3566:– are presented one at a time to an algorithm that must decide whether to include or exclude each element, without knowledge of the later elements. In this variant of the problem, which allows for interesting applications in several contexts, it is possible to devise an optimal selection procedure that, given a random sample of size
2589:
31:
in which the subsequence's elements are sorted in an ascending order and in which the subsequence is as long as possible. This subsequence is not necessarily contiguous or unique. The longest increasing subsequences are studied in the context of various disciplines related to
2624:
newL > L: // If we found a subsequence longer than any we've // found yet, update L L = newL // Reconstruct the longest increasing subsequence // It consists of the values of X at the L indices: // ..., P]], P], M S = array of length L k = M
2620:: // if X] < X lo = mid + 1 // After searching, lo == hi is 1 greater than the // length of the longest prefix of X newL = lo // The predecessor of X is the last index of // the subsequence of length newL-1 P = M M = i
3658:
after the usual centering and scaling. The same asymptotic results hold with more precise bounds for the corresponding problem in the setting of a
Poisson arrival process. A further refinement in the Poisson process setting is given through the proof of a
400:, the length of the first row of the tableau corresponding to a permutation equals the length of the longest increasing subsequence of the permutation, and the length of the first column equals the length of the longest decreasing subsequence.
3663:
for the optimal selection process which holds, with a suitable normalization, in a more complete sense than one would expect. The proof yields not only the "correct" functional limit theorem but also the (singular)
377:
corresponds to the longest decreasing subsequence of the permutation that defines the graph (assuming the original non-permuted sequence is sorted from lowest value to highest). Similarly, the maximum
2809:
1371:
1477:
140:
This subsequence has length six; the input sequence has no seven-member increasing subsequences. The longest increasing subsequence in this example is not the only solution: for instance,
1063:
2437:
1520:
365:
3652:
318:
3614:
3474:
1153:
1111:
981:
463:
2687:
1705:
91:
1775:
1205:
696:
3418:
2725:
can be used to extend the current longest increasing sequence, in constant time, prior to doing the binary search. With this modification, the algorithm uses at most
2463:
1832:
1572:
1546:
895:
775:
729:
2112:
2077:
4095:
3444:
2838:
2580:
2530:
2504:
862:
2313:
2261:
2049:
2004:
1798:
933:
827:
495:
277:
231:
3584:
3560:
3520:
3496:
2723:
2342:
2290:
2229:
2199:
2170:
2141:
2026:
1972:
1943:
1910:
1881:
1861:
1734:
1655:
1635:
1615:
1595:
1391:
1306:
1286:
1257:
1234:
795:
749:
667:
638:
618:
589:
569:
538:
515:
251:
208:
188:
111:
381:
in a permutation graph corresponds to the longest non-decreasing subsequence. Therefore, longest increasing subsequence algorithms can be used to solve the
3446:
For inputs in which each permutation of the input is equally likely, the expected length of the longest increasing subsequence is approximately
4262:
389:
4233:
3680: – Russian mathematician (1933–2024) who studied applications of group theory to longest increasing subsequences in random permutations.
4195:; Delbaen, Freddy (2004), "A central limit theorem for the optimal selection process for monotone subsequences of maximum expected length",
412:. It processes the sequence elements in order, maintaining the longest increasing subsequence found so far. Denote the sequence values as
1637:
but it is possible that not all elements in this array are used by the algorithm (in fact, if the longest increasing sequence has length
163:
3781:
2608:
N-1: //N-1 included // Binary search for the smallest positive l ≤ L // such that X] > X lo = 1 hi = L + 1
3805:
3683:
4002:; Kerov, C. V. (1977), "Asymptotics of the Plancheral measure of the symmetric group and a limiting form for Young tableaux",
4023:
3903:
378:
4163:; Delbaen, Freddy (2001), "Optimal rules for the sequential selection of monotone subsequences of maximum expected length",
3384:
2728:
1311:
3523:
4292:
4277:
3499:
1396:
2350:
4257:
3688:
2811:
comparisons in the worst case, which is optimal for a comparison-based algorithm up to the constant factor in the
2644:
Because the algorithm performs a single binary search per sequence element, its total time can be expressed using
4287:
3527:
3616:
The length of the increasing subsequence selected by this optimal procedure has variance approximately equal to
3707:− an algebraic system defined by transformations that preserve the length of the longest increasing subsequence
3697: – Sorting algorithm − an efficient technique for finding the length of the longest increasing subsequence
986:
3979:
3879:
122:
408:
The algorithm outlined below solves the longest increasing subsequence problem efficiently with arrays and
4282:
4026:(1999), "On the distribution of the length of the longest increasing subsequence of random permutations",
3660:
370:
45:
4252:
3732:(1999), "Longest increasing subsequences: from patience sorting to the Baik–Deift–Johansson theorem",
1484:
323:
3619:
2596:
P = array of length N M = array of length N + 1 M = -1 // undefined so can be set to any value L = 0
2208:
41:
282:
3655:
3589:
3542:, in which the elements of a sequence of independent random variables with continuous distribution
3449:
1116:
167:
3987:, IMA Volumes in Mathematics and its Applications, vol. 72, Springer-Verlag, pp. 111–131
3586:
as input, will generate an increasing sequence with maximal expected length of size approximately
1068:
938:
415:
4143:
4125:
4035:
3861:
3563:
2651:
1660:
55:
3502:
says, that the length of the longest increasing subsequence of a randomly permuted sequence of
1739:
4229:
4113:
4066:
3963:
3842:
Hunt, J.; Szymanski, T. (1977), "A fast algorithm for computing longest common subsequences",
3777:
3665:
1157:
672:
374:
3390:
2442:
1802:
1551:
1525:
867:
754:
701:
4204:
4172:
4135:
4085:
4045:
3912:
3851:
3814:
3800:
3769:
3741:
3694:
3539:
2082:
2056:
20:
3828:
4192:
4160:
3936:
3898:
3824:
3677:
3423:
2814:
2535:
2509:
2468:
832:
4228:. Institute of Mathematical Statistics Textbooks. New York: Cambridge University Press.
2295:
2234:
2031:
1977:
1780:
900:
800:
468:
259:
213:
4116:(2015), "Optimal online selection of a monotone subsequence: a central limit theorem",
3971:
3729:
3700:
3569:
3545:
3505:
3481:
2699:
2645:
2318:
2266:
2214:
2175:
2146:
2117:
2011:
1948:
1919:
1886:
1866:
1837:
1710:
1640:
1620:
1600:
1580:
1376:
1291:
1262:
1242:
1210:
780:
734:
643:
623:
594:
574:
545:
523:
500:
382:
236:
193:
173:
96:
4177:
4271:
3999:
3932:
3917:
409:
397:
4147:
3975:
3967:
3865:
3725:
2693:
37:
4050:
3940:
3746:
320:
this approach can be made much more efficient, leading to time bounds of the form
279:
However, for the special case in which the input is a permutation of the integers
3966:(1995), "Variations on the monotone subsequence theme of Erdős and Szekeres", in
2612:
lo < hi: mid = lo + floor((hi-lo)/2) // lo <= mid < hi
393:
33:
4070:
4209:
4139:
154:
are other increasing subsequences of equal length in the same input sequence.
