46:
947:
849:
35:
4713:. In 1977, Logan and Shepp, as well as Vershik and Kerov, showed that the Young diagram of a typical large partition becomes asympototically close to the graph of a certain analytic function minimizing a certain functional. In 1988, Baik, Deift and Johansson extended these results to determine the distribution of the longest increasing subsequence of a random permutation in terms of the
2753:
2737:
2732:
2659:
2654:
2649:
4556:
4550:
4544:
4539:
4501:
2759:
2743:
2722:
2696:
2690:
2685:
2680:
2675:
2670:
2607:
2602:
2597:
2592:
2587:
2582:
2577:
2572:
2567:
2550:
2544:
2538:
2533:
2518:
2513:
2503:
2498:
2446:
2440:
2434:
2429:
2418:
2413:
2407:
2397:
2391:
2386:
2381:
2366:
2355:
2350:
2344:
2339:
2329:
2323:
2318:
2313:
2308:
2303:
912:
906:
901:
896:
891:
885:
880:
875:
870:
865:
791:
785:
779:
773:
764:
758:
752:
747:
738:
733:
727:
722:
713:
707:
702:
697:
688:
683:
678:
673:
660:
654:
649:
644:
638:
633:
628:
623:
617:
612:
607:
602:
597:
592:
4533:
4528:
4523:
4517:
4512:
4507:
4496:
4491:
4486:
2764:
2748:
2727:
2716:
2711:
2706:
2701:
2643:
2638:
2633:
2628:
2623:
2618:
2613:
2528:
2508:
2423:
2402:
2376:
2360:
2334:
2298:
1886:
4288:
2469:
of one another. In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. Of particular interest are partitions, such as 2 + 2, which have
4460:. For example, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4. In the Ferrers diagram or Young diagram of a partition of rank
1481:
3089:
1669:
4050:
1597:
2038:
3602:
954:(40): A ruler with plus and minus signs (grey box) is slid downwards, the relevant parts added or subtracted. The positions of the signs are given by differences of alternating natural (blue) and odd (orange) numbers. In
530:
843:(often also called a Ferrers diagram). Rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. Thus, the Young diagram for the partition 5 + 4 + 1 is
4429:
1309:
3759:
275:
Some authors treat a partition as a decreasing sequence of summands, rather than an expression with plus signs. For example, the partition 2 + 2 + 1 might instead be written as the
2218:
2927:
358:
4693:, together with their branching properties, in characteristic zero. It also has received significant study for its purely combinatorial properties; notably, it is the motivating example of a
3942:
3691:
3464:
41:
associated to the partitions of the positive integers 1 through 8. They are arranged so that images under the reflection about the main diagonal of the square are conjugate partitions.
934:, and these tableaux have combinatorial and representation-theoretic significance. As a type of shape made by adjacent squares joined together, Young diagrams are a special kind of
2465:
By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14. Such partitions are said to be
4016:
1893:
1881:{\displaystyle p(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }A_{k}(n){\sqrt {k}}\cdot {\frac {d}{dn}}\left({{\frac {1}{\sqrt {n-{\frac {1}{24}}}}}\sinh \left}\right)}
4608:
1259:
4639:
1623:
2558:
One can then obtain a bijection between the set of partitions with distinct odd parts and the set of self-conjugate partitions, as illustrated by the following example:
1512:
4283:{\displaystyle {k+\ell \choose \ell }_{q}={k+\ell \choose k}_{q}={\frac {\prod _{j=1}^{k+\ell }(1-q^{j})}{\prod _{j=1}^{k}(1-q^{j})\prod _{j=1}^{\ell }(1-q^{j})}}.}
1111:
2073:
1053:
1143:
557:
4566:
The Durfee square has applications within combinatorics in the proofs of various partition identities. It also has some practical significance in the form of the
1659:
998:
2282:
In both combinatorics and number theory, families of partitions subject to various restrictions are often studied. This section surveys a few such restrictions.
1195:
1169:
2871:
For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction. An important example is
2272:
2252:
1301:
1215:
1073:
1018:
2852:
This is a general property. For each positive number, the number of partitions with odd parts equals the number of partitions with distinct parts, denoted by
1265:
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, ... (sequence
3483:
665:
The 14 circles are lined up in 4 rows, each having the size of a part of the partition. The diagrams for the 5 partitions of the number 4 are shown below:
4316:
2902:
1272:
930:: filling the boxes of Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a family of objects called
922:
While this seemingly trivial variation does not appear worthy of separate mention, Young diagrams turn out to be extremely useful in the study of
393:
4710:
5422:
5133:
5762:
4940:
3834:
1476:{\displaystyle \sum _{n=0}^{\infty }p(n)q^{n}=\prod _{j=1}^{\infty }\sum _{i=0}^{\infty }q^{ji}=\prod _{j=1}^{\infty }(1-q^{j})^{-1}.}
969:
963:
172:
5655:
5631:
5585:
5553:
5378:
5344:
5310:
5291:
5269:
3718:
5803:
4862:
4827:
2104:
5027:
4769:
3084:{\displaystyle \sum _{n=0}^{\infty }q(n)x^{n}=\prod _{k=1}^{\infty }(1+x^{k})=\prod _{k=1}^{\infty }{\frac {1}{1-x^{2k-1}}}.}
5720:
5332:
3785:
3216:
4714:
587:
The partition 6 + 4 + 3 + 1 of the number 14 can be represented by the following diagram:
572:
292:
4800:
5851:
5715:
927:
100:
4795:
3628:
4674:
3394:
2231:
2290:
If we flip the diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14:
3095:
2091:
2825:
Alternatively, we could count partitions in which no number occurs more than once. Such a partition is called a
4785:
24:
4821:
5150:
5545:
5414:
4705:
There is a deep theory of random partitions chosen according to the uniform probability distribution on the
2480:: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.
45:
4810:
2083:
1487:
568:
5666:
5385:(an elementary introduction to the topic of integer partitions, including a discussion of Ferrers graphs)
3947:
5260:
4759:
4670:
4642:
3697:
2865:
4744:
1499:
238:
230:
4580:
1220:
955:
567:
There are two common diagrammatic methods to represent partitions: as
Ferrers diagrams, named after
5406:
4735:
4574:
4291:
3704:
specifies the available coins). As two particular cases, one has that the number of partitions of
3620:
2910:
2223:
1495:
1491:
1280:
923:
848:
4860:
Josuat-Vergès, Matthieu (2010), "Bijections between pattern-avoiding fillings of Young diagrams",
4617:
1602:
1592:{\displaystyle p(n)\sim {\frac {1}{4n{\sqrt {3}}}}\exp \left({\pi {\sqrt {\frac {2n}{3}}}}\right)}
5826:
5641:
5516:(Has text, nearly complete bibliography, but they (and Abramowitz) missed the Selberg formula for
5229:
5203:
5172:
4960:
4897:
4871:
4764:
4694:
2227:
27:. For the problem of partitioning a multiset of integers so that each part has the same sum, see
20:
4665:
4654:
5710:
34:
5734:
5651:
5627:
5581:
5549:
5418:
5402:
5374:
5366:
5340:
5306:
5287:
5279:
5265:
5251:
5221:
5129:
2095:
1662:
946:
92:
28:
5778:
1078:
5688:
5678:
5615:
5591:
5559:
5537:
5483:
5465:
5436:
5350:
5213:
5162:
4952:
4881:
4663:
on partitions given by inclusion of Young diagrams. This partially ordered set is known as
2043:
1023:
5479:
5432:
4893:
1116:
535:
5753:
5698:(Provides the Selberg formula. The older form is the finite Fourier expansion of Selberg.)
5692:
5603:
5595:
5577:
5563:
5487:
5475:
5440:
5428:
5354:
5336:
4889:
4774:
4749:
4722:
4718:
4706:
4678:
4641:. This statistic (which is unrelated to the one described above) appears in the study of
2033:{\displaystyle A_{k}(n)=\sum _{0\leq m<k,\;(m,k)=1}e^{\pi i\left(s(m,k)-2nm/k\right)}.}
1635:
1629:
974:
234:
218:
1174:
1148:
5128:. Institute of Mathematical Statistics Textbooks. New York: Cambridge University Press.
5325:
5320:
5072:, volume 1, second edition. Cambridge University Press, 2012. Chapter 1, section 1.7.
