57:
958:
860:
46:
4724:. 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
2764:
2748:
2743:
2670:
2665:
2660:
4567:
4561:
4555:
4550:
4512:
2770:
2754:
2733:
2707:
2701:
2696:
2691:
2686:
2681:
2618:
2613:
2608:
2603:
2598:
2593:
2588:
2583:
2578:
2561:
2555:
2549:
2544:
2529:
2524:
2514:
2509:
2457:
2451:
2445:
2440:
2429:
2424:
2418:
2408:
2402:
2397:
2392:
2377:
2366:
2361:
2355:
2350:
2340:
2334:
2329:
2324:
2319:
2314:
923:
917:
912:
907:
902:
896:
891:
886:
881:
876:
802:
796:
790:
784:
775:
769:
763:
758:
749:
744:
738:
733:
724:
718:
713:
708:
699:
694:
689:
684:
671:
665:
660:
655:
649:
644:
639:
634:
628:
623:
618:
613:
608:
603:
4544:
4539:
4534:
4528:
4523:
4518:
4507:
4502:
4497:
2775:
2759:
2738:
2727:
2722:
2717:
2712:
2654:
2649:
2644:
2639:
2634:
2629:
2624:
2539:
2519:
2434:
2413:
2387:
2371:
2345:
2309:
1897:
4299:
2480:
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
4471:. 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
1492:
3100:
1680:
4061:
1608:
2049:
3613:
965:(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
541:
854:(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
4440:
1320:
3770:
286:
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
2229:
2938:
369:
4704:, 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
3953:
3702:
3475:
52:
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.
945:, 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
2476:
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
4027:
1904:
1892:{\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)}
4619:
1270:
4650:
1634:
2569:
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:
1523:
4294:{\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})}}.}
1122:
2084:
1064:
1154:
568:
4577:
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
1670:
1009:
2293:
In both combinatorics and number theory, families of partitions subject to various restrictions are often studied. This section surveys a few such restrictions.
1206:
1180:
2882:
For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction. An important example is
2283:
2263:
1312:
1226:
1084:
1029:
2863:
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
1276:
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
3494:
676:
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:
4327:
2913:
1283:
941:: filling the boxes of Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a family of objects called
933:
While this seemingly trivial variation does not appear worthy of separate mention, Young diagrams turn out to be extremely useful in the study of
404:
4721:
5433:
5144:
5773:
4951:
3845:
1487:{\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}.}
980:
974:
183:
5666:
5642:
5596:
5564:
5389:
5355:
5321:
5302:
5280:
3729:
5814:
4873:
4838:
2115:
5038:
4780:
3095:{\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}}}.}
5731:
5343:
3796:
3227:
4725:
598:
The partition 6 + 4 + 3 + 1 of the number 14 can be represented by the following diagram:
583:
303:
4811:
5862:
5726:
938:
111:
4806:
3639:
4685:
3405:
2242:
2301:
If we flip the diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14:
3106:
2102:
2836:
Alternatively, we could count partitions in which no number occurs more than once. Such a partition is called a
4796:
35:
4832:
5161:
5556:
5425:
4716:
There is a deep theory of random partitions chosen according to the uniform probability distribution on the
2491:: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.