4090:
3773:
3819:
3538:
The longest increasing subsequence has also been studied in the setting of
4071:"Optimal Sequential Selection of a Monotone Sequence From a Random Sample"
3856:
3420:
distinct integers has an increasing or a decreasing subsequence of length
2696:; in the variant that he studies, the algorithm tests whether each value
28:
3668:
of the three-dimensional process summarizing all interacting processes.
3886:, Computer Science and Applied Mathematics, Academic Press, p. 159
3526:, the distribution of the largest eigenvalue of a random matrix in the
540:— stores the length of the longest increasing subsequence found so far.
254:
49:
3901:(1975), "On computing the length of longest increasing subsequences",
2582:
Thus, we may do binary searches in this sequence in logarithmic time.
4040:
2439:
is increasing. For, if there is an increasing subsequence of length
162:
The longest increasing subsequence problem is closely related to the
2588:
4130:
52:. The longest increasing subsequence problem is solvable in time
1239:
To clarify, "there exists an increasing subsequence of length
4226:
The surprising mathematics of longest increasing subsequences
3766:
The
Surprising Mathematics of Longest Increasing Subsequences
3703: – monoid of positive integers modulo Knuth equivalence
2692:
discusses a variant of this algorithm, which he credits to
170:
solution: the longest increasing subsequence of a sequence
3803:(1961), "Longest increasing and decreasing subsequences",
1548:
represents the length of the increasing subsequence, and
2347:
Note that, at any point in the algorithm, the sequence
620:
such that there is an increasing subsequence of length
3622:
3592:
3572:
3548:
3508:
3484:
3452:
3426:
3393:
2817:
2804:{\displaystyle n\log _{2}n-n\log _{2}\log _{2}n+O(n)}
2731:
2702:
2654:
2538:
2512:
2471:
2445:
2353:
2321:
2298:
2269:
2237:
2217:
2178:
2149:
2120:
2085:
2059:
2034:
2014:
1980:
1951:
1922:
1889:
1869:
1840:
1805:
1783:
1742:
1713:
1663:
1643:
1623:
1603:
1583:
1554:
1528:
1487:
1399:
1379:
1314:
1294:
1265:
1245:
1213:
1160:
1119:
1071:
989:
941:
903:
870:
835:
803:
783:
777:
and there exists an increasing subsequence of length
757:
737:
704:
675:
646:
626:
597:
577:
548:
526:
503:
471:
418:
326:
285:
262:
239:
216:
196:
176:
99:
58:
3705:
Pages displaying wikidata descriptions as a fallback
2532:
ending at a smaller value: namely the one ending at
1366:{\displaystyle i_{1}<i_{2}<\cdots <i_{l}=k}
128:
0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15
1207:); if multiple indices satisfy this condition then
3646:
3608:
3578:
3554:
3514:
3490:
3468:
3438:
3412:
2832:
2803:
2717:
2681:
2574:
2524:
2498:
2457:
2431:
2336:
2307:
2284:
2255:
2223:
2193:
2164:
2135:
2106:
2071:
2043:
2020:
1998:
1966:
1937:
1904:
1875:
1855:
1826:
1792:
1769:
1728:
1699:
1649:
1629:
1609:
1589:
1566:
1540:
1514:
1471:
1385:
1365:
1300:
1280:
1251:
1228:
1199:
1147:
1105:
1057:
975:
927:
889:
856:
821:
789:
769:
743:
723:
690:
661:
632:
612:
583:
563:
532:
509:
489:
457:
359:
312:
271:
245:
225:
202:
182:
105:
85:
3691: – Algorithmic problem on pairs of sequences
1472:{\displaystyle X\left<X\left<\cdots <X.}
3654:and its limiting distribution is asymptotically
1974:in the longest increasing subsequence ending at
1015:
4263:Finding count of longest increased subsequences
132:one of the longest increasing subsequences is
27:problem aims to find a subsequence of a given
4253:Algorithmist's Longest Increasing Subsequence
3734:Bulletin of the American Mathematical Society
8:
4028:Journal of the American Mathematical Society
4197:Stochastic Processes and Their Applications
4165:Stochastic Processes and Their Applications
4118:Stochastic Processes and Their Applications
3884:Algorithmic Graph Theory and Perfect Graphs
3795:
3793:
2506:then there is also a subsequence of length
2585:The algorithm, then, proceeds as follows:
497:the algorithm will have stored an integer
113:denotes the length of the input sequence.
4258:Simplified Longest Increasing Subsequence
4208:
4176:
4129:
4089:
4049:
4039:
3916:
3855:
3818:
3745:
3633:
3623:
3621:
3593:
3591:
3571:
3547:
3522:items has a distribution approaching the
3507:
3483:
3456:
3451:
3425:
3398:
3392:
2816:
2774:
2761:
2739:
2730:
2701:
2653:
2537:
2511:
2470:
2444:
2352:
2320:
2297:
2268:
2236:
2216:
2177:
2148:
2119:
2084:
2058:
2033:
2013:
1979:
1950:
1945:— stores the index of the predecessor of
1921:
1888:
1868:
1839:
1804:
1782:
1741:
1712:
1662:
1642:
1622:
1602:
1582:
1553:
1527:
1486:
1435:
1411:
1398:
1378:
1351:
1332:
1319:
1313:
1293:
1264:
1244:
1212:
1159:
1130:
1118:
1091:
1070:
1029:
1018:
988:
961:
940:
902:
875:
869:
834:
802:
782:
756:
736:
709:
703:
674:
645:
625:
596:
576:
547:
525:
502:
470:
417:
325:
284:
261:
238:
215:
195:
175:
98:
57:
2847:
2616:X] >= X hi = mid
2587:
1574:represents the index of its termination.
1058:{\displaystyle X]=\min _{j\in K_{i,l}}X}
3759:
3757:
3717:
2689:
2292:corresponds to a subsequence of length
1863:is undefined since sequences of length
1707:are used by the algorithm). If however
158:Relations to other algorithmic problems
4112:Arlotto, Alessandro; Nguyen, Vinh V.;
3941:"A combinatorial problem in geometry"
3562:– or alternatively the elements of a
190:is the longest common subsequence of
7:
2637:0: //0 included S = X k = P
2344:and adjust the indices accordingly.
