:
2286:
2271:
2270:
2226:
2225:
2224:
2223:
2191:
2190:
2189:
2188:
2167:
2166:
2165:
2164:
2139:
2138:
2137:
2122:
2121:
2060:
2058:
2057:
2052:
2022:
2020:
2019:
2014:
2002:
2000:
1999:
1994:
1992:
1991:
1983:
1973:
1971:
1970:
1965:
1953:
1951:
1950:
1945:
1933:
1931:
1930:
1925:
1913:
1911:
1910:
1905:
1903:
1902:
1883:
1881:
1880:
1875:
1867:
1866:
1865:
1857:
1841:
1840:
1824:
1791:
1790:
1736:
1734:
1733:
1728:
1716:
1709:
1691:
1689:
1688:
1683:
1665:
1663:
1662:
1657:
1652:
1651:
1620:
1618:
1617:
1612:
1610:
1609:
1593:
1591:
1590:
1585:
1580:
1528:
1526:
1525:
1520:
1518:
1517:
1494:
1492:
1491:
1486:
1468:
1466:
1465:
1460:
1440:
1438:
1437:
1432:
1420:
1418:
1417:
1412:
1394:
1392:
1391:
1386:
1350:
1348:
1347:
1342:
1241:
1239:
1238:
1233:
1231:
1230:
1206:
1205:
1187:
1186:
1168:
1167:
1142:
1140:
1139:
1134:
1132:
1131:
1106:
1104:
1103:
1098:
1093:
1089:
1088:
1087:
1075:
1074:
1064:
1059:
1043:
1038:
1023:
1019:
1018:
1017:
1007:
1002:
981:
967:
965:
964:
946:
904:symmetric tensor
873:
871:
870:
865:
853:
851:
850:
845:
837:
836:
808:
806:
805:
800:
795:
794:
785:
784:
753:
751:
750:
745:
740:
739:
726:
710:
705:
690:
689:
688:
680:
660:
627:
626:
577:
575:
574:
569:
564:
563:
554:
553:
534:
523:
480:
478:
477:
472:
470:
469:
453:
451:
450:
445:
440:
439:
417:
415:
414:
409:
407:
406:
390:
388:
387:
382:
380:
379:
363:
361:
360:
355:
350:
349:
319:
317:
316:
311:
309:
308:
254:, i.e. over all
238:
236:
235:
230:
225:
224:
199:
194:
178:
177:
176:
129:) is defined as
21:
8055:
8054:
8050:
8049:
8048:
8046:
8045:
8044:
8010:
8009:
7972:
7969:
7967:Further reading
7964:
7955:
7922:
7917:
7884:
7877:
7872:
7852:
7847:
7834:
7821:10.2307/2316574
7804:
7788:
7779:
7757:
7735:
7726:
7721:
7708:
7682:
7669:
7648:
7627:Vazirani, Vijay
7623:Valiant, Leslie
7617:
7596:
7573:
7543:
7521:
7512:
7503:
7496:
7489:
7457:
7453:
7448:
7442:
7421:
7420:
7416:
7369:
7368:
7364:
7329:
7328:
7324:
7313:
7311:
7307:
7296:
7292:
7287:Vazirani (1988)
7281:
7277:
7270:
7266:
7259:
7255:
7245:
7241:
7233:
7229:
7225:
7181:
7176:
7175:
7148:
7143:
7142:
7097:
7096:
7059:
7058:
7030:
7029:
6996:
6982:
6969:
6968:
6943:
6942:
6923:
6922:
6903:
6902:
6883:
6882:
6863:
6862:
6834:
6833:
6821:is polynomial.
6811:Metropolis rule
6764:
6740:
6739:
6718:
6713:
6712:
6691:
6686:
6685:
6666:
6665:
6664:for the matrix
6646:
6645:
6626:
6625:
6594:
6560:
6555:
6522:
6500:
6496:
6472:
6450:
6446:
6416:
6373:
6347:
6346:
6336:
6308:
6295:
6293:
6265:
6262:
6261:
6251:
6231:
6206:
6204:
6185:
6164:
6136:
6135:
6125:
6123:
6113:
6110:
6109:
6099:
6097:
6087:
6080:
6078:
6067:
6066:
6042:
6008:
5981:
5980:
5958:
5952:
5951:
5929:
5923:
5922:
5897:
5892:
5891:
5870:
5865:
5864:
5843:
5838:
5837:
5811:
5750:
5717:
5695:
5691:
5690:
5660:
5617:
5594:
5581:
5565:
5512:
5496:
5491:
5490:
5471:
5470:
5439:
5434:
5433:
5424:-th column for
5400:
5395:
5394:
5361:
5356:
5355:
5336:
5335:
5302:
5297:
5296:
5273:
5272:
5253:
5252:
5233:
5232:
5199:
5167:
5140:
5118:
5114:
5096:
5074:
5070:
5040:
4997:
4974:
4961:
4899:
4885:
4884:
4856:
4822:
4795:
4794:
4775:
4774:
4755:
4754:
4727:
4726:
4578:
4577:
4567:
4539:
4526:
4524:
4496:
4493:
4492:
4482:
4462:
4437:
4417:
4401:
4391:
4390:
4380:
4378:
4368:
4365:
4364:
4354:
4352:
4342:
4335:
4323:
4322:
4295:
4290:
4289:
4268:
4263:
4262:
4241:
4236:
4235:
4215:
4174:
4160:
4159:
4140:
4139:
4113:
4112:
4093:
4092:
4020:
4015:
4014:
3992:
3991:
3959:
3954:
3953:
3934:
3933:
3914:
3913:
3853:
3837:
3823:
3822:
3801:
3800:
3772:
3771:
3741:
3736:
3735:
3699:
3657:
3620:
3608:
3607:
3588:
3587:
3541:
3536:
3535:
3512:
3511:
3482:
3460:
3434:
3422:
3421:
3402:
3401:
3359:
3354:
3353:
3324:
3302:
3279:
3278:
3251:
3250:
3218:
3213:
3212:
3193:
3192:
3154:
3132:
3069:
3068:
3032:
3003:
2981:
2918:
2917:
2896:
2891:
2890:
2854:
2821:
2787:
2786:
2732:
2703:
2693:
2663:
2658:
2634:
2629:
2604:
2583:
2567:
2533:
2528:
2527:
2506:
2505:
2461:
2460:
2432:
2411:
2405:
2404:
2386:
2385:
2364:
2359:
2332:
2327:
2299:
2278:
2262:
2246:
2231:
2230:
2209:
2204:
2180:
2175:
2150:
2129:
2113:
2097:
2073:
2072:
2025:
2024:
2005:
2004:
1976:
1975:
1956:
1955:
1936:
1935:
1916:
1915:
1894:
1889:
1888:
1851:
1832:
1782:
1750:
1749:
1719:
1718:
1714:
1707:
1668:
1667:
1634:
1623:
1622:
1601:
1596:
1595:
1542:
1541:
1535:
1503:
1498:
1497:
1471:
1470:
1451:
1450:
1423:
1422:
1397:
1396:
1353:
1352:
1333:
1332:
1319:
1268:bipartite graph
1260:
1257:
1248:
1222:
1197:
1178:
1159:
1145:
1144:
1117:
1112:
1111:
1076:
1066:
1009:
987:
983:
972:
968:
950:
922:
921:
884:
856:
855:
822:
811:
810:
786:
770:
759:
758:
728:
671:
618:
586:
585:
555:
545:
486:
485:
461:
456:
455:
431:
420:
419:
398:
393:
392:
371:
366:
365:
341:
330:
329:
300:
295:
294:
283:H. J. Ryser
279:
253:
244:symmetric group
201:
168:
134:
133:
128:
106:
59:polynomial time
28:
23:
22:
15:
12:
11:
5:
8053:
8051:
8043:
8042:
8037:
8032:
8027:
8025:Linear algebra
8022:
8012:
8011:
8008:
8007:
7968:
7965:
7963:
7962:
7953:
7920:
7915:
7882:
7875:
7870:
7850:
7845:
7832:
7815:(6): 649–650,
7802:
7786:
7777:
7766:(3): 376–381,
7755:
7746:(3): 187–208,
7733:
7724:
7719:
7706:
7680:
7667:
7646:
7615:
7595:978-1581133493
7594:
7571:
7554:(1–3): 39–47,
7541:
7519:
7510:
7501:
7487:
7475:10.1.1.51.3546
7454:
7452:
7449:
7447:
7446:
7440:
7414:
7362:
7322:
7319:, 22 July 2015
7305:
7290:
7275:
7264:
7253:
7239:
7226:
7224:
7221:
7217:quantum optics
7213:boson sampling
7119:
7116:
7113:
7110:
7107:
7104:
7084:
7081:
7078:
7075:
7072:
7069:
7066:
7046:
7043:
7040:
7037:
7014:
7011:
7008:
7005:
7002:
6999:
6994:
6991:
6988:
6985:
6979:
6976:
6956:
6953:
6950:
6930:
6910:
6890:
6870:
6850:
6847:
6844:
6841:
6763:
6760:
6747:
6725:
6721:
6698:
6694:
6673:
6653:
6633:
6607:
6604:
6601:
6597:
6591:
6586:
6583:
6580:
6576:
6572:
6567:
6563:
6558:
6554:
6551:
6548:
6544:
6540:
6535:
6532:
6529:
6525:
6519:
6514:
6511:
6508:
6504:
6499:
6494:
6490:
6485:
6482:
6479:
6475:
6469:
6464:
6461:
6458:
6454:
6449:
6443:
6440:
6437:
6432:
6425:
6422:
6415:
6412:
6409:
6406:
6403:
6400:
6397:
6394:
6389:
6382:
6379:
6372:
6369:
6366:
6363:
6360:
6357:
6351:
6343:
6339:
6333:
6330:
6325:
6321:
6315:
6311:
6307:
6302:
6298:
6294:
6290:
6287:
6282:
6278:
6272:
6268:
6264:
6263:
6258:
6254:
6248:
6245:
6240:
6236:
6232:
6228:
6225:
6220:
6216:
6212:
6211:
6209:
6203:
6200:
6197:
6192:
6188:
6184:
6179:
6176:
6173:
6168:
6163:
6158:
6155:
6152:
6149:
6146:
6140:
6132:
6128:
6124:
6120:
6116:
6112:
6111:
6106:
6102:
6098:
6094:
6090:
6086:
6085:
6083:
6077:
6074:
6049:
6045:
6039:
6034:
6031:
6028:
6024:
6020:
6015:
6011:
6005:
6000:
5997:
5994:
5990:
5965:
5961:
5936:
5932:
5904:
5900:
5890:being square,
5877:
5873:
5850:
5846:
5818:
5814:
5808:
5803:
5800:
5797:
5793:
5789:
5786:
5783:
5780:
5776:
5772:
5769:
5766:
5763:
5760:
5757:
5753:
5749:
5746:
5743:
5738:
5733:
5729:
5724:
5720:
5714:
5709:
5706:
5703:
5699:
5694:
5687:
5684:
5681:
5676:
5669:
5666:
5659:
5656:
5653:
5650:
5647:
5644:
5641:
5638:
5633:
5626:
5623:
5616:
5613:
5610:
5607:
5604:
5601:
5597:
5591:
5588:
5584:
5580:
5577:
5572:
5568:
5564:
5561:
5558:
5553:
5550:
5547:
5543:
5539:
5536:
5531:
5528:
5525:
5522:
5519:
5515:
5511:
5508:
5503:
5499:
5478:
5455:
5448:
5445:
5407:
5403:
5368:
5364:
5354:via repeating
5343:
5321:
5318:
5315:
5312:
5309:
5305:
5280:
5260:
5251:, an n-vector
5240:
5206:
5202:
5196:
5191:
5188:
5185:
5181:
5177:
5174:
5170:
5166:
5163:
5160:
5156:
5152:
5147:
5143:
5137:
5132:
5129:
5126:
5122:
5117:
5112:
5108:
5103:
5099:
5093:
5088:
5085:
5082:
5078:
5073:
5067:
5064:
5061:
5056:
5049:
5046:
5039:
5036:
5033:
5030:
5027:
5024:
5021:
5018:
5013:
5006:
5003:
4996:
4993:
4990:
4987:
4984:
4981:
4977:
4971:
4968:
4964:
4960:
4957:
4954:
4951:
4948:
4945:
4940:
4937:
4934:
4930:
4926:
4923:
4918:
4915:
4912:
4909:
4906:
4902:
4898:
4895:
4892:
4863:
4859:
4853:
4848:
4845:
4842:
4838:
4834:
4829:
4825:
4819:
4814:
4811:
4808:
4804:
4782:
4762:
4734:
4677:such that its
4653:) = 0 for any
4582:
4574:
4570:
4564:
4561:
4556:
4552:
4546:
4542:
4538:
4533:
4529:
4525:
4521:
4518:
4513:
4509:
4503:
4499:
4495:
4494:
4489:
4485:
4479:
4476:
4471:
4467:
4463:
4459:
4456:
4451:
4447:
4443:
4442:
4440:
4435:
4432:
4429:
4424:
4420:
4416:
4413:
4408:
4404:
4400:
4395:
4387:
4383:
4379:
4375:
4371:
4367:
4366:
4361:
4357:
4353:
4349:
4345:
4341:
4340:
4338:
4333:
4330:
4302:
4298:
4288:being square,
4275:
4271:
4248:
4244:
4203:
4200:
4197:
4194:
4191:
4188:
4177:
4173:
4170:
4167:
4147:
4120:
4100:
4089:
4088:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4027:
4023:
3999:
3979:
3976:
3973:
3962:
3941:
3921:
3907:
3906:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3860:
3856:
3852:
3847:
3844:
3840:
3836:
3833:
3830:
3808:
3779:
3759:
3756:
3753:
3748:
3744:
3721:
3718:
3715:
3712:
3709:
3706:
3702:
3696:
3691:
3688:
3685:
3681:
3675:
3672:
3669:
3664:
3660:
3656:
3653:
3649:
3645:
3642:
3639:
3636:
3633:
3627:
3623:
3619:
3616:
3595:
3553:
3549:
3544:
3519:
3499:
3494:
3490:
3485:
3481:
3478:
3475:
3472:
3467:
3463:
3459:
3456:
3453:
3450:
3447:
3441:
3437:
3433:
3430:
3409:
3371:
3367:
3362:
3341:
3336:
3332:
3327:
3323:
3320:
3317:
3314:
3309:
3305:
3301:
3298:
3295:
3292:
3289:
3286:
3258:
3238:
3235:
3232:
3221:
3200:
3173:
3166:
3163:
3157:
3153:
3150:
3147:
3144:
3139:
3135:
3131:
3128:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3095:
3091:
3088:
3085:
3082:
3079:
3076:
3054:
3048:
3044:
3040:
3035:
3031:
3028:
3025:
3022:
3015:
3012:
3006:
3002:
2999:
2996:
2993:
2988:
2984:
2980:
2977:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2944:
2940:
2937:
2934:
2931:
2928:
2925:
2903:
2899:
2876:
2873:
2870:
2867:
2864:
2861:
2857:
2851:
2846:
2843:
2840:
2836:
2828:
2824:
2820:
2817:
2813:
2809:
2806:
2803:
2800:
2797:
2794:
2765:
2759:
2755:
2751:
2748:
2745:
2742:
2739:
2735:
2730:
2726:
2723:
2720:
2716:
2710:
2706:
2701:
2696:
2692:
2689:
2686:
2683:
2676:
2673:
2670:
2666:
2661:
2657:
2654:
2651:
2648:
2641:
2637:
2632:
2628:
2625:
2617:
2614:
2611:
2607:
2602:
2599:
2596:
2590:
2586:
2580:
2574:
2570:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2540:
2536:
2524:and an odd p:
2513:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2445:
2442:
2439:
2435:
2430:
2427:
2424:
2418:
2414:
2384:
2377:
2374:
2371:
2367:
2362:
2358:
2355:
2352:
2349:
2346:
2339:
2335:
2330:
2326:
2323:
2320:
2312:
2309:
2306:
2302:
2297:
2294:
2291:
2285:
2281:
2275:
2269:
2265:
2261:
2258:
2255:
2252:
2249:
2247:
2245:
2242:
2239:
2236:
2233:
2232:
2229:
2222:
2219:
2216:
2212:
2207:
2203:
2200:
2197:
2194:
2187:
2183:
2178:
2174:
2171:
2163:
2160:
2157:
2153:
2148:
2145:
2142:
2136:
2132:
2126:
2120:
2116:
2112:
2109:
2106:
2103:
2100:
2098:
2096:
2093:
2090:
2087:
2084:
2081:
2080:
2066:characteristic
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2012:
1989:
1986:
1963:
1943:
1923:
1901:
1897:
1885:
1884:
1873:
1870:
1863:
1860:
1854:
1850:
1847:
1844:
1839:
1835:
1831:
1828:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1795:
1789:
1785:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1726:
1698:Valiant (1979)
1681:
1678:
1675:
1655:
1650:
1647:
1644:
1641:
1637:
1633:
1630:
1608:
1604:
1583:
1579:
1576:
1572:
1569:
1564:
1561:
1558:
1555:
1552:
1549:
1534:
1531:
1516:
1513:
1510:
1506:
1484:
1481:
1478:
1458:
1430:
1410:
1407:
1404:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1340:
1317:
1262:The number of
1259:
1255:
1249:
1247:
1244:
1229:
1225:
1221:
1218:
1215:
1212:
1209:
1204:
1200:
1196:
1193:
1190:
1185:
1181:
1177:
1174:
1171:
1166:
1162:
1158:
1155:
1152:
1130:
1127:
1124:
1120:
1108:
1107:
1096:
1092:
1086:
1083:
1079:
1073:
1069:
1063:
1058:
1055:
1052:
1048:
1042:
1037:
1034:
1031:
1027:
1022:
1016:
1012:
1006:
1001:
998:
995:
991:
986:
980:
976:
971:
963:
960:
957:
953:
949:
944:
941:
938:
935:
932:
929:
883:
880:
863:
843:
840:
835:
832:
829:
825:
821:
818:
798:
793:
789:
783:
780:
777:
773:
769:
766:
755:
754:
743:
738:
735:
731:
725:
722:
719:
715:
709:
704:
701:
698:
694:
687:
683:
679:
674:
670:
667:
664:
659:
656:
653:
650:
647:
644:
641:
638:
635:
631:
625:
621:
617:
614:
611:
608:
605:
602:
599:
596:
593:
579:
578:
567:
562:
558:
552:
548:
544:
541:
538:
533:
530:
527:
522:
519:
516:
512:
508:
505:
502:
499:
496:
493:
468:
464:
443:
438:
434:
430:
427:
405:
401:
378:
374:
353:
348:
344:
340:
337:
307:
303:
278:
275:
249:
240:
239:
228:
223:
220:
217:
214:
211:
208:
204:
198:
193:
190:
187:
183:
175:
171:
167:
164:
160:
156:
153:
150:
147:
144:
141:
124:
105:
102:
83:Valiant (1979)
32:linear algebra
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8052:
8041:
8038:
8036:
8033:
8031:
8030:Matrix theory
8028:
8026:
8023:
8021:
8018:
8017:
8015:
8004:
8000:
7996:
7992:
7987:
7982:
7978:
7977:
7971:
7970:
7966:
7959:
7956:"Permanent",
7954:
7951:
7947:
7943:
7939:
7935:
7931:
7930:
7925:
7921:
7918:
7912:
7907:
7902:
7898:
7894:
7890:
7889:
7880:
7876:
7873:
7867:
7863:
7859:
7855:
7851:
7848:
7842:
7838:
7833:
7830:
7826:
7822:
7818:
7814:
7810:
7809:
7803:
7799:
7795:
7791:
7787:
7783:
7778:
7774:
7769:
7765:
7761:
7756:
7753:
7749:
7745:
7741:
7740:
7734:
7730:
7725:
7722:
7716:
7712:
7707:
7704:
7700:
7695:
7690:
7686:
7681:
7678:
7674:
7670:
7668:0-8186-7594-2
7664:
7660:
7656:
7652:
7647:
7644:
7640:
7636:
7632:
7628:
7624:
7620:
7616:
7613:
7609:
7605:
7601:
7597:
7591:
7587:
7583:
7579:
7578:
7572:
7569:
7565:
7561:
7557:
7553:
7549:
7548:
7542:
7539:
7535:
7531:
7527:
7526:
7520:
7516:
7511:
7507:
7502:
7495:
7494:
7488:
7485:
7481:
7476:
7471:
7467:
7463:
7462:
7456:
7455:
7450:
7443:
7437:
7433:
7429:
7425:
7418:
7415:
7411:
7407:
7403:
7399:
7395:
7391:
7386:
7381:
7378:(2): 022329,
7377:
7373:
7366:
7363:
7358:
7353:
7348:
7343:
7339:
7335:
7334:
7326:
7323:
7318:
7317:
7309:
7306:
7303:
7299:
7294:
7291:
7288:
7284:
7283:Little (1974)
7279:
7276:
7273:
7268:
7265:
7262:
7257:
7254:
7251:
7248:
7243:
7240:
7237:
7231:
7228:
7222:
7220:
7218:
7214:
7210:
7140:
7136:
7131:
7114:
7108:
7102:
7079:
7073:
7067:
7064:
7041:
7035:
7028:. The number
7009:
7003:
6997:
6989:
6983:
6977:
6974:
6954:
6948:
6928:
6908:
6888:
6868:
6845:
6839:
6831:
6827:
6822:
6820:
6816:
6812:
6808:
6804:
6800:
6795:
6793:
6789:
6785:
6781:
6777:
6776:probabilistic
6773:
6772:approximately
6769:
6761:
6759:
6745:
6723:
6719:
6696:
6692:
6671:
6651:
6631:
6622:
6605:
6602:
6599:
6595:
6589:
6584:
6581:
6578:
6574:
6570:
6565:
6561:
6552:
6549:
6542:
6538:
6533:
6530:
6527:
6523:
6517:
6512:
6509:
6506:
6502:
6497:
6492:
6488:
6483:
6480:
6477:
6473:
6467:
6462:
6459:
6456:
6452:
6447:
6438:
6435:
6430:
6420:
6410:
6407:
6404:
6398:
6395:
6392:
6387:
6377:
6367:
6364:
6361:
6349:
6341:
6337:
6331:
6328:
6323:
6319:
6313:
6309:
6305:
6300:
6296:
6288:
6285:
6280:
6276:
6270:
6266:
6256:
6252:
6246:
6243:
6238:
6234:
6226:
6223:
6218:
6214:
6207:
6201:
6198:
6190:
6186:
6177:
6174:
6171:
6161:
6153:
6150:
6147:
6138:
6130:
6126:
6118:
6114:
6104:
6100:
6092:
6088:
6081:
6075:
6072:
6064:
6047:
6043:
6037:
6032:
6029:
6026:
6022:
6018:
6013:
6009:
6003:
5998:
5995:
5992:
5988:
5963:
5959:
5934:
5930:
5920:
5902:
5898:
5875:
5871:
5848:
5844:
5834:
5816:
5812:
5806:
5801:
5798:
5795:
5791:
5787:
5784:
5781:
5778:
5774:
5767:
5764:
5761:
5755:
5747:
5744:
5736:
5731:
5727:
5722:
5718:
5712:
5707:
5704:
5701:
5697:
5692:
5682:
5679:
5674:
5664:
5654:
5651:
5648:
5642:
5639:
5636:
5631:
5621:
5611:
5608:
5605:
5589:
5586:
5582:
5575:
5570:
5566:
5559:
5551:
5548:
5545:
5537:
5526:
5523:
5520:
5513:
5506:
5501:
5497:
5476:
5453:
5443:
5431:
5427:
5423:
5405:
5401:
5392:
5388:
5384:
5366:
5362:
5341:
5316:
5313:
5310:
5303:
5294:
5278:
5258:
5238:
5230:
5226:
5223:where for an
5221:
5204:
5200:
5194:
5189:
5186:
5183:
5179:
5175:
5172:
5164:
5161:
5154:
5150:
5145:
5141:
5135:
5130:
5127:
5124:
5120:
5115:
5110:
5106:
5101:
5097:
5091:
5086:
5083:
5080:
5076:
5071:
5062:
5059:
5054:
5044:
5034:
5031:
5028:
5022:
5019:
5016:
5011:
5001:
4991:
4988:
4985:
4969:
4966:
4962:
4955:
4952:
4946:
4938:
4935:
4932:
4924:
4913:
4910:
4907:
4900:
4893:
4890:
4882:
4880:
4861:
4857:
4851:
4846:
4843:
4840:
4836:
4832:
4827:
4823:
4817:
4812:
4809:
4806:
4802:
4780:
4760:
4752:
4748:
4732:
4724:
4720:
4716:
4711:
4709:
4708:#3-P-complete
4705:
4701:
4696:
4692:
4688:
4684:
4680:
4676:
4672:
4668:
4664:
4660:
4656:
4652:
4648:
4644:
4640:
4636:
4632:
4628:
4624:
4620:
4616:
4612:
4608:
4604:
4600:
4595:
4580:
4572:
4568:
4562:
4559:
4554:
4550:
4544:
4540:
4536:
4531:
4527:
4519:
4516:
4511:
4507:
4501:
4497:
4487:
4483:
4477:
4474:
4469:
4465:
4457:
4454:
4449:
4445:
4438:
4433:
4430:
4422:
4418:
4411:
4406:
4402:
4398:
4393:
4385:
4381:
4373:
4369:
4359:
4355:
4347:
4343:
4336:
4331:
4328:
4320:
4318:
4300:
4296:
4273:
4269:
4246:
4242:
4233:
4229:
4225:
4221:
4220:#3-P-complete
4201:
4198:
4192:
4189:
4186:
4175:
4168:
4165:
4145:
4137:
4132:
4118:
4098:
4072:
4069:
4057:
4054:
4051:
4042:
4036:
4030:
4025:
4021:
4013:
4012:
4011:
3997:
3977:
3974:
3971:
3960:
3939:
3919:
3912:
3893:
3887:
3881:
3878:
3875:
3869:
3863:
3858:
3854:
3845:
3842:
3838:
3831:
3828:
3821:
3820:
3819:
3806:
3799:
3794:
3791:
3777:
3754:
3746:
3742:
3716:
3710:
3707:
3704:
3700:
3694:
3689:
3686:
3683:
3679:
3670:
3662:
3658:
3654:
3651:
3647:
3643:
3637:
3631:
3593:
3586:belonging to
3585:
3581:
3577:
3573:
3569:
3551:
3547:
3542:
3533:
3517:
3492:
3488:
3483:
3479:
3476:
3470:
3465:
3461:
3457:
3451:
3445:
3407:
3399:
3395:
3391:
3387:
3369:
3365:
3360:
3334:
3330:
3325:
3321:
3318:
3312:
3307:
3303:
3299:
3293:
3287:
3284:
3276:
3272:
3256:
3236:
3233:
3230:
3219:
3198:
3191:
3187:
3161:
3155:
3148:
3145:
3137:
3133:
3118:
3115:
3112:
3109:
3106:
3100:
3097:
3093:
3089:
3083:
3077:
3074:
3065:
3052:
3042:
3029:
3026:
3010:
3004:
2997:
2994:
2986:
2982:
2967:
2964:
2961:
2958:
2955:
2949:
2946:
2942:
2938:
2932:
2926:
2923:
2901:
2897:
2871:
2865:
2862:
2859:
2855:
2849:
2844:
2841:
2838:
2834:
2826:
2822:
2818:
2815:
2811:
2807:
2801:
2795:
2792:
2784:
2779:
2757:
2753:
2746:
2743:
2740:
2733:
2724:
2721:
2718:
2708:
2704:
2690:
2687:
2674:
2671:
2668:
2664:
2659:
2649:
2639:
2635:
2630:
2615:
2612:
2609:
2605:
2600:
2597:
2594:
2588:
2584:
2578:
2572:
2564:
2561:
2555:
2549:
2543:
2538:
2534:
2525:
2511:
2502:
2500:
2481:
2478:
2475:
2472:
2469:
2443:
2440:
2437:
2433:
2428:
2425:
2422:
2416:
2412:
2402:
2375:
2372:
2369:
2365:
2360:
2353:
2350:
2347:
2337:
2333:
2328:
2321:
2318:
2310:
2307:
2304:
2300:
2295:
2292:
2289:
2283:
2279:
2273:
2267:
2259:
2256:
2250:
2248:
2240:
2220:
2217:
2214:
2210:
2205:
2195:
2185:
2181:
2176:
2161:
2158:
2155:
2151:
2146:
2143:
2140:
2134:
2130:
2124:
2118:
2110:
2107:
2101:
2099:
2091:
2085:
2082:
2070:
2067:
2062:
2045:
2042:
2039:
2036:
2033:
2010:
1984:
1961:
1941:
1921:
1899:
1895:
1871:
1858:
1852:
1837:
1833:
1818:
1815:
1812:
1809:
1806:
1800:
1797:
1793:
1787:
1779:
1776:
1770:
1764:
1758:
1755:
1748:
1747:
1746:
1744:
1740:
1724:
1711:
1705:
1700:
1699:
1695:
1679:
1676:
1673:
1648:
1645:
1642:
1639:
1635:
1628:
1606:
1602:
1581:
1574:
1570:
1562:
1559:
1553:
1550:
1539:
1532:
1530:
1514:
1511:
1508:
1504:
1482:
1476:
1456:
1448:
1444:
1443:Little (1975)
1428:
1408:
1405:
1402:
1382:
1376:
1370:
1364:
1361:
1358:
1338:
1330:
1325:
1321:
1316:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1284:FKT algorithm
1281:
1280:planar graphs
1277:
1273:
1269:
1265:
1254:
1250:
1246:Special cases
1245:
1243:
1227:
1219:
1216:
1210:
1202:
1198:
1194:
1191:
1188:
1183:
1179:
1175:
1172:
1169:
1164:
1160:
1153:
1150:
1128:
1125:
1122:
1118:
1094:
1090:
1084:
1081:
1077:
1071:
1067:
1061:
1056:
1053:
1050:
1046:
1040:
1035:
1032:
1029:
1025:
1020:
1014:
1010:
1004:
999:
996:
993:
989:
984:
978:
974:
969:
961:
958:
955:
951:
947:
942:
936:
930:
927:
920:
919:
918:
915:
913:
909:
905:
901:
897:
893:
889:
881:
879:
877:
861:
838:
833:
830:
827:
823:
816:
791:
787:
781:
778:
775:
771:
764:
741:
736:
733:
729:
723:
720:
717:
713:
707:
702:
699:
696:
692:
681:
668:
665:
654:
651:
648:
645:
642:
636:
633:
629:
623:
615:
612:
606:
600:
594:
591:
584:
583:
582:
565:
560:
550:
542:
539:
531:
528:
525:
520:
517:
514:
510:
506:
500:
494:
491:
484:
483:
482:
466:
462:
436:
432:
425:
403:
376:
372:
346:
342:
335:
328:columns, let
327:
323:
305:
301:
292:
288:
284:
277:Ryser formula
276:
274:
272:
269:
265:
261:
257:
252:
248:
245:
226:
218:
212:
209:
206:
202:
196:
191:
188:
185:
181:
173:
169:
165:
162:
158:
154:
148:
142:
139:
132:
131:
130:
127:
123:
119:
115:
111:
103:
101:
98:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
55:
53:
47:
45:
41:
37:
33:
19:
8035:Permutations
7974:
7957:
7933:
7927:
7887:
7857:
7836:
7812:
7806:
7797:
7793:
7781:
7763:
7759:
7743:
7742:, Series B,
7737:
7728:
7710:
7684:
7650:
7634:
7630:
7619:Jerrum, Mark
7576:
7551:
7545:
7529:
7523:
7514:
7505:
7492:
7465:
7459:
7423:
7417:
7375:
7372:Phys. Rev. A
7371:
7365:
7337:
7333:Algorithmica
7331:
7325:
7315:
7308:
7302:Reich (1971)
7298:Pólya (1913)
7293:
7278:
7267:
7256:
7242:
7230:
7132:
6823:
6815:Markov chain
6796:
6783:
6779:
6767:
6765:
6623:
6065:
5921:and of size
5835:
5429:
5425:
5421:
5390:
5386:
5385:-th row for
5382:
5292:
5228:
5224:
5222:
4883:
4878:
4750:
4746:
4722:
4718:
4712:
4690:
4686:
4682:
4678:
4674:
4670:
4666:
4662:
4658:
4654:
4650:
4642:
4638:
4634:
4630:
4626:
4622:
4618:
4614:
4610:
4606:
4602:
4598:
4596:
4321:
4227:
4136:Kogan (1996)
4133:
4090:
3908:
3795:
3792:
3583:
3579:
3575:
3571:
3567:
3531:
3397:
3393:
3389:
3385:
3274:
3270:
3066:
2780:
2526:
2503:
2498:
2403:− 1)-tuples
2400:
2068:
2063:
1886:
1743:Kogan (1996)
1713:Second, for
1712:
1704:Glynn (2010)
1701:
1536:
1529:, as above.