4805:
4790:
3597:{\displaystyle \sum _{n\geq 0}p_{k}(n)x^{n}=x^{k}\prod _{i=1}^{k}{\frac {1}{1-x^{i}}}.}
2861:
2257:
2237:
2087:
1286:
1200:
1058:
1003:
931:
5470:
5453:
5083:
5845:
5620:
5233:
4779:
4754:
4660:
4473:
4439:
834:
214:
65:
61:
38:
5176:
4901:
5449:
5410:
5394:
5255:
4815:
2076:
19:
This article is about partitioning an integer. For grouping elements of a set, see
5261:
Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical Tables
5822:
5794:
5390:
3768:
in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of
1503:
390:
may be omitted.) For example, in this notation, the partitions of 5 are written
222:
5737:
4885:
5815:
5167:
2752:
2736:
2731:
2658:
2653:
2648:
5371:
A Walk
Through Combinatorics: An Introduction to Enumeration and Graph Theory
5225:
4555:
4549:
4543:
4538:
4500:
4030:, and subtracting 1 from each part of such a partition yields a partition of
3821:. Equivalently, these are the partitions whose Young diagram fits inside an
3708:
in which all parts are 1 or 2 (or, equivalently, the number of partitions of
2758:
2742:
2721:
2695:
2689:
2684:
2679:
2674:
2669:
2606:
2601:
2596:
2591:
2586:
2581:
2576:
2571:
2566:
2549:
2543:
2537:
2532:
2517:
2512:
2502:
2497:
2445:
2439:
2433:
2428:
2417:
2412:
2406:
2396:
2390:
2385:
2380:
2365:
2354:
2349:
2343:
2338:
2328:
2322:
2317:
2312:
2307:
2302:
911:
905:
900:
895:
890:
884:
879:
874:
869:
864:
790:
784:
778:
772:
763:
757:
751:
746:
737:
732:
726:
721:
712:
706:
701:
696:
687:
682:
677:
672:
659:
653:
648:
643:
637:
632:
627:
622:
616:
611:
606:
601:
596:
591:
5784:
5742:
4532:
4527:
4522:
4516:
4511:
4506:
4495:
4490:
4485:
2763:
2747:
2726:
2715:
2710:
2705:
2700:
2642:
2637:
2632:
2627:
2622:
2617:
2612:
2527:
2507:
2422:
2401:
2375:
2359:
2333:
2297:
935:
525:{\displaystyle 5^{1},1^{1}4^{1},2^{1}3^{1},1^{2}3^{1},1^{1}2^{2},1^{3}2^{1}}
88:
5683:
5454:"On the remainder and convergence of the series for the partition function"
289:
This multiplicity notation for a partition can be written alternatively as
5492:
Provides the main formula (no derivatives), remainder, and older form for
5727:
4820:
A Goldbach partition is the partition of an even number into primes (see
3790:
One may also simultaneously limit the number and size of the parts. Let
2829:. If we count the partitions of 8 with distinct parts, we also obtain 6:
99:
are considered the same partition. (If order matters, the sum becomes a
5217:
4964:
4567:
2798:
Among the 22 partitions of the number 8, there are 6 that contain only
226:
96:
73:
5781:
with reference tables to the On-Line
Encyclopedia of Integer Sequences
5360:(See chapter 5 for a modern pedagogical intro to Rademacher's formula)
5264:. United States Department of Commerce, National Bureau of Standards.
5645:
5208:
5191:
286:
where the superscript indicates the number of repetitions of a part.
4956:
839:
An alternative visual representation of an integer partition is its
5808:
4876:
4424:{\displaystyle \sum _{n=0}^{MN}p(N,M;n)q^{n}={M+N \choose M}_{q}.}
2226:
discovered that the partition function has nontrivial patterns in
945:
276:
44:
33:
5798:
3611:
is a set of positive integers then the number of partitions of
137:
The only partition of zero is the empty sum, having no parts.
5610:. Vol. v II. MIT Press. pp. 100–07, 108–22, 460–75.
5126:
The surprising mathematics of longest increasing subsequences
3469:
One possible generating function for such partitions, taking
2879:) (partitions into distinct parts). The first few values of
5667:"A sum connected with the series for the partition function"
3780:
Partitions in a rectangle and
Gaussian binomial coefficients
3754:{\displaystyle \left\lfloor {\frac {n}{2}}+1\right\rfloor ,}
5788:
2897:
1267:
2486:: The crucial observation is that every odd part can be "
16:
Decomposition of an integer as a sum of positive integers
5809:
Generating All
Partitions: A Comparison Of Two Encodings
5034:. Philadelphia: W. B. Saunders Company. pp. 149–50.
4669:. The lattice was originally defined in the context of
2470:
themselves as conjugate. Such partitions are said to be
2213:{\displaystyle p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\cdots }
575:. Both have several possible conventions; here, we use
5327:
Modular functions and
Dirichlet series in number theory
2234:. For instance, whenever the decimal representation of
5509:
Gupta, Hansraj; Gwyther, C.E.; Miller, J.C.P. (1962).
2254:
ends in the digit 4 or 9, the number of partitions of
1498:
by which it can be calculated exactly. It grows as an
4620:
4583:
4319:
4053:
3950:
3837:
3772:
into at most three parts) is the nearest integer to (
3721:
3631:
3486:
3397:
2930:
2260:
2240:
2107:
2046:
1896:
1672:
1638:
1605:
1515:
1490:
for the partition function is known, but it has both
1312:
1289:
1223:
1203:
1177:
1151:
1119:
1081:
1061:
1026:
1006:
977:
538:
396:
295:
4472:
square of entries in the upper-left is known as the
4290:
The
Gaussian binomial coefficient is related to the
855:
while the
Ferrers diagram for the same partition is
4573:A different statistic is also sometimes called the
5619:
5324:
4941:"Partition identities - from Euler to the present"
4633:
4602:
4423:
4282:
4010:
3936:
3753:
3685:
3596:
3458:
3083:
2490:" in the middle to form a self-conjugate diagram:
2266:
2246:
2212:
2067:
2032:
1880:
1653:
1617:
1591:
1475:
1295:
1253:
1209:
1189:
1163:
1137:
1105:
1067:
1047:
1012:
992:
579:, with diagrams aligned in the upper-left corner.
551:
524:
353:{\displaystyle 1^{m_{1}}2^{m_{2}}3^{m_{3}}\cdots }
352:
95:. Two sums that differ only in the order of their
5804:Fast Algorithms For Generating Integer Partitions
5650:. Vol. 1 and 2. Cambridge University Press.
4782:, defined by partitions into consecutive integers
4406:
4385:
4119:
4098:
4079:
4058:
4047:The Gaussian binomial coefficient is defined as:
163:An individual summand in a partition is called a
5576:. Graduate Texts in Mathematics. Vol. 195.
5056:
5044:
5014:
3937:{\displaystyle p(N,M;n)=p(N,M-1;n)+p(N-1,M;n-M)}
2895:1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... (sequence
1632:found a way to represent the partition function
1000:counts the partitions of a non-negative integer
5335:. Vol. 41 (2nd ed.). New York etc.:
4721:related these results to the combinatorics of
3686:{\displaystyle \prod _{t\in T}(1-x^{t})^{-1}.}
3247:parts is equal to the number of partitions of
213:Partitions can be graphically visualized with
5085:Some Famous Problems of the Theory of Numbers
3459:{\displaystyle p(n)=\sum _{k=0}^{n}p_{k}(n).}
8:
5301:Andrews, George E.; Eriksson, Kimmo (2004).
378:is the number of 2's, etc. (Components with
5001:
4989:
4977:
4926:
3831:rectangle. There is a recurrence relation
5155:International Mathematics Research Notices
4452:such that the partition contains at least
1942:
563:Diagrammatic representations of partitions
107:can be partitioned in five distinct ways:
23:. For the partition calculus of sets, see
5682:
5469:
5207:
5166:
5151:"Random matrices and random permutations"
4875:
4625:
4619:
4588:
4582:
4412:
4405:
4384:
4382:
4372:
4335:
4324:
4318:
4265:
4246:
4235:
4222:
4203:
4192:
4177:
4152:
4141:
4134:
4125:
4118:
4097:
4095:
4085:
4078:
4057:
4055:
4052:
3949:
3836:
3727:
3720:
3671:
3661:
3636:
3630:
3582:
3566:
3560:
3549:
3539:
3526:
3507:
3491:
3485:
3438:
3428:
3417:
3396:
3060:
3044:
3038:
3027:
3011:
2992:
2981:
2968:
2946:
2935:
2929:
2259:
2239:
2106:
2045:
2012:
1970:
1923:
1901:
1895:
1867:
1866:
1865:
1847:
1826:
1824:
1814:
1813:
1789:
1777:
1776:
1757:
1747:
1732:
1722:
1711:
1697:
1688:
1671:
1637:
1604:
1567:
1563:
1543:
1531:
1514:
1461:
1451:
1432:
1421:
1405:
1395:
1384:
1374:
1363:
1350:
1328:
1317:
1311:
1288:
1222:
1202:
1176:
1150:
1118:
1080:
1060:
1025:
1005:
976:
543:
537:
516:
506:
493:
483:
470:
460:
447:
437:
424:
414:
401:
395:
339:
334:
322:
317:
305:
300:
294:
5542:Symmetric functions and Hall polynomials
5399:An Introduction to the Theory of Numbers
4577:(or Dyson rank), namely, the difference
221:. They occur in a number of branches of
5111:
5098:
4914:
4847:
4840:
3211:Restricted part size or number of parts
2286:Conjugate and self-conjugate partitions
2094:this function is an alternating sum of
958:hover over the image to move the ruler.