121:In the first 16 terms of the binary
3981:Discrete Probability and Algorithms
385:efficiently in permutation graphs.
4101:from the original on July 30, 2018
164:longest common subsequence problem
14:
2207:Because the algorithm below uses
390:Robinson–Schensted correspondence
2432:{\displaystyle X],X],\ldots ,X]}
1515:{\displaystyle 1\leq l\leq i+1,}
360:{\displaystyle O(n\log \log n).}
3806:Canadian Journal of Mathematics
3684:Longest alternating subsequence
3647:{\displaystyle {\sqrt {2n}}/3,}
2315:A real implementation can skip
731:denotes the set of all indices
2827:
2821:
2798:
2792:
2712:
2706:
2673:
2658:
2566:
2563:
2560:
2554:
2548:
2542:
2490:
2487:
2481:
2475:
2426:
2423:
2417:
2411:
2396:
2393:
2387:
2381:
2372:
2369:
2363:
2357:
2331:
2325:
2279:
2273:
2247:
2241:
2188:
2182:
2159:
2153:
2130:
2124:
2095:
2089:
1990:
1984:
1961:
1955:
1932:
1926:
1899:
1893:
1850:
1844:
1815:
1809:
1764:
1758:
1723:
1717:
1694:
1688:
1673:
1667:
1463:
1457:
1275:
1269:
1223:
1217:
1194:
1188:
1179:
1176:
1170:
1164:
1081:
1075:
1052:
1046:
1008:
1005:
999:
993:
951:
945:
922:
919:
913:
907:
851:
845:
813:
807:
656:
650:
607:
601:
558:
552:
481:
475:
443:
437:
428:
422:
351:
330:
313:{\displaystyle 1,2,\ldots ,n,}
77:
62:
25:longest increasing subsequence
1:
4178:10.1016/S0304-4149(01)00122-3
4051:10.1090/S0894-0347-99-00307-0
3747:10.1090/S0273-0979-99-00796-X
3609:{\displaystyle {\sqrt {2n}}.}
3469:{\displaystyle 2{\sqrt {n}}.}
1148:{\displaystyle j\in K_{i,l},}
166:, which has a quadratic time
3918:10.1016/0012-365X(75)90103-X
3500:Baik-Deift-Johansson theorem
1777:(and moreover, at iteration
1106:{\displaystyle M\in K_{i,l}}
976:{\displaystyle M\in K_{i,l}}
465:etc. Then, after processing
458:{\displaystyle X,X,\ldots ,}
4022:Baik, Jinho; Deift, Percy;
2682:{\displaystyle O(n\log n).}
1700:{\displaystyle M,\ldots ,M}
935:is minimized; meaning that
86:{\displaystyle O(n\log n),}
4309:
3689:Longest common subsequence
3224:Values stored in variables
2860:Values stored in variables
2263:which goes unused so that
517:and values in two arrays:
4210:10.1016/j.spa.2004.09.002
4140:10.1016/j.spa.2015.03.009
3844:Communications of the ACM
3528:Gaussian unitary ensemble
3498:approaches infinity, the
3355:
3330:
3305:
3280:
3251:
3227:
1770:{\displaystyle l-1\leq M}
1288:" means that there exist
698:Explicitly, suppose that
3774:10.1017/CBO9781139872003
3524:Tracy–Widom distribution
1617:more than the length of
1200:{\displaystyle X]\leq X}
691:{\displaystyle k\leq i.}
16:Computer science problem
3413:{\displaystyle n^{2}+1}
2458:{\displaystyle l\geq 2}
1827:{\displaystyle M\leq i}
1567:{\displaystyle k\geq 0}
1541:{\displaystyle l\geq 1}
890:{\displaystyle K_{i,l}}
770:{\displaystyle j\leq i}
724:{\displaystyle K_{i,l}}
123:Van der Corput sequence
4091:10.1214/aop/1176994265
4065:Samuels, Stephen. M.;
3978:; et al. (eds.),
3945:Compositio Mathematica
3820:10.4153/CJM-1961-015-3
3648:
3610:
3580:
3556:
3516:
3492:
3470:
3440:
3414:
3385:Erdős–Szekeres theorem
2834:
2805:
2719:
2683:
2593:
2576:
2526:
2500:
2459:
2433:
2338:
2309:
2286:
2257:
2225:
2195:
2166:
2137:
2108:
2107:{\displaystyle P<k}
2073:
2072:{\displaystyle k>0}
2045:
2022:
2000:
1968:
1939:
1906:
1883:have no ending index (
1877:
1857:
1828:
1794:
1771:
1730:
1701:
1651:
1631:
1611:
1591:
1568:
1542:
1516:
1473:
1387:
1367:
1302:
1282:
1253:
1230:
1201:
1149:
1107:
1059:
977:
929:
891:
858:
823:
791:
771:
745:
725:
692:
663:
634:
614:
591:of the smallest value
585:
565:
534:
511:
491:
459:
361:
314:
273:
247:
227:
204:
184:
107:
87:
4078:Annals of Probability
4004:Dokl. Akad. Nauk SSSR
3857:10.1145/359581.359603
3661:central limit theorem
3649:
3611:
3581:
3557:
3517:
3493:
3471:
3441:
3415:
2835:
2806:
2720:
2684:
2591:
2577:
2527:
2501:
2460:
2434:
2339:
2310:
2287:
2258:
2226:
2196:
2167:
2138:
2109:
2074:
2046:
2023:
2001:
1969:
1940:
1907:
1878:
1858:
1829:
1795:
1772:
1736:is used/defined then
1731:
1702:
1652:
1632:
1612:
1592:
1569:
1543:
1517:
1474:
1388:
1368:
1303:
1283:
1254:
1231:
1202:
1150:
1108:
1060:
978:
930:
892:
859:
824:
792:
772:
746:
726:
693:
664:
635:
615:
586:
566:
535:
512:
492:
460:
362:
315:
274:
248:
228:
205:
185:
108:
88:
46:representation theory
3904:Discrete Mathematics
3620:
3590:
3570:
3546:
3506:
3482:
3450:
3439:{\displaystyle n+1.}
3424:
3391:
2833:{\displaystyle O(n)}
2815:
2729:
2700:
2652:
2575:{\displaystyle X]].}
2536:
2510:
2469:
2443:
2351:
2319:
2296:
2267:
2235:
2215:
2209:zero-based numbering
2176:
2172:has no predecessor (
2147:
2118:
2083:
2057:
2032:
2028:is equal to that of
2012:
1978:
1949:
1920:
1887:
1867:
1838:
1803:
1781:
1740:
1711:
1661:
1641:
1621:
1601:
1581:
1552:
1526:
1485:
1397:
1377:
1312:
1292:
1263:
1243:
1236:is the largest one.