1324:George Pólya
1322:
1314:
1308:homeomorphic
1288:Tutte matrix
1261:
1252:
1109:
916:
885:
756:
580:
325:
324:by deleting
321:
280:
270:
267:
259:
256:permutations
250:
246:
241:
125:
121:
117:
113:
109:
107:
99:
56:
48:
29:
7637:: 169–188,
6819:mixing time
4700:P versus NP
1954:indexed by
1304:determinant
1300:square root
1251:Planar and
391:, and let
79:#P-complete
77:, and even
44:determinant
8014:Categories
7986:1601.07518
7694:1710.01783
7451:References
7385:1609.02416
7347:2111.03142
5919:invertible
5428:= 1, ...,
5420:times its
5389:= 1, ...,
5381:times its
4793:such that
4317:invertible
3952:such that
3798:invertible
1469:for which
1351:such that
912:Glynn 2013
908:Glynn 2010
93:circuits.(
7906:1813/6700
7790:Pólya, G.
7470:CiteSeerX
7112:∖
7077:∖
7065:ρ
7007:∖
6975:ρ
6952:∖
6803:uniformly
6790:(FPRAS) (
6720:β
6693:α
6652:β
6632:α
6596:α
6575:∑
6550:−
6524:β
6503:∏
6474:α
6453:∏
6439:α
6436:−
6424:→
6408:−
6396:β
6393:−
6381:→
6365:−
6329:−
6306:−
6286:−
6244:−
6224:−
6202:
6175:−
6154:β
6148:α
6076:
6044:β
6023:∑
6010:α
5989:∑
5813:α
5792:∑
5765:−
5745:−
5719:α
5698:∏
5683:α
5680:−
5668:→
5652:−
5640:α
5637:−
5625:→
5609:−
5587:−
5576:
5549:−
5527:α
5521:α
5507:
5447:→
5201:α
5180:∑
5162:−
5142:β
5121:∏
5098:α
5077:∏
5063:α
5060:−
5048:→
5032:−
5020:β
5017:−
5005:→
4989:−
4967:−
4956:
4936:−
4914:β
4908:α
4894:
4858:β
4837:∑
4824:α
4803:∑
4781:β
4761:α
4560:−
4537:−
4517:−
4475:−
4455:−
4434:
4412:
4332:
4190:−
4169:
4070:−
4031:
3882:
3864:
3843:−
3832:
3711:σ
3680:∏
3655:∈
3652:σ
3648:∑
3632:
3471:
3446:
3313:
3288:
3165:¯
3149:
3113:…
3101:⊆
3094:∑
3078:
3027:−
3014:¯
2998:
2962:…
2950:⊆
2943:∑
2927:
2866:σ
2835:∏
2819:∈
2816:σ
2812:∑
2796:
2744:−
2722:⋯
2688:−
2672:−
2650:⋯
2613:−
2598:…
2579:∑
2562:−
2544:
2476:…
2441:−
2426:…
2373:−
2354:
2348:⋯
2322:
2308:−
2293:…
2274:∑
2257:−
2218:−
2196:⋯
2159:−
2144:…
2125:∑
2108:−
2086:
2040:…
1988:¯
1862:¯
1813:…
1801:⊆
1794:∑
1777:−
1759:
1677:≥
1646:−
1560:≡
1551:−
1480:∖
1421:matrices
1406:×
1362:
1217:±
1211:∈
1199:δ
1192:…
1180:δ
1161:δ
1151:δ
1126:−
1068:δ
1047:∑
1026:∏
1011:δ
990:∏
979:δ
975:∑
959:−
931:
894:); also (
876:Gray code
831:−
779:−
721:∈
714:∑
693:∏
666:−
649:…
637:⊆
630:∑
613:−
595:
557:Σ
540:−
529:−
511:∑
495:
400:Σ
213:σ
182:∏
166:∈
163:σ
159:∑
143:
36:permanent
7856:(1963),
7677:39024286
7612:TR00-079
7568:36911503
7410:54194194
7209:spectrum
7095:, where
7057:is then
6782:, where
5231:-matrix
4745:and two
4689:-th and
4681:-th and
4661:-matrix
3909:for any
3796:for any
3400:-matrix
1621:in time
1395:for all
1292:Pfaffian
1143:vectors
892:Bax 1998
7950:1637832
7829:2316574
7800:(3): 27
7699:Bibcode
7604:8368245
7390:Bibcode
6801:almost
4725:matrix
4695:unitary
4629:)- or (
3911:unitary
3190:unitary
1694:UP-hard
1310:to the
1302:of its
878:order.
481:. Then
285: (
116:matrix
75:#P-hard
8003:397350
8001:
7948:
7913:
7868:
7843:
7827:
7717:
7675:
7665:
7610:
7602:
7592:
7566:
7472:
7438:
7408:
6832:. Let
6799:sample
5979:, and
5917:being
5295:− 1},
4753:− 1})
4637:+ 1)×(
4315:being
4091:where
3990:where
3734:where
3510:where
3352:where
3249:where
2889:where
1974:, and
1887:where
1739:minors
1538:Modulo
902:for a
40:matrix
7999:S2CID
7981:arXiv
7946:S2CID
7825:JSTOR
7689:arXiv
7673:S2CID
7600:S2CID
7564:S2CID
7497:(PDF)
7406:S2CID
7380:arXiv
7342:arXiv
7250:p. 99
7223:Notes
2497:into
1298:(the
1266:in a
1258:-free
38:of a
7911:ISBN
7866:ISBN
7841:ISBN
7715:ISBN
7663:ISBN
7608:ECCC
7590:ISBN
7436:ISBN
6711:and
6644:and
5393:and
4773:and
4261:and
4166:rank
1745:):
1666:for
928:perm
890:), (
592:perm
492:perm
287:1963
140:perm
112:-by-
7991:doi
7938:doi
7901:hdl
7893:doi
7883:3,3
7817:doi
7768:doi
7748:doi
7655:doi
7639:doi
7582:doi
7556:doi
7534:doi
7480:doi
7428:doi
7398:doi
7352:doi
7185:BPP
7152:BPP
6901:in
6794:).