5192:"Infinite wedge and random partitions"
5626:. New York: Perennial-HarperCollins.
4448:of a partition is the largest number
2864:in 1748 and later was generalized as
571:, and as Young diagrams, named after
7:
5728:Partition and composition calculator
5513:. Vol. 4, Tables of partitions.
148:, and the two distinct compositions
5608:Collected Papers of Hans Rademacher
5574:Elementary Methods in Number Theory
4673:, where it is used to describe the
4011:{\displaystyle p(N,M;n)-p(N,M-1;n)}
3809:denote the number of partitions of
3253:in which the largest part has size
1494:that accurately approximate it and
5544:. Oxford Mathematical Monographs.
4389:
4102:
4062:
3039:
2993:
2947:
2086:of its generating function is the
1723:
1612:
1433:
1396:
1375:
1329:
1217:. The values of this function for
964:Partition function (number theory)
14:
5471:10.1090/S0002-9947-1939-0000410-9
4711:Robinson–Schensted correspondence
3221:By taking conjugates, the number
282:or in the even more compact form
4554:
4548:
4542:
4537:
4531:
4526:
4521:
4515:
4510:
4505:
4499:
4494:
4489:
4484:
3764:and the number of partitions of
2762:
2757:
2751:
2746:
2741:
2735:
2730:
2725:
2720:
2714:
2709:
2704:
2699:
2694:
2688:
2683:
2678:
2673:
2668:
2657:
2652:
2647:
2641:
2636:
2631:
2626:
2621:
2616:
2611:
2605:
2600:
2595:
2590:
2585:
2580:
2575:
2570:
2565:
2548:
2542:
2536:
2531:
2526:
2516:
2511:
2506:
2501:
2496:
2444:
2438:
2432:
2427:
2421:
2416:
2411:
2405:
2400:
2395:
2389:
2384:
2379:
2374:
2364:
2358:
2353:
2348:
2342:
2337:
2332:
2327:
2321:
2316:
2311:
2306:
2301:
2296:
910:
904:
899:
894:
889:
883:
878:
873:
868:
863:
847:
789:
783:
777:
771:
762:
756:
750:
745:
736:
731:
725:
720:
711:
705:
700:
695:
686:
681:
676:
671:
658:
652:
647:
642:
636:
631:
626:
621:
615:
610:
605:
600:
595:
590:
156:represent the same partition as
140:The order-dependent composition
5829:from the original on 2021-12-11
5814:Grime, James (April 28, 2016).
5768:from the original on 2021-02-24
5373:. World Scientific Publishing.
5059:, p. 826, 24.2.2 eq. II(A)
4863:Journal of Combinatorial Theory
3615:, all of whose parts belong to
167:. The number of partitions of
5795:Integer::Partition Perl module
5755:Lectures on Integer Partitions
5671:Pacific Journal of Mathematics
5305:. Cambridge University Press.
5286:. Cambridge University Press.
5047:, p. 825, 24.2.2 eq. I(B)
4770:Stars and bars (combinatorics)
4603:{\displaystyle \lambda _{k}-k}
4365:
4347:
4271:
4252:
4228:
4209:
4183:
4164:
4005:
3981:
3972:
3954:
3931:
3901:
3892:
3868:
3859:
3841:
3668:
3648:
3519:
3513:
3450:
3444:
3407:
3401:
3017:
2998:
2961:
2955:
2201:
2189:
2180:
2168:
2159:
2147:
2138:
2126:
2117:
2111:
2062:
2050:
1997:
1985:
1955:
1943:
1913:
1907:
1744:
1738:
1682:
1676:
1648:
1642:
1609:
1525:
1519:
1506:of its argument., as follows:
1458:
1438:
1343:
1337:
1254:{\displaystyle n=0,1,2,\dots }
1036:
1030:
987:
981:
249:The seven partitions of 5 are
1:
5511:Royal Society of Math. Tables
5333:Graduate Texts in Mathematics
4945:American Mathematical Monthly
3786:Gaussian binomial coefficient
3217:Triangle of partition numbers
2860:). This result was proved by
2827:partition with distinct parts
2821:1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
2794:Odd parts and distinct parts
950:Using Euler's method to find
5057:Abramowitz & Stegun 1964
5045:Abramowitz & Stegun 1964
5015:Abramowitz & Stegun 1964
4828:Kostant's partition function
4634:{\displaystyle \lambda _{k}}
3817:parts, each of size at most
1618:{\displaystyle n\to \infty }
5716:Encyclopedia of Mathematics
5190:Okounkov, A. (2001-04-01).
4725:and representation theory.
4675:irreducible representations
3944:obtained by observing that
928:group representation theory
239:group representation theory
5868:
5816:"Partitions - Numberphile"
5413:. (6th ed.). Oxford:
4886:10.1016/j.jcta.2010.03.006
4652:
4437:
3783:
3696:This can be used to solve
3380:One recovers the function
3214:
2667:
2563:
961:
832:
18:
5647:Enumerative Combinatorics
5168:10.1155/S1073792800000532
5149:Okounkov, Andrei (2000).
5070:Enumerative Combinatorics
4018:counts the partitions of
3275:satisfies the recurrence
3096:pentagonal number theorem
2092:pentagonal number theorem
229:, including the study of
144:is the same partition as
5779:Counting with partitions
5665:Whiteman, A. L. (1956).
5572:Nathanson, M.B. (2000).
5284:The Theory of Partitions
4939:Alder, Henry L. (1969).
4796:Ewens's sampling formula
4786:Multiplicative partition
4738:, a different notion of
4715:Tracy–Widom distribution
4614:parts with largest part
2274:will be divisible by 5.
2098:powers of its argument.
1075:has the five partitions
25:Infinitary combinatorics
5622:The Music of the Primes
5546:Oxford University Press
5415:Oxford University Press
5002:Hardy & Wright 2008
4990:Hardy & Wright 2008
4978:Hardy & Wright 2008
4927:Hardy & Wright 2008
4816:Smallest-parts function
4648:
4456:parts of size at least
3178:− 22) − ...
3098:gives a recurrence for
2232:Ramanujan's congruences
1106:{\displaystyle 1+1+1+1}
5684:10.2140/pjm.1956.6.159
5532:which is in Whiteman.)
5458:Trans. Amer. Math. Soc
4801:Faà di Bruno's formula
4635:
4604:
4434:Rank and Durfee square
4425:
4343:
4284:
4251:
4208:
4163:
4026:parts of size at most
4012:
3938:
3755:
3712:into 1 or 2 parts) is
3698:change-making problems
3687:
3598:
3565:
3460:
3433:
3085:
3043:
2997:
2951:
2459:4 + 3 + 3 + 2 + 1 + 1
2268:
2248:
2214:
2084:multiplicative inverse
2069:
2068:{\displaystyle s(m,k)}
2034:
1882:
1727:
1655:
1619:
1593:
1488:closed-form expression
1477:
1437:
1400:
1379:
1333:
1297:
1255:
1211:
1191:
1165:
1139:
1107:
1069:
1049:
1048:{\displaystyle p(4)=5}
1014:
994:
959:
569:Norman Macleod Ferrers
553:
526:
369:is the number of 1's,
354:
83:, is a way of writing
57:
42:
4822:Goldbach's conjecture
4760:Integer factorization
4671:representation theory
4643:Ramanujan congruences
4636:
4605:
4426:
4320:
4285:
4231:
4188:
4137:
4013:
3939:
3756:
3688:
3599:
3545:
3461:
3413:
3086:
3023:
2977:
2931:
2887:) are (starting with
2818:3 + 1 + 1 + 1 + 1 + 1
2278:Restricted partitions
2269:
2249:
2215:
2070:
2035:
1883:
1707:
1656:
1620:
1594:
1492:asymptotic expansions
1478:
1417:
1380:
1359:
1313:
1298:
1256:
1212:
1192:
1166:
1140:
1138:{\displaystyle 1+1+2}
1108:
1070:
1050:
1015:
995:
949:
554:
552:{\displaystyle 1^{5}}
527:
355:
231:symmetric polynomials
48:
37:
5082:Hardy, G.H. (1920).
4745:Crank of a partition
4618:
4581:
4317:
4051:
3948:
3835:
3719:
3629:
3484:
3395:
3328:with initial values
3207:and is 0 otherwise.