1211:
1158:
1117:
1069:
987:
939:
901:
868:
833:
801:
781:
755:
735:
702:
673:
644:
624:
595:
575:
546:
524:
501:
469:
416:
404:Efficient algorithms
324:
283:
260:
237:
214:
194:
174:
97:
56:
42:random matrix theory
4293:Dynamic programming
4278:Problems on strings
4224:Romik, Dan (2015).
3899:Fredman, Michael L.
3764:Romik, Dan (2015).
2855:
2592:A demo of the code.
2525:{\displaystyle l-1}
2499:{\displaystyle X],}
2143:is undefined since
857:{\displaystyle k=M}
571:— stores the index
168:dynamic programming
136:0, 2, 6, 9, 11, 15.
4114:Steele, J. Michael
4067:Steele, J. Michael
3964:Steele, J. Michael
3644:
3606:
3576:
3564:random permutation
3552:
3512:
3488:
3466:
3436:
3410:
3387:, any sequence of
2848:
2830:
2801:
2715:
2679:
2594:
2572:
2522:
2496:
2455:
2429:
2334:
2308:{\displaystyle l.}
2305:
2282:
2256:{\displaystyle M,}
2253:
2221:
2201:can be any value).
2191:
2162:
2133:
2104:
2069:
2044:{\displaystyle X,}
2041:
2018:
1999:{\displaystyle X.}
1996:
1964:
1935:
1912:can be any value).
1902:
1873:
1853:
1824:
1793:{\displaystyle i,}
1790:
1767:
1726:
1697:
1647:
1627:
1607:
1587:
1564:
1538:
1512:
1469:
1383:
1363:
1298:
1278:
1249:
1226:
1197:
1145:
1103:
1065:(or equivalently,
1055:
1042:
973:
928:{\displaystyle X]}
925:
887:
854:
822:{\displaystyle X.}
819:
787:
767:
741:
721:
688:
659:
630:
610:
581:
561:
530:
507:
490:{\displaystyle X,}
487:
455:
357:
310:
272:{\displaystyle S.}
269:
243:
226:{\displaystyle T,}
223:
200:
180:
150:0, 4, 6, 9, 13, 15
147:0, 2, 6, 9, 13, 15
144:0, 4, 6, 9, 11, 15
103:
83:
4235:978-1-107-42882-9
3666:covariance matrix
3631:
3601:
3579:{\displaystyle n}
3555:{\displaystyle F}
3540:online algorithms
3534:Online algorithms
3515:{\displaystyle n}
3491:{\displaystyle n}
3461:
3383:According to the
3376:
3375:
2718:{\displaystyle X}
2337:{\displaystyle M}
2285:{\displaystyle M}
2224:{\displaystyle M}
2194:{\displaystyle P}
2165:{\displaystyle X}
2136:{\displaystyle P}
2021:{\displaystyle P}
1967:{\displaystyle X}
1938:{\displaystyle P}
1905:{\displaystyle M}
1876:{\displaystyle 0}
1856:{\displaystyle M}
1834:will also hold).
1729:{\displaystyle M}
1650:{\displaystyle L}
1630:{\displaystyle X}
1610:{\displaystyle 1}
1590:{\displaystyle M}
1386:{\displaystyle k}
1301:{\displaystyle l}
1281:{\displaystyle X}
1252:{\displaystyle l}
1229:{\displaystyle M}
1014:
790:{\displaystyle l}
744:{\displaystyle j}
662:{\displaystyle X}
633:{\displaystyle l}
613:{\displaystyle X}
584:{\displaystyle k}
564:{\displaystyle M}
533:{\displaystyle L}
510:{\displaystyle L}
375:permutation graph
253:is the result of
246:{\displaystyle T}
203:{\displaystyle S}
183:{\displaystyle S}
106:{\displaystyle n}
4300:
4288:Formal languages
4240:
4239:
4221:
4215:
4213:
4212:
4193:Bruss, F. Thomas
4189:
4183:
4181:
4180:
4161:Bruss, F. Thomas
4157:
4151:
4150:
4133:
4124:(9): 3596–3622,
4109:
4103:
4102:
4100:
4093:
4075:
4062:
4056:
4054:
4053:
4043:
4034:(4): 1119–1178,
4019:
4013:
4011:
3996:
3990:
3988:
3986:
3960:
3954:
3952:
3937:Szekeres, George
3929:
3923:
3921:
3920:
3895:
3889:
3887:
3876:
3870:
3869:
3859:
3839:
3833:
3831:
3822:
3797:
3788:
3787:
3761:
3752:
3750:
3749:
3722:
3706:
3695:Patience sorting
3653:
3651:
3650:
3645:
3637:
3632:
3624:
3615:
3613:
3612:
3607:
3602:
3594:
3585:
3583:
3582:
3577:
3561:
3559:
3558:
3553:
3521:
3519:
3518:
3513:
3497:
3495:
3494:
3489:
3478:In the limit as
3475:
3473:
3472:
3467:
3462:
3457:
3445:
3443:
3442:
3437:
3419:
3417:
3416:
3411:
3403:
3402:
3372:
3371:
3365:
3364:
3358:
3353:
3352:
3343:
3338:
3333:
3328:
3327:
3318:
3313:
3308:
3303:
3302:
3293:
3288:
3283:
3278:
3277:
3268:
3263:
3262:
3256:
3255:
3248:
3240:
3235:
3230:
3225:
3218:
3217:
3211:
3210:
3204:
3203:
3197:
3196:
3190:
3189:
3183:
3182:
3176:
3175:
3166:
3161:
3156:
3151:
3146:
3141:
3136:
3135:
3126:
3121:
3116:
3111:
3106:
3101:
3096:
3095:
3086:
3081:
3076:
3071:
3066:
3061:
3056:
3055:
3046:
3041:
3036:
3031:
3026:
3021:
3016:
3015:
3006:
3001:
2996:
2991:
2986:
2981:
2976:
2975:
2966:
2961:
2956:
2951:
2946:
2941:
2940:
2934:
2933:
2924:
2919:
2914:
2909:
2899:
2891:
2886:
2881:
2876:
2871:
2866:
2861:
2856:
2854:
2853:
2839:
2837:
2836:
2831:
2810:
2808:
2807:
2802:
2779:
2778:
2766:
2765:
2744:
2743:
2724:
2722:
2721:
2716:
2688:
2686:
2685:
2680:
2581:
2579:
2578:
2573:
2531:
2529:
2528:
2523:
2505:
2503:
2502:
2497:
2464:
2462:
2461:
2456:
2438:
2436:
2435:
2430:
2343:
2341:
2340:
2335:
2314:
2312:
2311:
2306:
2291:
2289:
2288:
2283:
2262:
2260:
2259:
2254:
2230:
2228:
2227:
2222:
2200:
2198:
2197:
2192:
2171:
2169:
2168:
2163:
2142:
2140:
2139:
2134:
2113:
2111:
2110:
2105:
2078:
2076:
2075:
2070:
2050:
2048:
2047:
2042:
2027:
2025:
2024:
2019:
2005:
2003:
2002:
1997:
1973:
1971:
1970:
1965:
1944:
1942:
1941:
1936:
1911:
1909:
1908:
1903:
1882:
1880:
1879:
1874:
1862:
1860:
1859:
1854:
1833:
1831:
1830:
1825:
1799:
1797:
1796:
1791:
1776:
1774:
1773:
1768:
1735:
1733:
1732:
1727:
1706:
1704:
1703:
1698:
1656:
1654:
1653:
1648:
1636:
1634:
1633:
1628:
1616:
1614:
1613:
1608:
1596:
1594:
1593:
1588:
1573:
1571:
1570:
1565:
1547:
1545:
1544:
1539:
1521:
1519:
1518:
1513:
1478:
1476:
1475:
1470:
1444:
1440:
1439:
1420:
1416:
1415:
1392:
1390:
1389:
1384:
1372:
1370:
1369:
1364:
1356:
1355:
1337:
1336:
1324:
1323:
1307:
1305:
1304:
1299:
1287:
1285:
1284:
1279:
1258:
1256:
1255:
1250:
1235:
1233:
1232:
1227:
1206:
1204:
1203:
1198:
1154:
1152:
1151:
1146:
1141:
1140:
1112:
1110:
1109:
1104:
1102:
1101:
1064:
1062:
1061:
1056:
1041:
1040:
1039:
982:
980:
979:
974:
972:
971:
934:
932:
931:
926:
896:
894:
893:
888:
886:
885:
864:is the index in
863:
861:
860:
855:
828:
826:
825:
820:
796:
794:
793:
788:
776:
774:
773:
768:
750:
748:
747:
742:
730:
728:
727:
722:
720:
719:
697:
695:
694:
689:
668:
666:
665:
660:
639:
637:
636:
631:
619:
617:
616:
611:
590:
588:
587:
582:
570:
568:
567:
562:
539:
537:
536:
531:
516:
514:
513:
508:
496:
494:
493:
488:
464:
462:
461:
456:
410:binary searching
366:
364:
363:
358:
319:
317:
316:
311:
278:
276:
275:
270:
252:
250:
249:
244:
232:
230:
229:
224:
209:
207:
206:
201:
189:
187:
186:
181:
112:
110:
109:
104:
92:
90:
89:
84:
21:computer science
4308:
4307:
4303:
4302:
4301:
4299:
4298:
4297:
4268:
4267:
4249:
4244:
4243:
4236:
4223:
4222:
4218:
4191:
4190:
4186:
4159:
4158:
4154:
4111:
4110:
4106:
4098:
4073:
4064:
4063:
4059:
4024:Johansson, Kurt
4021:
4020:
4016:
3998:
3997:
3993:
3984:
3972:Diaconis, Persi
3962:
3961:
3957:
3931:
3930:
3926:
3897:
3896:
3892:
3880:Golumbic, M. C.
3878:
3877:
3873:
3841:
3840:
3836:
3799:
3798:
3791:
3784:
3763:
3762:
3755:
3730:Diaconis, Persi
3724:
3723:
3719:
3714:
3704:
3678:Anatoly Vershik
3674:
3618:
3617:
3588:
3587:
3568:
3567:
3544:
3543:
3536:
3504:
3503:
3480:
3479:
3448:
3447:
3422:
3421:
3394:
3389:
3388:
3381:
3369:
3368:
3362:
3361:
3356:
3350:
3349:At end of loop
3348:
3341:
3336:
3331:
3325:
3324:At end of loop
3323:
3316:
3311:
3306:
3300:
3299:At end of loop
3298:
3291:
3286:
3281:
3275:
3274:At end of loop
3273:
3266:
3260:
3259:
3253:
3252:
3246:
3238:
3233:
3228:
3223:
3215:
3214:
3208:
3207:
3201:
3200:
3194:
3193:
3187:
3186:
3180:
3179:
3173:
3172:At end of loop
3171:
3164:
3159:
3154:
3149:
3144:
3139:
3133:
3132:At end of loop
3131:
3124:
3119:
3114:
3109:
3104:
3099:
3093:
3092:At end of loop
3091:
3084:
3079:
3074:
3069:
3064:
3059:
3053:
3052:At end of loop
3051:
3044:
3039:
3034:
3029:
3024:
3019:
3013:
3012:At end of loop
3011:
3004:
2999:
2994:
2989:
2984:
2979:
2973:
2972:At end of loop
2971:
2964:
2959:
2954:
2949:
2944:
2938:
2937:
2931:
2930:At end of loop
2929:
2922:
2917:
2912:
2907:
2897:
2889:
2884:
2879:
2874:
2869:
2864:
2859:
2851:
2849:
2813:
2812:
2770:
2757:
2735:
2727:
2726:
2698:
2697:
2650:
2649:
2642:
2534:
2533:
2508:
2507:
2467:
2466:
2441:
2440:
2349:
2348:
2317:
2316:
2294:
2293:
2265:
2264:
2233:
2232:
2231:is padded with
2213:
2212:
2174:
2173:
2145:
2144:
2116:
2115:
2081:
2080:
2055:
2054:
2030:
2029:
2010:
2009:
1976:
1975:
1947:
1946:
1918:
1917:
1885:
1884:
1865:
1864:
1836:
1835:
1801:
1800:
1779:
1778:
1738:
1737:
1709:
1708:
1659:
1658:
1639:
1638:
1619:
1618:
1599:
1598:
1579:
1578:
1550:
1549:
1524:
1523:
1483:
1482:
1431:
1427:
1407:
1403:
1395:
1394:
1375:
1374:
1347:
1328:
1315:
1310:
1309:
1290:
1289:
1261:
1260:
1241:
1240:
1209:
1208:
1156:
1155:
1126:
1115:
1114:
1087:
1067:
1066:
1025:
985:
984:
957:
937:
936:
899:
898:
871:
866:
865:
831:
830:
799:
798:
779:
778:
753:
752:
733:
732:
705:
700:
699:
671:
670:
642:
641:
622:
621:
593:
592:
573:
572:
544:
543:
522:
521:
499:
498:
467:
466:
414:
413:
406:
379:independent set
322:
321:
281:
280:
258:
257:
235:
234:
212:
211:
192:
191:
172:
171:
160:
119:
95:
94:
54:
53:
17:
12:
11:
5:
4306:
4304:
4296:
4295:
4290:
4285:
4280:
4270:
4269:
4266:
4265:
4260:
4255:
4248:
4247:External links
4245:
4242:
4241:
4234:
4216:
4203:(2): 287–311,
4184:
4171:(2): 313–342,
4152:
4104:
4084:(6): 937–947,
4057:
4014:
4000:Vershik, A. M.