56:
4821:
2094:
1498:
579:
5677:
5396:(an elementary introduction to the topic of integer partitions, including a discussion of Ferrers graphs)
3958:
5271:
4770:
4681:
4653:
3708:
2876:
4755:
1510:
249:
241:
4591:
1231:
966:
578:
There are two common diagrammatic methods to represent partitions: as
Ferrers diagrams, named after
5417:
4746:
4585:
4302:
3715:
specifies the available coins). As two particular cases, one has that the number of partitions of
3631:
2921:
2234:
1506:
1502:
1291:
934:
859:
4871:
Josuat-Vergès, Matthieu (2010), "Bijections between pattern-avoiding fillings of Young diagrams",
4628:
1613:
1603:{\displaystyle p(n)\sim {\frac {1}{4n{\sqrt {3}}}}\exp \left({\pi {\sqrt {\frac {2n}{3}}}}\right)}
5837:
5652:
5527:(Has text, nearly complete bibliography, but they (and Abramowitz) missed the Selberg formula for
5240:
5214:
5183:
4971:
4908:
4882:
4775:
4705:
2238:
38:. For the problem of partitioning a multiset of integers so that each part has the same sum, see
31:
4676:
4665:
5721:
45:
5745:
5662:
5638:
5592:
5560:
5429:
5413:
5385:
5377:
5351:
5317:
5298:
5290:
5276:
5262:
5232:
5140:
2106:
1673:
957:
103:
39:
5789:
1089:
5699:
5689:
5626:
5602:
5570:
5548:
5494:
5476:
5447:
5361:
5224:
5173:
4963:
4892:
4674:
on partitions given by inclusion of Young diagrams. This partially ordered set is known as
2054:
1034:
5490:
5443:
4904:
1127:
546:
5764:
5709:(Provides the Selberg formula. The older form is the finite Fourier expansion of Selberg.)
5703:
5614:
5606:
5588:
5574:
5498:
5486:
5451:
5439:
5365:
5347:
4900:
4785:
4760:
4733:
4729:
4717:
4689:
4652:. This statistic (which is unrelated to the one described above) appears in the study of
2044:{\displaystyle A_{k}(n)=\sum _{0\leq m<k,\;(m,k)=1}e^{\pi i\left(s(m,k)-2nm/k\right)}.}
1646:
1640:
985:
245:
229:
1185:
1159:
5139:. Institute of Mathematical Statistics Textbooks. New York: Cambridge University Press.
5336:
5331:
5083:, volume 1, second edition. Cambridge University Press, 2012. Chapter 1, section 1.7.
4816:
4801:
3608:{\displaystyle \sum _{n\geq 0}p_{k}(n)x^{n}=x^{k}\prod _{i=1}^{k}{\frac {1}{1-x^{i}}}.}
2872:
2268:
2248:
2098:
1297:
1211:
1069:
1014:
942:
17:
5481:
5464:
5094:
5856:
5631:
5244:
4790:
4765:
4671:
4484:
4450:
845:
225:
76:
72:
49:
5187:
4912:
5460:
5421:
5405:
5266:
4826:
2087:
30:
This article is about partitioning an integer. For grouping elements of a set, see
5272:
Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical Tables
5833:
5805:
5401:
3779:
in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of
1514:
401:
may be omitted.) For example, in this notation, the partitions of 5 are written
233:
5748:
4896:
5826:
5178:
2763:
2747:
2742:
2669:
2664:
2659:
5382:
A Walk
Through Combinatorics: An Introduction to Enumeration and Graph Theory
5236:
4566:
4560:
4554:
4549:
4511:
4041:, and subtracting 1 from each part of such a partition yields a partition of
3832:. Equivalently, these are the partitions whose Young diagram fits inside an
3719:
in which all parts are 1 or 2 (or, equivalently, the number of partitions of
2769:
2753:
2732:
2706:
2700:
2695:
2690:
2685:
2680:
2617:
2612:
2607:
2602:
2597:
2592:
2587:
2582:
2577:
2560:
2554:
2548:
2543:
2528:
2523:
2513:
2508:
2456:
2450:
2444:
2439:
2428:
2423:
2417:
2407:
2401:
2396:
2391:
2376:
2365:
2360:
2354:
2349:
2339:
2333:
2328:
2323:
2318:
2313:
922:
916:
911:
906:
901:
895:
890:
885:
880:
875:
801:
795:
789:
783:
774:
768:
762:
757:
748:
743:
737:
732:
723:
717:
712:
707:
698:
693:
688:
683:
670:
664:
659:
654:
648:
643:
638:
633:
627:
622:
617:
612:
607:
602:
5795:
5753:
4543:
4538:
4533:
4527:
4522:
4517:
4506:
4501:
4496:
2774:
2758:
2737:
2726:
2721:
2716:
2711:
2653:
2648:
2643:
2638:
2633:
2628:
2623:
2538:
2518:
2433:
2412:
2386:
2370:
2344:
2308:
946:
536:{\displaystyle 5^{1},1^{1}4^{1},2^{1}3^{1},1^{2}3^{1},1^{1}2^{2},1^{3}2^{1}}
99:
5694:
5465:"On the remainder and convergence of the series for the partition function"
300:
This multiplicity notation for a partition can be written alternatively as
5503:
Provides the main formula (no derivatives), remainder, and older form for
5738:
4831:
A Goldbach partition is the partition of an even number into primes (see
3801:
One may also simultaneously limit the number and size of the parts. Let
2840:. If we count the partitions of 8 with distinct parts, we also obtain 6:
110:
are considered the same partition. (If order matters, the sum becomes a
5228:
4975:
4578:
2809:
Among the 22 partitions of the number 8, there are 6 that contain only
237:
107:
84:
5792:
with reference tables to the On-Line
Encyclopedia of Integer Sequences
5371:(See chapter 5 for a modern pedagogical intro to Rademacher's formula)
5275:. United States Department of Commerce, National Bureau of Standards.
5656:
5219:
5202:
297:
where the superscript indicates the number of repetitions of a part.
4967:
850:
An alternative visual representation of an integer partition is its
5819:
4887:
4435:{\displaystyle \sum _{n=0}^{MN}p(N,M;n)q^{n}={M+N \choose M}_{q}.}
2237:
discovered that the partition function has nontrivial patterns in
956:
287:
55:
44:
5809:
3622:
is a set of positive integers then the number of partitions of
148:
The only partition of zero is the empty sum, having no parts.
5621:. Vol. v II. MIT Press. pp. 100–07, 108–22, 460–75.
5137:
The surprising mathematics of longest increasing subsequences
3480:
One possible generating function for such partitions, taking
2890:) (partitions into distinct parts). The first few values of
5678:"A sum connected with the series for the partition function"
3791:
Partitions in a rectangle and
Gaussian binomial coefficients
3765:{\displaystyle \left\lfloor {\frac {n}{2}}+1\right\rfloor ,}
5799:
2908:
1278:
2497:: The crucial observation is that every odd part can be "
27:
Decomposition of an integer as a sum of positive integers
5820:
Generating All
Partitions: A Comparison Of Two Encodings
5045:. Philadelphia: W. B. Saunders Company. pp. 149–50.
4680:. The lattice was originally defined in the context of
2481:
themselves as conjugate. Such partitions are said to be
2224:{\displaystyle p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\cdots }
586:. Both have several possible conventions; here, we use
5338:
Modular functions and
Dirichlet series in number theory
2245:. For instance, whenever the decimal representation of
5520:
Gupta, Hansraj; Gwyther, C.E.; Miller, J.C.P. (1962).
2265:
ends in the digit 4 or 9, the number of partitions of
1509:
by which it can be calculated exactly. It grows as an
4631:
4594:
4330:
4064:
3961:
3848:
3783:
into at most three parts) is the nearest integer to (
3732:
3642:
3497:
3408:
2941:
2271:
2251:
2118:
2057:
1907:
1683:
1649:
1616:
1526:
1501:
for the partition function is known, but it has both
1323:
1300:
1234:
1214:
1188:
1162:
1130:
1092:
1072:
1037:
1017:
988:
549:
407:
306:
4483:
square of entries in the upper-left is known as the
4301:
The
Gaussian binomial coefficient is related to the
866:
while the
Ferrers diagram for the same partition is
4584:A different statistic is also sometimes called the
5630:
5335:
4952:"Partition identities - from Euler to the present"
4644:
4613:
4434:
4293:
4021:
3947:
3764:
3696:
3607:
3469:
3094:
2501:" in the middle to form a self-conjugate diagram:
2277:
2257:
2223:
2078:
2043:
1891:
1664:
1628:
1602:
1486:
1306:
1264:
1220:
1200:
1174:
1148:
1116:
1078:
1058:
1023:
1003:
590:, with diagrams aligned in the upper-left corner.