6774:in
6199:per
6167:det
6073:per
5833:).
5567:haf
5542:det
5498:haf
4953:per
4929:det
4891:per
4621:)×(
4609:or
4431:per
4403:det
4329:per
4319:):
4064:det
4046:det
4022:per
3879:per
3855:det
3829:per
3534:},
3462:det
3420:as
3304:det
3285:ham
3146:det
3127:det
3075:ham
2995:per
2976:det
2924:ham
2793:ham
2653:det
2624:det
2535:haf
2351:per
2319:per
2235:det
2199:det
2170:det
2083:per
2023:in
1846:det
1827:det
1756:per
1571:mod
1380:det
1359:per
1318:3,3
1256:3,3
914:).
874:in
126:i,j
120:= (
91:ACC
61:by
30:In
8016::
7997:,
7989:,
7979:,
7944:,
7932:,
7909:,
7899:,
7864:,
7823:,
7813:78
7811:,
7798:20
7796:,
7762:,
7744:18
7697:,
7687:,
7671:,
7661:,
7635:43
7633:,
7625:;
7621:;
7606:,
7598:,
7588:,
7562:,
7552:68
7550:,
7530:31
7528:,
7478:,
7466:23
7464:,
7434:,
7404:,
7396:,
7388:,
7376:96
7374:,
7350:,
7338:85
7336:,
7300:,
7285:,
7192:NP
7159:NP
6342:12
6324:11
6314:21
6301:22
6281:11
6271:21
6257:12
6239:11
6219:11
6191:11
6131:22
6119:21
6105:12
6093:11
5903:11
5876:22
5863:,
5849:11
4881::
4641:−
4633:−
4625:−
4617:−
4573:12
4555:11
4545:21
4532:22
4512:11
4502:21
4488:12
4470:11
4450:11
4423:11
4386:22
4374:21
4360:12
4348:11
4301:11
4274:22
4247:11
4131:.
3790:.
1320:.
1242:.
268:n!
97:)
87:NP
69:,
8006:.
7993::
7983::
7940::
7934:8
7903::
7895::
7819::
7770::
7764:5
7750::
7701::
7691::
7657::
7641::
7584::
7558::
7536::
7482::
7430::
7400::
7392::
7382::
7354::
7344::
7118:)
7115:e
7109:G
7106:(
7103:M
7083:)
7080:e
7074:G
7071:(
7068:M
7045:)
7042:G
7039:(
7036:M
7013:)
7010:e
7004:G
7001:(
6998:M
6993:)
6990:G
6987:(
6984:M
6978:=
6955:e
6949:G
6929:G
6909:G
6889:e
6869:G
6849:)
6846:G
6843:(
6840:M
6784:M
6780:M
6768:A
6746:A
6724:t
6697:s
6672:A
6606:i
6603:,
6600:1
6590:n
6585:1
6582:=
6579:i
6571:+
6566:1
6562:n
6557:)
6553:1
6547:(
6543:)
6539:!
6534:j
6531:,
6528:1
6518:n
6513:1
6510:=
6507:j
6498:(
6493:)
6489:!
6484:i
6481:,
6478:1
6468:n
6463:1
6460:=
6457:i
6448:(
6442:)
6431:n
6421:1
6414:)
6411:1
6405:p
6402:(
6399:,
6388:n
6378:1
6371:)
6368:1
6362:p
6359:(
6356:(
6350:)
6338:A
6332:1
6320:A
6310:A
6297:A
6289:1
6277:A
6267:A
6253:A
6247:1
6235:A
6227:1
6215:A
6208:(
6196:)
6187:A
6183:(
6178:1
6172:p
6162:=
6157:)
6151:,
6145:(
6139:)
6127:A
6115:A
6101:A
6089:A
6082:(
6048:j
6038:n
6033:1
6030:=
6027:j
6019:=
6014:i
6004:n
5999:1
5996:=
5993:i
5964:1
5960:n
5950:x
5935:1
5931:n
5899:A
5872:A
5845:A
5817:i
5807:n
5802:1
5799:=
5796:i
5788:+
5785:n
5782:+
5779:2
5775:/
5771:)
5768:1
5762:p
5759:(
5756:n
5752:)
5748:1
5742:(
5737:2
5732:)
5728:!
5723:i
5713:n
5708:1
5705:=
5702:i
5693:(
5686:)
5675:n
5665:1
5658:)
5655:1
5649:p
5646:(
5643:,
5632:n
5622:1
5615:)
5612:1
5606:p
5603:(
5600:(
5596:)
5590:1
5583:A
5579:(
5571:2
5563:)
5560:A
5557:(
5552:1
5546:p
5538:=
5535:)
5530:)
5524:,
5518:(
5514:A
5510:(
5502:2
5477:A
5454:n
5444:1
5430:m
5426:j
5422:j
5406:j
5402:y
5391:n
5387:i
5383:i
5367:i
5363:x
5342:M
5320:)
5317:y
5314:,
5311:x
5308:(
5304:M
5293:p
5279:y
5259:x
5239:M
5229:m
5227:×
5225:n
5205:i
5195:n
5190:1
5187:=
5184:i
5176:+
5173:n
5169:)
5165:1
5159:(
5155:)
5151:!
5146:j
5136:n
5131:1
5128:=
5125:j
5116:(
5111:)
5107:!