3991:
3955:
3924:
3890:
3871:
3850:(5): 350–353,
3834:
3789:
3782:
3753:
3740:(4): 413–432,
3716:
3715:
3713:
3710:
3709:
3708:
3701:Plactic monoid
3698:
3692:
3686:
3681:
3673:
3670:
3643:
3640:
3636:
3630:
3627:
3605:
3600:
3597:
3575:
3551:
3535:
3532:
3511:
3487:
3465:
3460:
3455:
3435:
3432:
3429:
3409:
3406:
3401:
3397:
3380:
3377:
3374:
3373:
3366:
3359:
3354:
3345:
3344:
3339:
3334:
3329:
3320:
3319:
3314:
3309:
3304:
3295:
3294:
3289:
3284:
3279:
3270:
3269:
3264:
3257:
3250:
3242:
3241:
3236:
3231:
3226:
3220:
3219:
3212:
3205:
3198:
3191:
3184:
3177:
3168:
3167:
3162:
3157:
3152:
3147:
3142:
3137:
3128:
3127:
3122:
3117:
3112:
3107:
3102:
3097:
3088:
3087:
3082:
3077:
3072:
3067:
3062:
3057:
3048:
3047:
3042:
3037:
3032:
3027:
3022:
3017:
3008:
3007:
3002:
2997:
2992:
2987:
2982:
2977:
2968:
2967:
2962:
2957:
2952:
2947:
2942:
2935:
2926:
2925:
2920:
2915:
2910:
2905:
2903:
2901:
2893:
2892:
2887:
2882:
2877:
2872:
2867:
2862:
2829:
2826:
2823:
2820:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2777:
2773:
2769:
2764:
2760:
2756:
2753:
2750:
2747:
2742:
2738:
2734:
2714:
2711:
2708:
2705:
2690:Fredman (1975)
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2646:Big O notation
2595:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2521:
2518:
2515:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2454:
2451:
2448:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2333:
2330:
2327:
2324:
2304:
2301:
2281:
2278:
2275:
2272:
2252:
2249:
2246:
2243:
2240:
2220:
2211:, for clarity
2205:
2204:
2203:
2202:
2190:
2187:
2184:
2181:
2161:
2158:
2155:
2152:
2132:
2129:
2126:
2123:
2103:
2100:
2097:
2094:
2091:
2088:
2068:
2065:
2062:
2051:
2040:
2037:
2017:
2008:The length of
1995:
1992:
1989:
1986:
1983:
1963:
1960:
1957:
1954:
1934:
1931:
1928:
1925:
1915:
1914:
1913:
1901:
1898:
1895:
1892:
1872:
1852:
1849:
1846:
1843:
1823:
1820:
1817:
1814:
1811:
1808:
1789:
1786:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1725:
1722:
1719:
1716:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1646:
1626:
1606:
1586:
1577:The length of
1575:
1563:
1560:
1557:
1537:
1534:
1531:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1479:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1443:
1438:
1434:
1430:
1426:
1423:
1419:
1414:
1410:
1406:
1402:
1382:
1362:
1359:
1354:
1350:
1346:
1343:
1340:
1335:
1331:
1327:
1322:
1318:
1297:
1277:
1274:
1271:
1268:
1248:
1225:
1222:
1219:
1216:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1144:
1139:
1136:
1133:
1129:
1125:
1122:
1113:and for every
1100:
1097:
1094:
1090:
1086:
1083:
1080:
1077:
1074:
1054:
1051:
1048:
1045:
1038:
1035:
1032:
1028:
1024:
1021:
1017:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
970:
967:
964:
960:
956:
953:
950:
947:
944:
924:
921:
918:
915:
912:
909:
906:
884:
881:
878:
874:
853:
850:
847:
844:
841:
838:
818:
815:
812:
809:
806:
786:
766:
763:
760:
740:
718:
715:
712:
708:
687:
684:
681:
678:
658:
655:
652:
649:
629:
609:
606:
603:
600:
580:
560:
557:
554:
551:
541:
529:
506:
486:
483:
480:
477:
474:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
405:
402:
398:Young tableaux
383:clique problem
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
309:
306:
303:
300:
297:
294:
291:
288:
268:
265:
242:
222:
219:
199:
179:
159:
156:
152:
151:
148:
145:
138:
137:
130:
129:
118:
115:
102:
82:
79:
76:
73:
70:
67:
64:
61:
15:
13:
10:
9:
6:
4:
3:
2:
4305:
4294:
4291:
4289:
4286:
4284:
4283:Combinatorics
4281:
4279:
4276:
4275:
4273:
4264:
4261:
4259:
4256:
4254:
4251:
4250:
4246:
4237:
4231:
4227:
4220:
4217:
4211:
4206:
4202:
4198:
4194:
4188:
4185:
4179:
4174:
4170:
4166:
4162:
4156:
4153:
4149:
4145:
4141:
4137:
4132:
4127:
4123:
4119:
4115:
4108:
4105:
4097:
4092:
4087:
4083:
4079:
4072:
4068:
4061:
4058:
4052:
4047:
4042:
4037:
4033:
4029:
4025:
4018:
4015:
4009:
4005:
4001:
3995:
3992:
3983:
3982:
3977:
3976:Spencer, Joel
3973:
3969:
3968:Aldous, David
3965:
3959:
3956:
3950:
3946:
3942:
3938:
3934:
3928:
3925:
3919:
3914:
3910:
3906:
3905:
3900:
3894:
3891:
3885:
3881:
3875:
3872:
3867:
3863:
3858:
3853:
3849:
3845:
3838:
3835:
3830:
3826:
3821:
3816:
3812:
3808:
3807:
3802:
3801:Schensted, C.