3170:− 15) −
2928:
2258:
2238:
2105:
2044:
1894:
1670:
1654:{\displaystyle p(n)}
1636:
1603:
1513:
1500:exponential function
1496:recurrence relations
1310:
1287:
1221:
1201:
1175:
1149:
1117:
1079:
1059:
1055:because the integer
1024:
1004:
993:{\displaystyle p(n)}
975:
536:
394:
293:
5642:Stanley, Richard P.
5196:Selecta Mathematica
5124:Romik, Dan (2015).
4811:Newton's identities
4736:Rank of a partition
4659:There is a natural
4610:for a partition of
4575:rank of a partition
4292:generating function
3621:generating function
3607:More generally, if
3377:are not both zero.
3146:− 5) −
3138:− 2) −
2911:generating function
2224:Srinivasa Ramanujan
1281:generating function
1190:{\displaystyle 2+2}
1164:{\displaystyle 1+3}
924:symmetric functions
5852:Integer partitions
5785:Integer partitions
5735:Weisstein, Eric W.
5616:Sautoy, Marcus Du.
5303:Integer Partitions
5280:Andrews, George E.
5252:Abramowitz, Milton
5218:10.1007/PL00001398
5088:. Clarendon Press.
5028:Andrews, George E.
4765:Partition of a set
4695:differential poset
4631:
4600:
4421:
4280:
4008:
3934:
3751:
3683:
3647:
3594:
3502:
3456:
3081:
2866:Glaisher's theorem
2777:5 + 5 + 4 + 3 + 2
2264:
2244:
2228:modular arithmetic
2210:
2065:
2030:
1965:
1878:
1651:
1615:
1589:
1473:
1293:
1251:
1207:
1187:
1161:
1135:
1103:
1065:
1045:
1010:
990:
970:partition function
960:
942:Partition function
549:
522:
350:
206:is a partition of
173:partition function
72:of a non-negative
58:
53:with largest part
43:
21:Partition of a set
5751:Wilf, Herbert S.
5568:(See section I.1)
5538:Macdonald, Ian G.
5424:978-0-19-921986-5
5403:D. R. Heath-Brown
5135:978-1-107-42882-9
5101:, pp. 33–34.
5068:Richard Stanley,
5013:Notation follows
4701:Random partitions
4562:
4561:
4404:
4275:
4117:
4077:
3735:
3632:
3589:
3487:
3237:of partitions of
3203:for some integer
3191:is (−1) if
3076:
2791:
2790:
2556:
2555:
2463:
2462:
2267:{\displaystyle n}
2247:{\displaystyle n}
2096:pentagonal number
1919:
1862:
1855:
1834:
1822:
1800:
1799:
1797:
1770:
1752:
1705:
1702:
1663:convergent series
1582:
1581:
1551:
1548:
1296:{\displaystyle p}
1210:{\displaystyle 4}
1068:{\displaystyle 4}
1013:{\displaystyle n}
918:
917:
826:
825:
271:1 + 1 + 1 + 1 + 1
103:.) For example,
93:positive integers
81:integer partition
79:, also called an
29:Partition problem
5859:
5838:
5836:
5834:
5820:
5775:
5774:
5773:
5767:
5760:
5748:
5747:
5724:
5696:
5686:
5661:
5637:
5625:
5611:
5604:Rademacher, Hans
5599:
5567:
5514:
5491:
5473:
5445:
5444:
5384:
5358:
5330:
5316:
5297:
5275:
5238:
5237:
5211:
5187:
5181:
5180:
5170:
5146:
5140:
5139:
5121:
5115:
5108:
5102:
5096:
5090:
5089:
5079:
5073:
5066:
5060:
5054:
5048:
5042:
5036:
5035:
5024:
5018:
5011:
5005:
4999:
4993:
4987:
4981:
4975:
4969:
4968:
4936:
4930:
4924:
4918:
4912:
4906:
4904:
4879:
4870:(8): 1218–1230,
4857:
4851:
4845:
4723:Riemann surfaces
4679:symmetric groups
4640:
4638:
4637:
4632:
4630:
4629:
4609:
4607:
4606:
4601:
4593:
4592:
4558:
4552:
4546:
4541:
4535:
4530:
4525:
4519:
4514:
4509:
4503:
4498:
4493:
4488:
4481:
4480:
4430:
4428:
4427:
4422:
4417:
4416:
4411:
4410:
4409:
4400:
4388:
4377:
4376:
4342:
4334:
4313:by the equality
4312:
4289:
4287:
4286:
4281:
4276:
4274:
4270:
4269:
4250:
4245:
4227:
4226:
4207:
4202:
4186:
4182:
4181:
4162:
4151:
4135:
4130:
4129:
4124:
4123:
4122:
4113:
4101:
4090:
4089:
4084:
4083:
4082:
4073:
4061:
4043:
4039:
4029:
4025:
4021:
4017:
4015:
4014:
4009:
3943:
3941:
3940:
3935:
3830:
3820:
3816:
3812:
3808:
3760:
3758:
3757:
3752:
3747:
3743:
3736:
3728:
3692:
3690:
3689:
3684:
3679:
3678:
3666:
3665:
3646:
3603:
3601:
3600:
3595:
3590:
3588:
3587:
3586:
3567:
3564:
3559:
3544:
3543:
3531:
3530:
3512:
3511:
3501:
3465:
3463:
3462:
3457:
3443:
3442:
3432:
3427:
3376:
3370:
3364:
3353:
3337:
3324:
3274:
3259:. The function
3258:
3252:
3242:
3236:
3090:
3088:
3087:
3082:
3077:
3075:
3074:
3073:
3045:
3042:
3037:
3016:
3015:
2996:
2991:
2973:
2972:
2950:
2945:
2900:
2766:
2761:
2755:
2750:
2745:
2739:
2734:
2729:
2724:
2718:
2713:
2708:
2703:
2698:
2692:
2687:
2682:
2677:
2672:
2661:
2656:
2651:
2645:
2640:
2635:
2630:
2625:
2620:
2615:
2609:
2604:
2599:
2594:
2589:
2584:
2579:
2574:
2569:
2561:
2560:
2552:
2546:
2540:
2535:
2530:
2523: ↔
2520:
2515:
2510:
2505:
2500:
2493:
2492:
2448:
2442:
2436:
2431:
2425:
2420:
2415:
2409:
2404:
2399:
2393:
2388:
2383:
2378:
2368:
2362:
2357:
2352:
2346:
2341:
2336:
2331:
2325:
2320:
2315:
2310:
2305:
2300:
2293:
2292:
2273:
2271:
2270:
2265:
2253:
2251:
2250:
2245:
2219:
2217:
2216:
2211:
2074:
2072:
2071:
2066:
2039:
2037:
2036:
2031:
2026:
2025:
2024:
2020:
2016:
1964:
1906:
1905:
1887:
1885:
1884:
1879:
1877:
1873:
1872:
1868:
1864:
1863:
1861:
1857:
1856:
1848:
1835:
1827:
1825:
1823:
1815:
1801:
1798:
1790:
1782:
1778:
1771:
1769:
1758:
1753:
1748:
1737:
1736:
1726:
1721:
1706:
1704:
1703:
1698:
1689:
1660:
1658:
1657:
1652:
1624:
1622:
1621:
1616:
1598:
1596:
1595:
1590:
1588:
1584:
1583:
1577:
1569:
1568:
1552:
1550:
1549:
1544:
1532:
1482:
1480:
1479:
1474:
1469:
1468:
1456:
1455:
1436:
1431:
1413:
1412:
1399:
1394:
1378:
1373:
1355:
1354:
1332:
1327:
1302:
1300:
1299:
1294:
1270:
1260:
1258:
1257:
1252:
1216:
1214:
1213:
1208:
1196:
1194:
1193:
1188:
1170:
1168:
1167:
1162:
1144:
1142:
1141:
1136:
1112:
1110:
1109:
1104:
1074:
1072:
1071:
1066:
1054:
1052:
1051:
1046:
1020:. For instance,
1019:
1017:
1016:
1011:
999:
997:
996:
991:
914:
908:
903:
898:
893:
887:
882:
877:
872:
867:
860:
859:
851:
793:
787:
781:
775:
766:
760:
754:
749:
740:
735:
729:
724:
715:
709:
704:
699:
690:
685:
680:
675:
668:
667:
662:
656:
651:
646:
640:
635:
630:
625:
619:
614:
609:
604:
599:
594:
577:English notation
558:
556:
555:
550:
548:
547:
531:
529:
528:
523:
521:
520:
511:
510:
498:
497:
488:
487:
475:
474:
465:
464:
452:
451:
442:
441:
429:
428:
419:
418:
406:
405:
389:
377:
368:
359:
357:
356:
351:
346:
345:
344:
343:
329:
328:
327:
326:
312:
311:
310:
309:
285:
281:
219:Ferrers diagrams
209:
205:
201:
191:
184:
171:is given by the
170:
159:
155:
151:
147:
143:
133:
128:
123:
118:
113:
106:
86:
78:
56:
52:
5867:
5866:
5862:
5861:
5860:
5858:
5857:
5856:
5842:
5841:
5832:
5830:
5818:
5813:
5771:
5769:
5765:
5758:
5752:
5733:
5732:
5709:
5706:
5664:
5658:
5640:
5634:
5614:
5602:
5588:
5578:Springer-Verlag
5571:
5556:
5536:
5525:
5508:
5500:
5448:
5425:
5409:. Foreword by
5407:J. H. Silverman
5389:
5388:
5381:
5365:
5347:
5337:Springer-Verlag
5321:Apostol, Tom M.