5102:i
5092:n
5087:1
5084:=
5081:i
5072:(
5066:)
5055:n
5045:1
5038:)
5035:1
5029:p
5026:(
5023:,
5012:n
5002:1
4995:)
4992:1
4986:p
4983:(
4980:(
4976:)
4970:1
4963:A
4959:(
4950:)
4947:A
4944:(
4939:1
4933:p
4925:=
4922:)
4917:)
4911:,
4905:(
4901:A
4897:(
4879:p
4862:j
4852:n
4847:1
4844:=
4841:j
4833:=
4828:i
4818:n
4813:1
4810:=
4807:i
4751:p
4747:n
4733:A
4723:n
4721:×
4719:n
4691:j
4687:i
4683:j
4679:i
4675:j
4673:,
4671:i
4667:n
4663:A
4659:n
4657:×
4655:n
4651:A
4643:k
4639:n
4635:k
4631:n
4627:k
4623:n
4619:k
4615:n
4611:k
4607:k
4603:n
4601:×
4599:n
4581:)
4569:A
4563:1
4551:A
4541:A
4528:A
4520:1
4508:A
4498:A
4484:A
4478:1
4466:A
4458:1
4446:A
4439:(
4428:)
4419:A
4415:(
4407:2
4399:=
4394:)
4382:A
4370:A
4356:A
4344:A
4337:(
4297:A
4270:A
4243:A
4228:k
4216:k
4202:k
4199:=
4196:)
4193:I
4187:A
4181:T
4176:A
4172:(
4146:A
4119:U
4099:V
4076:)
4073:U
4067:(
4061:)
4058:V
4055:+
4052:U
4049:(
4043:=
4040:)
4037:U
4034:(
4026:2
3998:I
3978:I
3975:=
3972:U
3966:T
3961:U
3940:U
3920:U
3894:;
3891:)
3888:A
3885:(
3876:=
3873:)
3870:A
3867:(
3859:2
3851:)
3846:1
3839:A
3835:(
3807:A
3778:K
3758:)
3755:K
3752:(
3747:n
3743:H
3720:)
3717:i
3714:(
3708:,
3705:i
3701:a
3695:n
3690:1
3687:=
3684:i
3674:)
3671:K
3668:(
3663:n
3659:H
3644:=
3641:)
3638:A
3635:(
3626:K
3622:m
3618:a
3615:h
3594:K
3584:k
3580:k
3578:,
3576:k
3572:n
3570:×
3568:n
3552:K
3548:/
3543:I
3532:n
3518:K
3498:)
3493:K
3489:/
3484:I
3480:+
3477:U
3474:(
3466:2
3458:=
3455:)
3452:U
3449:(
3440:K
3436:m
3432:a
3429:h
3408:U
3398:n
3396:×
3394:n
3390:n
3388:×
3386:n
3370:1
3366:/
3361:I
3340:)
3335:1
3331:/
3326:I
3322:+
3319:U
3316:(
3308:2
3300:=
3297:)
3294:U
3291:(
3275:n
3273:×
3271:n
3257:I
3237:I
3234:=
3231:U
3225:T
3220:U
3199:U
3172:)
3162:J
3156:A
3152:(
3143:)
3138:J
3134:A
3130:(
3122:}
3119:n
3116:,
3110:,
3107:2
3104:{
3098:J
3090:=
3087:)
3084:A
3081:(
3053:.
3047:|
3043:J
3039:|
3034:)
3030:1
3024:(
3021:)
3011:J
3005:A
3001:(
2992:)
2987:J
2983:A
2979:(
2971:}
2968:n
2965:,
2959:,
2956:2
2953:{
2947:J
2939:=
2936:)
2933:A
2930:(
2902:n
2898:H
2875:)
2872:i
2869:(
2863:,
2860:i
2856:a
2850:n
2845:1
2842:=
2839:i
2827:n
2823:H
2808:=
2805:)
2802:A
2799:(
2764:|
2758:2
2754:/
2750:)
2747:1
2741:p
2738:(
2734:J
2729:|
2725:+
2719:+
2715:|
2709:1
2705:J
2700:|
2695:)
2691:1
2685:(
2682:)
2675:1
2669:p
2665:J
2660:A
2656:(
2647:)
2640:1
2636:J
2631:A
2627:(
2616:1
2610:p
2606:J
2601:,
2595:,
2589:1
2585:J
2573:n
2569:)
2565:1
2559:(
2556:=
2553:)
2550:A
2547:(
2539:2
2512:A
2499:p
2485:}
2482:n
2479:,
2473:,
2470:1
2467:{
2444:1
2438:p
2434:J
2429:,
2423:,
2417:1
2413:J
2401:p
2383:)
2376:1
2370:p
2366:J
2361:A
2357:(
2345:)
2338:1
2334:J
2329:A
2325:(
2311:1
2305:p
2301:J
2296:,
2290:,
2284:1
2280:J
2268:n
2264:)
2260:1
2254:(
2251:=
2244:)
2241:A
2238:(
2228:)
2221:1
2215:p
2211:J
2206:A
2202:(
2193:)
2186:1
2182:J
2177:A
2173:(
2162:1
2156:p
2152:J
2147:,
2141:,
2135:1
2131:J
2119:n
2115:)
2111:1
2105:(
2102:=
2095:)
2092:A
2089:(
2069:p
2049:}
2046:n
2043:,
2037:,
2034:1
2031:{
2011:J
1985:J
1962:J
1942:A
1922:A
1900:J
1896:A
1872:,
1869:)
1859:J
1853:A
1849:(
1843:)
1838:J
1834:A
1830:(
1822:}
1819:n
1816:,
1810:,
1807:1
1804:{
1798:J
1788:n
1784:)
1780:1
1774:(
1771:=
1768:)
1765:A
1762:(
1741:(
1725:A
1715:p
1708:p
1680:2
1674:k
1654:)
1649:3
1643:k
1640:4
1636:n
1632:(
1629:O
1607:k
1603:2
1582:.
1578:)
1575:2
1568:(
1563:1
1557:)
1554:1
1548:(
1515:3
1512:,
1509:3
1505:K
1483:C
1477:G
1457:C
1429:A
1409:n
1403:n
1383:A
1377:=
1374:)
1371:A
1368:(
1365:T
1339:T
1327:(
1315:K
1253:K
1228:n
1224:}
1220:1
1214:{
1208:)
1203:n
1195:,
1189:,
1184:2
1176:,
1173:1
1170:=
1165:1
1157:(
1154:=
1129:1
1123:n
1119:2
1095:,
1091:]
1085:j
1082:i
1078:a
1072:i
1062:n
1057:1
1054:=
1051:i
1041:n
1036:1
1033:=
1030:j
1021:)
1015:k
1005:n
1000:1
997:=
994:k
985:(
970:[
962:1
956:n
952:2
948:1
943:=
940:)
937:A
934:(
906:(
862:S
842:)
839:n
834:1
828:n
824:2
820:(
817:O
797:)
792:2
788:n
782:1
776:n
772:2
768:(
765:O
742:.
737:j
734:i
730:a
724:S
718:j
708:n
703:1
700:=
697:i
686:|
682:S
678:|
673:)
669:1
663:(
658:}
655:n
652:,
646:,
643:1
640:{
634:S
624:n
620:)
616:1
610:(
607:=
604:)
601:A
598:(
566:.
561:k
551:k
547:)
543:1
537:(
532:1
526:n
521:0
518:=
515:k
507:=
504:)
501:A
498:(
467:k
463:A
442:)
437:k
433:A
429:(
426:P
404:k
377:k
373:A
352:)
347:k
343:A
339:(
336:P
326:k
322:A
306:k
302:A
271:n
260:n
251:n
247:S
227:.
222:)
219:i
216:(
210:,
207:i
203:a
197:n
192:1
189:=
186:i
174:n
170:S
155:=
152:)
149:A
146:(
122:a
118:A
114:n
110:n
20:)
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