3796:
3794:
3790:
3785:
3783:9781107075832
3779:
3775:
3771:
3767:
3760:
3758:
3754:
3748:
3743:
3739:
3735:
3731:
3727:
3726:Aldous, David
3721:
3718:
3711:
3702:
3699:
3696:
3693:
3690:
3687:
3685:
3682:
3679:
3676:
3675:
3671:
3669:
3667:
3662:
3657:
3641:
3638:
3634:
3628:
3625:
3603:
3598:
3595:
3573:
3565:
3549:
3541:
3533:
3531:
3529:
3525:
3509:
3501:
3485:
3476:
3463:
3458:
3453:
3433:
3430:
3427:
3407:
3404:
3399:
3395:
3386:
3379:Length bounds
3378:
3367:
3360:
3347:
3346:
3340:
3335:
3322:
3321:
3315:
3310:
3297:
3296:
3290:
3285:
3272:
3271:
3265:
3258:
3244:
3243:
3237:
3232:
3222:
3221:
3213:
3206:
3199:
3192:
3185:
3178:
3170:
3169:
3163:
3158:
3153:
3148:
3143:
3138:
3130:
3129:
3123:
3118:
3113:
3108:
3103:
3098:
3090:
3089:
3083:
3078:
3073:
3068:
3063:
3058:
3050:
3049:
3043:
3038:
3033:
3028:
3023:
3018:
3010:
3009:
3003:
2998:
2993:
2988:
2983:
2978:
2970:
2969:
2963:
2958:
2953:
2948:
2943:
2936:
2928:
2927:
2921:
2916:
2911:
2906:
2904:
2902:
2895:
2894:
2888:
2883:
2878:
2873:
2868:
2863:
2858:
2857:
2846:
2845:
2841:
2824:
2818:
2795:
2789:
2786:
2783:
2780:
2775:
2771:
2767:
2762:
2758:
2754:
2751:
2748:
2745:
2740:
2736:
2732:
2709:
2703:
2695:
2691:
2676:
2670:
2667:
2664:
2661:
2655:
2647:
2640:
2636:
2632:
2628:
2623:
2619:
2615:
2611:
2607:
2603:
2599:
2590:
2586:
2583:
2569:
2557:
2551:
2545:
2539:
2519:
2516:
2513:
2493:
2484:
2478:
2472:
2452:
2449:
2446:
2420:
2414:
2408:
2405:
2402:
2399:
2390:
2384:
2378:
2375:
2366:
2360:
2354:
2345:
2328:
2322:
2302:
2299:
2276:
2270:
2250:
2244:
2238:
2218:
2210:
2185:
2179:
2156:
2150:
2127:
2121:
2101:
2098:
2092:
2086:
2066:
2063:
2060:
2052:
2038:
2035:
2015:
2007:
2006:
1993:
1987:
1981:
1958:
1952:
1929:
1923:
1916:
1896:
1890:
1870:
1847:
1841:
1821:
1818:
1812:
1806:
1787:
1784:
1761:
1755:
1752:
1749:
1746:
1743:
1720:
1714:
1691:
1685:
1682:
1679:
1676:
1670:
1664:
1644:
1624:
1604:
1584:
1576:
1561:
1558:
1555:
1535:
1532:
1529:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1480:
1466:
1460:
1454:
1451:
1448:
1445:
1441:
1436:
1432:
1428:
1424:
1421:
1417:
1412:
1408:
1404:
1400:
1380:
1360:
1357:
1352:
1348:
1344:
1341:
1338:
1333:
1329:
1325:
1320:
1316:
1295:
1272:
1266:
1246:
1238:
1237:
1220:
1214:
1191:
1185:
1182:
1173:
1167:
1161:
1142:
1137:
1134:
1131:
1127:
1123:
1120:
1098:
1095:
1092:
1088:
1084:
1078:
1072:
1049:
1043:
1036:
1033:
1030:
1026:
1022:
1019:
1011:
1002:
996:
990:
968:
965:
962:
958:
954:
948:
942:
916:
910:
904:
882:
879:
876:
872:
848:
842:
839:
836:
816:
810:
804:
784:
764:
761:
758:
738:
716:
713:
710:
706:
685:
682:
679:
676:
669:in the range
653:
647:
627:
604:
598:
578:
555:
549:
542:
527:
520:
519:
518:
504:
484:
478:
472:
452:
449:
446:
440:
434:
431:
425:
419:
411:
403:
401:
399:
395:
391:
386:
384:
380:
376:
372:
367:
354:
348:
345:
342:
339:
336:
333:
327:
307:
304:
301:
298:
295:
292:
289:
286:
266:
263:
256:
240:
220:
217:
197:
177:
169:
165:
157:
155:
149:
146:
143:
142:
141:
135:
134:
133:
127:
126:
125:
124:
116:
114:
100:
80:
74:
71:
68:
65:
59:
51:
47:
43:
39:
35:
30:
26:
22:
4225:
4219:
4200:
4196:
4187:
4168:
4164:
4155:
4121:
4117:
4107:
4081:
4077:
4060:
4041:math/9810105
4031:
4027:
4017:
4007:
4003:
3994:
3980:
3958:
3948:
3944:
3927:
3911:(1): 29–35,
3908:
3902:
3893:
3883:
3874:
3847:
3843:
3837:
3810:
3804:
3765:
3737:
3733:
3720:
3537:
3477:
3382:
2843:
2842:
2694:Donald Knuth
2643:
2638:
2634:
2630:
2626:
2621:
2617:
2613:
2609:
2605:
2601:
2597:
2584:
2346:
2206:
407:
394:permutations
387:
369:The largest
368:
161:
153:
139:
131:
120:
38:algorithmics
36:, including
24:
18:
4010:: 1024–1027
3933:Erdős, Paul
3813:: 179–191,
2844:Example run
34:mathematics
4272:Categories
3712:References
3363:k = P = -1
2465:ending at
1657:then only
1481:Note that
1393:such that
1373:ending at
1259:ending at
897:for which
797:ending at
751:such that
640:ending at
4131:1408.6750
3951:: 463–470
3337:k = P = 0
3312:k = P = 3
3287:k = P = 4
3261:k = M = 5
2781:
2768:
2752:−
2746:
2668:
2517:−
2450:≥
2403:…
1819:≤
1753:≤
1747:−
1680:…
1559:≥
1533:≥
1498:≤
1492:≤
1449:⋯
1342:⋯
1183:≤
1124:∈
1085:∈
1023:∈
955:∈
762:≤
680:≤
450:…
346:
340:
299:…
72:
4148:15900488
4096:archived
4069:(1981),
3939:(1935),
3882:(1980),
3672:See also
3188:newL = 1
3145:newL = 4
3105:newL = 3
3065:newL = 2
3025:newL = 3
2985:newL = 2
2945:newL = 1
2631:in range
2602:in range
1522:because
1308:indices
392:between
29:sequence
3866:3226080
3829:0121305
3370:X = N/A
3245:Before
2896:Before
388:In the
255:sorting
117:Example
50:physics
4232:
4146:
3864:
3827:
3780:
3656:normal
2850:Using
2840:term.