5319:
5313:
5300:
5294:
5278:
5272:
5250:
5247:
5242:
5241:
5189:
5188:
5184:
5148:
5147:
5143:
5136:
5123:
5122:
5118:
5109:
5105:
5097:
5093:
5081:
5080:
5076:
5067:
5063:
5055:
5051:
5043:
5039:
5026:
5025:
5021:
5012:
5008:
5000:
4996:
4988:
4984:
4976:
4972:
4957:10.2307/2317861
4938:
4937:
4933:
4925:
4921:
4913:
4909:
4859:
4858:
4854:
4846:
4842:
4837:
4832:
4775:Plane partition
4750:Dominance order
4731:
4707:symmetric group
4703:
4688:
4666:Young's lattice
4657:
4655:Young's lattice
4651:
4649:Young's lattice
4621:
4616:
4615:
4584:
4579:
4578:
4553:
4547:
4536:
4520:
4504:
4442:
4436:
4390:
4383:
4381:
4368:
4315:
4314:
4295:
4261:
4218:
4187:
4173:
4136:
4103:
4096:
4094:
4063:
4056:
4054:
4049:
4048:
4041:
4031:
4027:
4023:
4019:
3946:
3945:
3833:
3832:
3822:
3818:
3814:
3810:
3791:
3788:
3782:
3726:
3722:
3717:
3716:
3700:(where the set
3667:
3657:
3627:
3626:
3578:
3571:
3535:
3522:
3503:
3482:
3481:
3434:
3393:
3392:
3372:
3366:
3355:
3347:
3339:
3335:
3329:
3318:
3300:
3287:
3279:
3268:
3260:
3254:
3248:
3238:
3230:
3222:
3219:
3213:
3190:
3121:
3056:
3049:
3007:
2964:
2926:
2925:
2921:) is given by
2896:
2796:
2787:self-conjugate
2756:
2740:
2719:
2693:
2646:
2610:
2547:
2541:
2484:Proof (outline)
2443:
2437:
2426:
2410:
2394:
2363:
2347:
2326:
2288:
2280:
2256:
2255:
2236:
2235:
2230:, now known as
2103:
2102:
2042:
2041:
1981:
1977:
1966:
1897:
1892:
1891:
1840:
1836:
1812:
1808:
1772:
1762:
1728:
1693:
1668:
1667:
1634:
1633:
1630:Hans Rademacher
1601:
1600:
1570:
1559:
1536:
1511:
1510:
1457:
1447:
1401:
1346:
1308:
1307:
1285:
1284:
1266:
1219:
1218:
1199:
1198:
1173:
1172:
1147:
1146:
1115:
1114:
1077:
1076:
1057:
1056:
1022:
1021:
1002:
1001:
973:
972:
966:
944:
909:
888:
837:
831:
788:
782:
776:
761:
755:
730:
710:
657:
641:
620:
585:
583:Ferrers diagram
565:
539:
534:
533:
512:
502:
489:
479:
466:
456:
443:
433:
420:
410:
397:
392:
391:
387:
379:
376:
370:
367:
361:
335:
330:
318:
313:
301:
296:
291:
290:
283:
279:
247:
235:symmetric group
207:
203:
193:
192:. The notation
186:
175:
168:
157:
153:
149:
145:
141:
131:
126:
121:
116:
111:
104:
84:
76:
54:
50:
32:
17:
12:
11:
5:
5865:
5863:
5855:
5854:
5844:
5843:
5840:
5839:
5811:
5806:
5801:
5792:
5782:
5776:
5749:
5730:
5725:
5705:
5704:External links
5702:
5701:
5700:
5677:(1): 159–176.
5662:
5656:
5638:
5632:
5612:
5600:
5586:
5569:
5554:
5534:
5521:
5506:
5496:
5446:
5423:
5386:
5379:
5363:
5345:
5317:
5311:
5298:
5292:
5276:
5270:
5246:
5243:
5240:
5239:
5182:
5141:
5134:
5116:
5103:
5091:
5074:
5061:
5049:
5037:
5019:
5006:
5004:, p. 365.
4994:
4992:, p. 368.
4982:
4980:, p. 362.
4970:
4951:(7): 733–746.
4931:
4929:, p. 380.
4919:
4907:
4852:
4850:, p. 199.
4839:
4838:
4836:
4833:
4831:
4830:
4825:
4818:
4813:
4808:
4806:Multipartition
4803:
4798:
4793:
4791:Twelvefold way
4788:
4783:
4777:
4772:
4767:
4762:
4757:
4752:
4747:
4742:
4732:
4730:
4727:
4702:
4699:
4684:
4653:Main article:
4650:
4647:
4628:
4624:
4599:
4596:
4591:
4587:
4564:
4563:
4560:
4559:
4438:Main article:
4435:
4432:
4420:
4415:
4408:
4403:
4399:
4396:
4393:
4387:
4380:
4375:
4371:
4367:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4341:
4338:
4333:
4330:
4327:
4323:
4279:
4273:
4268:
4264:
4260:
4257:
4254:
4249:
4244:
4241:
4238:
4234:
4230:
4225:
4221:
4217:
4214:
4211:
4206:
4201:
4198:
4195:
4191:
4185:
4180:
4176:
4172:
4169:
4166:
4161:
4158:
4155:
4150:
4147:
4144:
4140:
4133:
4128:
4121:
4116:
4112:
4109:
4106:
4100:
4093:
4088:
4081:
4076:
4072:
4069:
4066:
4060:
4007:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3855:
3852:
3849:
3846:
3843:
3840:
3784:Main article:
3781:
3778:
3762:
3761:
3750:
3746:
3742:
3739:
3734:
3731:
3725:
3694:
3693:
3682:
3677:
3674:
3670:
3664:
3660:
3656:
3653:
3650:
3645:
3642:
3639:
3635:
3605:
3604:
3593:
3585:
3581:
3577:
3574:
3570:
3563:
3558:
3555:
3552:
3548:
3542:
3538:
3534:
3529:
3525:
3521:
3518:
3515:
3510:
3506:
3500:
3497:
3494:
3490:
3477:variable, is
3467:
3466:
3455:
3452:
3449:
3446:
3441:
3437:
3431:
3426:
3423:
3420:
3416:
3412:
3409:
3406:
3403:
3400:
3343:
3333:
3326:
3325:
3313:
3296:
3283:
3264:
3226:
3215:Main article:
3212:
3209:
3186:
3180:
3179:
3162:− 12) +
3117:
3092:
3091:
3080:
3072:
3069:
3066:
3063:
3059:
3055:
3052:
3048:
3041:
3036:
3033:
3030:
3026:
3022:
3019:
3014:
3010:
3006:
3003:
3000:
2995:
2990:
2987:
2984:
2980:
2976:
2971:
2967:
2963:
2960:
2957:
2954:
2949:
2944:
2941:
2938:
2934:
2907:
2906:
2862:Leonhard Euler
2850:
2849:
2846:
2843:
2840:
2837:
2834:
2823:
2822:
2819:
2816:
2813:
2810:
2807:
2795:
2792:
2789:
2788:
2785:
2783:
2779:
2778:
2775:
2772:
2768:
2767:
2666:
2663:
2554:
2553:
2524:
2521:
2472:self-conjugate
2461:
2460:
2457:
2454:
2453:6 + 4 + 3 + 1
2450:
2449:
2372:
2369:
2287:
2284:
2279:
2276:
2263:
2243:
2221:
2220:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2088:Euler function
2064:
2061:
2058:
2055:
2052:
2049:
2029:
2023:
2019:
2015:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1980:
1976:
1973:
1969:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1941:
1938:
1935:
1932:
1929:
1926:
1922:
1918:
1915:
1912:
1909:
1904:
1900:
1876:
1871:
1860:
1854:
1851:
1846:
1843:
1839:
1833:
1830:
1821:
1818:
1811:
1807:
1804:
1796:
1793:
1788:
1785:
1781:
1775:
1768:
1765:
1761:
1756:
1751:
1746:
1743:
1740:
1735:
1731:
1725:
1720:
1717:
1714:
1710:
1701:
1696:
1692:
1687:
1684:
1681:
1678:
1675:
1650:
1647:
1644:
1641:
1626:
1625:
1614:
1611:
1608:
1587:
1580:
1576:
1573:
1566:
1562:
1558:
1555:
1547:
1542:
1539:
1535:
1530:
1527:
1524:
1521:
1518:
1484:
1483:
1472:
1467:
1464:
1460:
1454:
1450:
1446:
1443:
1440:
1435:
1430:
1427:
1424:
1420:
1416:
1411:
1408:
1404:
1398:
1393:
1390:
1387:
1383:
1377:
1372:
1369:
1366:
1362:
1358:
1353:
1349:
1345:
1342:
1339:
1336:
1331:
1326:
1323:
1320:
1316:
1292:
1277:
1276:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1206:
1186:
1183:
1180:
1160:
1157:
1154:
1134:
1131:
1128:
1125:
1122:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1064:
1044:
1041:
1038:
1035:
1032:
1029:
1009:
989:
986:
983:
980:
962:Main article:
943:
940:
932:Young tableaux
920:
919:
916:
915:
853:
852:
833:Main article:
830:
827:
824:
823:
822:1 + 1 + 1 + 1
820:
817:
814:
811:
808:
805:
802:
799:
795:
794:
769:
767:
743:
741:
718:
716:
693:
691:
584:
581:
564:
561:
546:
542:
519:
515:
509:
505:
501:
496:
492:
486:
482:
478:
473:
469:
463:
459:
455:
450:
446:
440:
436:
432:
427:
423:
417:
413:
409:
404:
400:
383:
374:
365:
349:
342:
338:
333:
325:
321:
316:
308:
304:
299:
273:
272:
269:
266:
263:
260:
257:
254:
246:
243:
215:Young diagrams
135:
134:
129:
124:
119:
114:
49:Partitions of
39:Young diagrams
15:
13:
10:
9:
6:
4:
3:
2:
5864:
5853:
5850:
5849:
5847:
5828:
5824:
5817:
5812:
5810:
5807:
5805:
5802:
5800:
5796:
5793:
5790:
5787:entry in the
5786:
5783:
5780:
5777:
5764:
5757:
5756:
5750:
5745:
5744:
5739:
5736:
5731:
5729:
5726:
5722:
5718:
5717:
5712:
5708:
5707:
5703:
5699:
5694:
5690:
5685:
5680:
5676:
5672:
5668:
5663:
5659:
5657:0-521-56069-1
5653:
5649:
5648:
5643:
5639:
5635:
5633:9780066210704
5629:
5624:
5623:
5617:
5613:
5609:
5605:
5601:
5597:
5593:
5589:
5587:0-387-98912-9
5583:
5579:
5575:
5570:
5565:
5561:
5557:
5555:0-19-853530-9
5551:
5547:
5543:
5539:
5535:
5533:
5529:
5524:
5520:
5517:
5512:
5507:
5504:
5499:
5495:
5489:
5485:
5481:
5477:
5472:
5467:
5463:
5459:
5455:
5451:
5450:Lehmer, D. H.
5447:
5442:
5438:
5434:
5430:
5426:
5420:
5416:
5412:
5408:
5404:
5401:. Revised by
5400:
5396:
5395:Wright, E. M.
5392:
5387:
5382:
5380:981-02-4900-4
5376:
5372:
5368:
5364:
5361:
5356:
5352:
5348:
5346:0-387-97127-0
5342:
5338:
5334:
5329:
5328:
5322:
5318:
5314:
5312:0-521-60090-1
5308:
5304:
5299:
5295:
5293:0-521-63766-X
5289:
5285:
5281:
5277:
5273:
5271:0-486-61272-4
5267:
5263:
5262:
5257:
5256:Stegun, Irene
5253:
5249:
5248:
5244:
5235:
5231:
5227:
5223:
5219:
5215:
5210:
5205:
5201:
5197:
5193:
5186:
5183:
5178:
5174:
5169:
5164:
5160:
5156:
5152:
5145:
5142:
5137:
5131:
5127:
5120:
5117:
5113:
5107:
5104:
5100:
5095:
5092:
5087:
5086:
5078:
5075:
5071:
5065:
5062:
5058:
5053:
5050:
5046:
5041:
5038:
5033:
5032:Number Theory
5029:
5023:
5020:
5017:, p. 825
5016:
5010:
5007:
5003:
4998:
4995:
4991:
4986:
4983:
4979:
4974:
4971:
4966:
4962:
4958:
4954:
4950:
4946:
4942:
4935:
4932:
4928:
4923:
4920:
4917:, p. 69.
4916:
4911:
4908:
4903:
4899:
4895:
4891:
4887:
4883:
4878:
4873:
4869:
4865:
4864:
4856:
4853:
4849:
4844:
4841:
4834:
4829:
4826:
4823:
4819:
4817:
4814:
4812:
4809:
4807:
4804:
4802:
4799:
4797:
4794:
4792:
4789:
4787:
4784:
4781:
4780:Polite number
4778:
4776:
4773:
4771:
4768:
4766:
4763:
4761:
4758:
4756:
4755:Factorization
4753:
4751:
4748:
4746:
4743:
4741:
4737:
4734:
4733:
4728:
4726:
4724:
4720:
4716:
4712:
4708:
4700:
4698:
4696:
4692:
4687:
4683:
4680:
4676:
4672:
4668:
4667:
4662:
4661:partial order
4656:
4646:
4644:
4626:
4622:
4613:
4597:
4594:
4589:
4585:
4576:
4571:
4569:
4557:
4551:
4545:
4540:
4534:
4529:
4524:
4518:
4513:
4508:
4502:
4497:
4492:
4487:
4483:
4482:
4479:
4478:
4477:
4475:
4474:Durfee square
4471:
4467:
4463:
4459:
4455:
4451:
4447:
4441:
4440:Durfee square
4433:
4431:
4418:
4413:
4401:
4397:
4394:
4391:
4378:
4373:
4369:
4362:
4359:
4356:
4353:
4350:
4344:
4339:
4336:
4331:
4328:
4325:
4321:
4310:
4306:
4302:
4298:
4293:
4277:
4266:
4262:
4258:
4255:
4247:
4242:
4239:
4236:
4232:
4223:
4219:
4215:
4212:
4204:
4199:
4196:
4193:
4189:
4178:
4174:
4170:
4167:
4159:
4156:
4153:
4148:
4145:
4142:
4138:
4131:
4126:
4114:
4110:
4107:
4104:
4091:
4086:
4074:
4070:
4067:
4064:
4045:
4040:into at most
4038:
4034:
4022:into exactly
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3978:
3975:
3969:
3966:
3963:
3960:
3957:
3951:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3898:
3895:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3865:
3862:
3856:
3853:
3850:
3847:
3844:
3838:
3829:
3825:
3813:with at most
3806:
3802:
3798:
3794:
3787:
3779:
3777:
3775:
3771:
3767:
3748:
3744:
3740:
3737:
3732:
3729:
3723:
3715:
3714:
3713:
3711:
3707:
3703:
3699:
3680:
3675:
3672:
3662:
3658:
3654:
3651:
3643:
3640:
3637:
3633:
3625:
3624:
3623:
3622:
3618:
3614:
3610:
3591:
3583:
3579:
3575:
3572:
3568:
3561:
3556:
3553:
3550:
3546:
3540:
3536:
3532:
3527:
3523:
3516:
3508:
3504:
3498:
3495:
3492:
3488:
3480:
3479:
3478:
3476:
3472:
3453:
3447:
3439:
3435:
3429:
3424:
3421:
3418:
3414:
3410:
3404:
3398:
3391:
3390:
3389:
3387:
3383:
3378:
3375:
3369:
3362:
3359:≤ 0 or
3358:
3351:
3346:
3342:
3332:
3322:
3316:
3312:
3308:
3304:
3299:
3295:
3291:
3286:
3282:
3278:
3277:
3276:
3272:
3267:
3263:
3257:
3251:
3246:
3243:into exactly
3241:
3234:
3229:
3225:
3218:
3210:
3208:
3206:
3202:
3198:
3194:
3189:
3185:
3177:
3173:
3169:
3165:
3161:
3157:
3154:− 7) +
3153:
3149:
3145:
3141:
3137:
3133:
3130:− 1) +
3129:
3125:
3120:
3116:
3112:
3108:
3105:
3104:
3103:
3101:
3097:
3078:
3070:
3067:
3064:
3061:
3057:
3053:
3050:
3046:
3034:
3031:
3028:
3024:
3020:
3012:
3008:
3004:
3001:
2988:
2985:
2982:
2978:
2974:
2969:
2965:
2958:
2952:
2942:
2939:
2936:
2932:
2924:
2923:
2922:
2920:
2916:
2912:
2904:
2899:
2894:
2893:
2892:
2890:
2886:
2882:
2878:
2874:
2869:
2867:
2863:
2859:
2855:
2847:
2844:
2841:
2838:
2835:
2832:
2831:
2830:
2828:
2820:
2817:
2815:3 + 3 + 1 + 1
2814:
2812:5 + 1 + 1 + 1
2811:
2808:
2805:
2804:
2803:
2801:
2793:
2786:
2784:
2781:
2780:
2776:
2773:
2770:
2769:
2765:
2760:
2754:
2749:
2744:
2738:
2733:
2728:
2723:
2717:
2712:
2707:
2702:
2697:
2691:
2686:
2681:
2676:
2671:
2664:
2662:
2660:
2655:
2650:
2644:
2639:
2634:
2629:
2624:
2619:
2614:
2608:
2603:
2598:
2593:
2588:
2583:
2578:
2573:
2568:
2562:
2559:
2551:
2545:
2539:
2534:
2529:
2525:
2522:
2519:
2514:
2509:
2504:
2499:
2495:
2494:
2491:
2489:
2485:
2481:
2479:
2475:
2473:
2468:
2458:
2455:
2452:
2451:
2447:
2441:
2435:
2430:
2424:
2419:
2414:
2408:
2403:
2398:
2392:
2387:
2382:
2377:
2373:
2370:
2367:
2361:
2356:
2351:
2345:
2340:
2335:
2330:
2324:
2319:
2314:
2309:
2304:
2299:
2295:
2294:
2291:
2285:
2283:
2277:
2275:
2261:
2241:
2233:
2229:
2225:
2207:
2204:
2198:
2195:
2192:
2186:
2183:
2177:
2174:
2171:
2165:
2162:
2156:
2153:
2150:
2144:
2141:
2135:
2132:
2129:
2123:
2120:
2114:
2108:
2101:
2100:
2099:
2097:
2093:
2090:; by Euler's
2089:
2085:
2080:
2078:
2059:
2056:
2053:
2047:
2027:
2021:
2017:
2013:
2009:
2006:
2003:
2000:
1994:
1991:
1988:
1982:
1978:
1974:
1971:
1967:
1961:
1958:
1952:
1949:
1946:
1939:
1936:
1933:
1930:
1927:
1924:
1920:
1916:
1910:
1902:
1898:
1889:
1874:
1869:
1858:
1852:
1849:
1844:
1841:
1837:
1831:
1828:
1819:
1816:
1809:
1805:
1802:
1794:
1791:
1786:
1783:
1779:
1773:
1766:
1763:
1759:
1754:
1749:
1741:
1733:
1729:
1718:
1715:
1712:
1708:
1699:
1694:
1690:
1685:
1679:
1673:
1665:
1664:
1645:
1639:
1631:
1606:
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1578:
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1571:
1564:
1560:
1556:
1553:
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1540:
1537:
1533:
1528:
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1516:
1509:
1508:
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1497:
1493:
1489:
1470:
1465:
1462:
1452:
1448:
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1441:
1428:
1425:
1422:
1418:
1414:
1409:
1406:
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1391:
1388:
1385:
1381:
1370:
1367:
1364:
1360:
1356:
1351:
1347:
1340:
1334:
1324:
1321:
1318:
1314:
1306:
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1304:
1290:
1282:
1274:
1269:
1264:
1263:
1262:
1248:
1245:
1242:
1239:
1236:
1233:
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1227:
1224:
1204:
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1181:
1178:
1158:
1155:
1152:
1132:
1129:
1126:
1123:
1120:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1062:
1042:
1039:
1033:
1027:
1007:
984:
978:
971:
965:
957:
956:the SVG file,
953:
948:
941:
939:
937:
933:
929:
925:
913:
907:
902:
897:
892:
886:
881:
876:
871:
866:
862:
861:
858:
857:
856:
850:
846:
845:
844:
842:
841:Young diagram
836:
835:Young diagram
829:Young diagram
828:
821:
818:
815:
812:
809:
806:
803:
800:
797:
796:
792:
786:
780:
774:
770:
768:
765:
759:
753:
748:
744:
742:
739:
734:
728:
723:
719:
717:
714:
708:
703:
698:
694:
692:
689:
684:
679:
674:
670:
669:
666:
663:
661:
655:
650:
645:
639:
634:
629:
624:
618:
613:
608:
603:
598:
593:
588:
582:
580:
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574:
570:
562:
560:
544:
540:
517:
513:
507:
503:
499:
494:
490:
484:
480:
476:
471:
467:
461:
457:
453:
448:
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438:
434:
430:
425:
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415:
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407:
402:
398:
386:
382:
373:
364:
347:
340:
336:
331:
323:
319:
314:
306:
302:
297:
287:
278:
270:
268:2 + 1 + 1 + 1
267:
264:
261:
258:
255:
252:
251:
250:
244:
242:
240:
236:
232:
228:
224:
220:
216:
211:
200:
196:
189:
182:
178:
174:
166:
161:
138:
132:1 + 1 + 1 + 1
130:
125:
120:
115:
110:
109:
108:
102:
98:
94:
90:
82:
75:
71:
67:
66:combinatorics
63:
62:number theory
47:
40:
36:
30:
26:
22:
5831:. Retrieved
5770:, retrieved
5754:
5741:
5714:
5697:
5674:
5670:
5646:
5621:
5607:
5573:
5541:
5531:
5527:
5522:
5518:
5515:
5510:
5502:
5497:
5493:
5461:
5457:
5411:Andrew Wiles
5398:
5391:Hardy, G. H.
5370:
5367:Bóna, Miklós
5359:
5326:
5302:
5283:
5259:
5209:math/9907127
5202:(1): 57–81.
5199:
5195:
5185:
5161:(20): 1043.
5158:
5154:
5144:
5125:
5119:
5114:, p. 58
5112:Stanley 1999
5106:
5099:Andrews 1976
5094:
5084:
5077:
5069:
5064:
5052:
5040:
5031:
5022:
5009:
4997:
4985:
4973:
4948:
4944:
4934:
4922:
4915:Andrews 1976
4910:
4867:
4866:, Series A,
4861:
4855:
4848:Andrews 1976
4843:
4739:
4704:
4690:
4685:
4681:
4664:
4658:
4611:
4572:
4565:
4469:
4465:
4461:
4457:
4453:
4449:
4445:
4443:
4308:
4304:
4300:
4296:
4046:
4036:
4032:
3827:
3823:
3804:
3800:
3796:
3792:
3789:
3773:
3769:
3765:
3763:
3709:
3705:
3701:
3695:
3616:
3612:
3608:
3606:
3474:
3470:
3468:
3385:
3381:
3379:
3373:
3367:
3360:
3356:
3349:
3344:
3340:
3330:
3327:
3320:
3314:
3310:
3306:
3302:
3297:
3293:
3289:
3284:
3280:
3270:
3265:
3261:
3255:
3249:
3244:
3239:
3232:
3227:
3223:
3220:
3204:
3200:
3196:
3192:
3187:
3183:
3181:
3175:
3171:
3167:
3163:
3159:
3155:
3151:
3147:
3143:
3139:
3135:
3131:
3127:
3123:
3118:
3114:
3110:
3106:
3099:
3093:
2918:
2914:
2908:
2888:
2884:
2880:
2876:
2872:
2870:
2857:
2853:
2851:
2826:
2824:
2799:
2797:
2564:
2557:
2487:
2483:
2482:
2477:
2476:
2471:
2466:
2464:
2289:
2281:
2222:
2081:
2077:Dedekind sum
1890:
1666:
1627:
1485:
1278:
967:
951:
921:
854:
840:
838:
664:
589:
586:
576:
573:Alfred Young
566:
384:
380:
371:
362:
288:
274:
248:
241:in general.
212:
198:
194:
187:
180:
176:
164:
162:
139:
136:
80:
69:
59:
5823:Brady Haran
5738:"Partition"
5711:"Partition"
5464:: 362–373.
5110:see, e.g.,
3776:+ 3) / 12.
1504:square root
233:and of the
223:mathematics
202:means that
101:composition
5772:2021-02-28
5693:0071.04004
5596:0953.11002
5564:0487.20007
5488:0022.20401
5441:1159.11001
5355:0697.10023
5245:References
3473:fixed and
3323:− 1)
2782:Dist. odd
2771:9 + 7 + 3
5743:MathWorld
5721:EMS Press
5397:(2008) .
5323:(1990) .