2639:return
2114:while
371:clique
233:where
93:where
48:, and
23:, the
4144:S2CID
4126:arXiv
4099:(PDF)
4074:(PDF)
4036:arXiv
3985:(PDF)
3862:S2CID
3351:j = 0
3342:X = 2
3326:j = 1
3317:X = 5
3301:j = 2
3292:X = 6
3276:j = 3
3267:X = 7
3249:loop
3247:for j
3216:L = 4
3181:X = 1
3174:i = 6
3165:L = 4
3140:X = 7
3134:i = 5
3125:L = 3
3100:X = 6
3094:i = 4
3085:L = 3
3060:X = 5
3054:i = 3
3045:L = 3
3020:X = 9
3014:i = 2
3005:L = 2
2980:X = 8
2974:i = 1
2965:L = 1
2939:X = 2
2932:i = 0
2923:L = 0
2900:loop
2898:for i
2610:while
2079:then
829:Then
373:in a
4230:ISBN
3778:ISBN
3357:S =
3332:S =
3307:S =
3282:S =
3254:S =
3209:X =
3202:M =
3195:P =
3160:X =
3155:M =
3150:P =
3120:X =
3115:M =
3110:P =
3080:X =
3075:M =
3070:P =
3040:X =
3035:M =
3030:P =
3000:X =
2995:M =
2990:P =
2960:X =
2955:M =
2950:P =
2918:X =
2913:M =
2908:P =
2870:newL
2852:X =
2633:L-1
2618:else
2099:<
2064:>
1452:<
1446:<
1422:<
1345:<
1339:<
1326:<
983:and
396:and
210:and
4205:doi
4201:114
4173:doi
4136:doi
4122:125
4086:doi
4046:doi
4008:233
3913:doi
3852:doi
3815:doi
3770:doi
3742:doi
2772:log
2759:log
2737:log
2665:log
2648:as
2627:for
2598:for
2053:If
1597:is
1016:min
343:log
337:log
69:log
19:In
4274::
4199:,
4169:96
4167:,
4142:,
4134:,
4120:,
4094:,
4080:,
4076:,
4044:,
4032:12
4030:,
4006:,
3974:;
3970:;
3947:,
3943:,
3935:;
3909:11
3907:,
3860:,
3848:20
3846:,
3825:MR
3823:,
3811:13
3809:,
3792:^
3776:.
3768:.
3756:^
3738:36
3736:,
3728:;
3530:.
3434:1.
2641:S
2635:to
2629:j
2622:if
2614:if
2606:to
2604:0
2600:i
44:,
40:,
4238:.
4214:.
4207::
4182:.
4175::
4138::
4128::
4088::
4082:9
4055:.
4048::
4038::
4012:.
3989:.
3953:.
3949:2
3922:.
3915::
3888:.
3868:.
3854::
3832:.
3817::
3786:.
3772::
3751:.
3744::
3642:,
3639:3
3635:/
3629:n
3626:2
3604:.
3599:n
3596:2
3574:n
3550:F
3510:n
3486:n
3464:.
3459:n
3454:2
3431:+
3428:n
3408:1
3405:+
3400:2
3396:n
3239:X
3234:k
3229:S
2890:L
2885:X
2880:M
2875:P
2865:X
2828:)
2825:n
2822:(
2819:O
2799:)
2796:n
2793:(
2790:O
2787:+
2784:n
2776:2
2763:2
2755:n
2749:n
2741:2
2733:n
2713:]
2710:i
2707:[
2704:X
2677:.
2674:)
2671:n
2662:n
2659:(
2656:O
2570:.
2567:]
2564:]
2561:]
2558:l
2555:[
2552:M
2549:[
2546:P
2543:[
2540:X
2520:1
2514:l
2494:,
2491:]
2488:]
2485:l
2482:[
2479:M
2476:[
2473:X
2453:2
2447:l
2427:]
2424:]
2421:L
2418:[
2415:M
2412:[
2409:X
2406:,
2400:,
2397:]
2394:]
2391:2
2388:[
2385:M
2382:[
2379:X
2376:,
2373:]
2370:]
2367:1
2364:[
2361:M
2358:[
2355:X
2332:]
2329:0
2326:[
2323:M
2303:.
2300:l
2280:]
2277:l
2274:[
2271:M
2251:,
2248:]
2245:0
2242:[
2239:M
2219:M
2189:]
2186:0
2183:[
2180:P
2160:]
2157:0
2154:[
2151:X
2131:]
2128:0
2125:[
2122:P
2102:k
2096:]
2093:k
2090:[
2087:P
2067:0
2061:k
2039:,
2036:X
2016:P
1994:.
1991:]
1988:k
1985:[
1982:X
1962:]
1959:k
1956:[
1953:X
1933:]
1930:k
1927:[
1924:P
1900:]
1897:0
1894:[
1891:M
1871:0
1851:]
1848:0
1845:[
1842:M
1822:i
1816:]
1813:l
1810:[
1807:M
1788:,
1785:i
1765:]
1762:l
1759:[
1756:M
1750:1
1744:l
1724:]
1721:l
1718:[
1715:M
1695:]
1692:L
1689:[
1686:M
1683:,
1677:,
1674:]
1671:1
1668:[
1665:M
1645:L
1625:X
1605:1
1585:M
1562:0
1556:k
1536:1
1530:l
1510:,
1507:1
1504:+
1501:i
1495:l
1489:1
1467:.
1464:]
1461:k
1458:[
1455:X
1442:]
1437:2
1433:i
1429:[
1425:X
1418:]
1413:1
1409:i
1405:[
1401:X
1381:k
1361:k
1358:=
1353:l
1349:i
1334:2
1330:i
1321:1
1317:i
1296:l
1276:]
1273:k
1270:[
1267:X
1247:l
1224:]
1221:l
1218:[
1215:M
1195:]
1192:j
1189:[
1186:X
1180:]
1177:]
1174:l
1171:[
1168:M
1165:[
1162:X
1143:,
1138:l
1135:,
1132:i
1128:K
1121:j
1099:l
1096:,
1093:i
1089:K
1082:]
1079:l
1076:[
1073:M
1053:]
1050:j
1047:[
1044:X
1037:l
1034:,
1031:i
1027:K
1020:j
1012:=
1009:]
1006:]
1003:l
1000:[
997:M
994:[
991:X
969:l
966:,
963:i
959:K
952:]
949:l
946:[
943:M
923:]
920:]
917:l
914:[
911:M
908:[
905:X
883:l
880:,
877:i
873:K
852:]
849:l
846:[
843:M
840:=
837:k
817:.
814:]
811:j
808:[
805:X
785:l
765:i
759:j
739:j
717:l
714:,
711:i
707:K
686:.
683:i
677:k
657:]
654:k
651:[
648:X
628:l
608:]
605:k
602:[
599:X
579:k
559:]
556:l
553:[
550:M
528:L
505:L
485:,
482:]
479:i
476:[
473:X
453:,
447:,
444:]
441:1
438:[
435:X
432:,
429:]
426:0
423:[
420:X
355:.
352:)
349:n
334:n
331:(
328:O
308:,
305:n
302:,
296:,
293:2
290:,
287:1
267:.
264:S
241:T
221:,
218:T
198:S
178:S
101:n
81:,
78:)
75:n
66:n
63:(
60:O
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