5234:119176413
5226:1420-9020
4877:0801.4928
4623:λ
4595:−
4586:λ
4322:∑
4259:−
4248:ℓ
4233:∏
4216:−
4190:∏
4171:−
4160:ℓ
4139:∏
4111:ℓ
4075:ℓ
4071:ℓ
3994:−
3976:−
3926:−
3908:−
3881:−
3673:−
3655:−
3641:∈
3634:∏
3576:−
3547:∏
3496:≥
3489:∑
3415:∑
3363:≤ 0
3068:−
3054:−
3040:∞
3025:∏
2994:∞
2979:∏
2948:∞
2933:∑
2848:4 + 3 + 1
2845:5 + 2 + 1
2800:odd parts
2467:conjugate
2208:⋯
2196:−
2184:−
2175:−
2163:−
2154:−
2133:−
2001:−
1972:π
1928:≤
1921:∑
1845:−
1817:π
1806:
1787:−
1755:⋅
1724:∞
1709:∑
1695:π
1628:In 1937,
1613:∞
1610:→
1565:π
1557:
1529:∼
1463:−
1445:−
1434:∞
1419:∏
1397:∞
1382:∑
1376:∞
1361:∏
1330:∞
1315:∑
1249:…
936:polyomino
816:2 + 1 + 1
348:⋯
280:(2, 2, 1)
265:2 + 2 + 1
262:3 + 1 + 1
158:2 + 1 + 1
154:1 + 1 + 2
150:1 + 2 + 1
127:2 + 1 + 1
70:partition
5846:Category
5827:Archived
5791:database
5789:FindStat
5763:archived
5644:(1999).
5618:(2003).
5606:(1974).
5540:(1979).
5452:(1939).
5369:(2002).
5282:(1976).
5258:(1964).
5177:14308256
5030:(1971).
4902:15392503
4729:See also
4719:Okounkov
4709:via the
4689:for all
3745:⌋
3724:⌊
3199:−
2891:(0)=1):
360:, where
245:Examples
97:summands
5819:(video)
5723:, 2001
5480:0000410
5433:2445243
4965:2317861
4894:2677686
4568:h-index
4044:parts.
3336:(0) = 1
2901:in the
2898:A000009
2075:is the
1661:by the
1502:of the
1271:in the
1268:A000041
237:and in
227:physics
190:(4) = 5
74:integer
5691:
5654:
5630:
5594:
5584:
5562:
5552:
5486:
5478:
5439:
5431:
5421:
5377:
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5343:
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5290:
5268:
5232:
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4963:
4900:
4892:
4464:, the
3619:, has
3182:where
2488:folded
1888:where
1197:, and
532:, and
284:(2, 1)
204:λ
195:λ
5833:5 May
5797:from
5766:(PDF)
5759:(PDF)
5230:S2CID
5204:arXiv
5173:S2CID
4961:JSTOR
4898:S2CID
4872:arXiv
4835:Notes
3388:) by
3352:) = 0
2842:5 + 3
2839:6 + 2
2836:7 + 1
2809:5 + 3
2806:7 + 1
2478:Claim
1261:are:
810:2 + 2
804:3 + 1
277:tuple
259:3 + 2
256:4 + 1
185:. So
146:3 + 1
142:1 + 3
122:2 + 2
117:3 + 1
87:as a
5835:2016
5799:CPAN
5652:ISBN
5628:ISBN
5582:ISBN
5550:ISBN
5419:ISBN
5405:and
5375:ISBN
5341:ISBN
5307:ISBN
5288:ISBN
5266:ISBN
5222:ISSN
5159:2000
5130:ISBN
4740:rank
4446:rank
4444:The
3371:and
3365:and
3338:and
3309:) +
3292:) =
3113:) =
3094:The
2913:for
2909:The
2903:OEIS
2082:The
2040:and
1934:<
1803:sinh
1303:is
1279:The
1273:OEIS
968:The
926:and
225:and
165:part
152:and
68:, a
64:and
5689:Zbl
5679:doi
5592:Zbl
5560:Zbl
5530:),
5505:).)
5484:Zbl
5466:doi
5437:Zbl
5351:Zbl
5214:doi
5163:doi
4953:doi
4882:doi
4868:117
4677:of
4294:of
3354:if
3195:= 3
3102::
1599:as
1554:exp
1486:No
1283:of
388:= 0
217:or
91:of
89:sum
60:In
5848::
5825:.
5821:.
5761:,
5740:.
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5687:.
5673:.
5669:.
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5460:.
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5435:.
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813:=
807:=
801:=
559:.
210:.
197:⊢
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5837:.
5746:.
5695:.
5681::
5675:6
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5636:.
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5468::
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5216::
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5165::
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4967:.
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4297:p
4278:.
4272:)
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3869:(
3866:p
3863:=
3860:)
3857:n
3854:;
3851:M
3848:,
3845:N
3842:(
3839:p
3828:N
3824:M
3819:N
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3811:n
3807:)
3805:n
3801:M
3797:N
3795:(
3793:p
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3710:n
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3681:.
3676:1
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3644:T
3638:t
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3609:T
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3533:=
3528:n
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3520:)
3517:n
3514:(
3509:k
3505:p
3499:0
3493:n
3475:n
3471:k
3454:.
3451:)
3448:n
3445:(
3440:k
3436:p
3430:n
3425:0
3422:=
3419:k
3411:=
3408:)
3405:n
3402:(
3399:p
3386:n
3384:(
3382:p
3374:k
3368:n
3361:k
3357:n
3350:n
3348:(
3345:k
3341:p
3334:0
3331:p
3321:n
3319:(
3315:k
3311:p
3307:k
3303:n
3301:(
3298:k
3294:p
3290:n
3288:(
3285:k
3281:p
3273:)
3271:n
3269:(
3266:k
3262:p
3256:k
3250:n
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3233:n
3231:(
3228:k
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3205:m
3201:m
3197:m
3193:k
3188:k
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3176:k
3174:(
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3168:k
3166:(
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3111:k
3109:(
3107:q
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3079:.
3071:1
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3021:=
3018:)
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3009:x
3005:+
3002:1
2999:(
2989:1
2986:=
2983:k
2975:=
2970:n
2966:x
2962:)
2959:n
2956:(
2953:q
2943:0
2940:=
2937:n
2919:n
2917:(
2915:q
2889:q
2885:n
2883:(
2881:q
2877:n
2875:(
2873:q
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2856:(
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2262:n
2242:n
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2202:)
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2193:n
2190:(
2187:p
2181:)
2178:5
2172:n
2169:(
2166:p
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2151:n
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2145:p
2142:+
2139:)
2136:1
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2127:(
2124:p
2121:=
2118:)
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2112:(
2109:p
2063:)
2060:k
2057:,
2054:m
2051:(
2048:s
2028:.
2022:)
2018:k
2014:/
2010:m
2007:n
2004:2
1998:)
1995:k
1992:,
1989:m
1986:(
1983:s
1979:(
1975:i
1968:e
1962:1
1959:=
1956:)
1953:k
1950:,
1947:m
1944:(
1940:,
1937:k
1931:m
1925:0
1917:=
1914:)
1911:n
1908:(
1903:k
1899:A
1875:)
1870:]
1859:)
1850:1
1842:n
1838:(
1832:3
1829:2
1820:k
1810:[
1792:1
1784:n
1780:1
1774:(
1767:n
1764:d
1760:d
1750:k
1745:)
1742:n
1739:(
1734:k
1730:A
1719:1
1716:=
1713:k
1700:2
1691:1
1686:=
1683:)
1680:n
1677:(
1674:p
1649:)
1646:n
1643:(
1640:p
1607:n
1586:)
1579:3
1575:n
1572:2
1561:(
1546:3
1541:n
1538:4
1534:1
1526:)
1523:n
1520:(
1517:p
1471:.
1466:1
1459:)
1453:j
1449:q
1442:1
1439:(
1429:1
1426:=
1423:j
1415:=
1410:i
1407:j
1403:q
1392:0
1389:=
1386:i
1371:1
1368:=
1365:j
1357:=
1352:n
1348:q
1344:)
1341:n
1338:(
1335:p
1325:0
1322:=
1319:n
1291:p
1246:,
1243:2
1240:,
1237:1
1234:,
1231:0
1228:=
1225:n
1205:4
1185:2
1182:+
1179:2
1159:3
1156:+
1153:1
1133:2
1130:+
1127:1
1124:+
1121:1
1101:1
1098:+
1095:1
1092:+
1089:1
1086:+
1083:1
1063:4
1043:5
1040:=
1037:)
1034:4
1031:(
1028:p
1008:n
988:)
985:n
982:(
979:p
952:p
798:4
545:5
541:1
518:1
514:2
508:3
504:1
500:,
495:2
491:2
485:1
481:1
477:,
472:1
468:3
462:2
458:1
454:,
449:1
445:3
439:1
435:2
431:,
426:1
422:4
416:1
412:1
408:,
403:1
399:5
385:i
381:m
375:2
372:m
366:1
363:m
341:3
337:m
332:3
324:2
320:m
315:2
307:1
303:m
298:1
253:5
208:n
199:n
188:p
183:)
181:n
179:(
177:p
169:n
112:4
105:4
85:n
77:n
55:k
51:n
31:.
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