562:
535:
364:{\displaystyle 1^{m_{1}}2^{m_{2}}3^{m_{3}}\cdots }
363:
106:. Two sums that differ only in the order of their
5815:Fast Algorithms For Generating Integer Partitions
5661:. Vol. 1 and 2. Cambridge University Press.
4793:, defined by partitions into consecutive integers
4417:
4396:
4130:
4109:
4090:
4069:
4058:The Gaussian binomial coefficient is defined as:
174:An individual summand in a partition is called a
5587:. Graduate Texts in Mathematics. Vol. 195.
5067:
5055:
5025:
3948:{\displaystyle p(N,M;n)=p(N,M-1;n)+p(N-1,M;n-M)}
2906:1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... (sequence
1643:found a way to represent the partition function
1011:counts the partitions of a non-negative integer
5346:. Vol. 41 (2nd ed.). New York etc.:
4732:related these results to the combinatorics of
3697:{\displaystyle \prod _{t\in T}(1-x^{t})^{-1}.}
3258:parts is equal to the number of partitions of
224:Partitions can be graphically visualized with
5096:Some Famous Problems of the Theory of Numbers
3470:{\displaystyle p(n)=\sum _{k=0}^{n}p_{k}(n).}
8:
5312:Andrews, George E.; Eriksson, Kimmo (2004).
389:is the number of 2's, etc. (Components with
5012:
5000:
4988:
4937:
3842:rectangle. There is a recurrence relation
5166:International Mathematics Research Notices
4463:such that the partition contains at least
1953:
574:Diagrammatic representations of partitions
118:can be partitioned in five distinct ways:
34:. For the partition calculus of sets, see
5693:
5480:
5218:
5177:
5162:"Random matrices and random permutations"
4886:
4636:
4630:
4599:
4593:
4423:
4416:
4395:
4393:
4383:
4346:
4335:
4329:
4276:
4257:
4246:
4233:
4214:
4203:
4188:
4163:
4152:
4145:
4136:
4129:
4108:
4106:
4096:
4089:
4068:
4066:
4063:
3960:
3847:
3738:
3731:
3682:
3672:
3647:
3641:
3593:
3577:
3571:
3560:
3550:
3537:
3518:
3502:
3496:
3449:
3439:
3428:
3407:
3071:
3055:
3049:
3038:
3022:
3003:
2992:
2979:
2957:
2946:
2940:
2270:
2250:
2117:
2056:
2023:
1981:
1934:
1912:
1906:
1878:
1877:
1876:
1858:
1837:
1835:
1825:
1824:
1800:
1788:
1787:
1768:
1758:
1743:
1733:
1722:
1708:
1699:
1682:
1648:
1615:
1578:
1574:
1554:
1542:
1525:
1472:
1462:
1443:
1432:
1416:
1406:
1395:
1385:
1374:
1361:
1339:
1328:
1322:
1299:
1233:
1213:
1187:
1161:
1129:
1091:
1071:
1036:
1016:
987:
554:
548:
527:
517:
504:
494:
481:
471:
458:
448:
435:
425:
412:
406:
350:
345:
333:
328:
316:
311:
305:
5553:Symmetric functions and Hall polynomials
5410:An Introduction to the Theory of Numbers
4588:(or Dyson rank), namely, the difference
232:. They occur in a number of branches of
5122:
5109:
4925:
4858:
4851:
3222:Restricted part size or number of parts
2297:Conjugate and self-conjugate partitions
2105:this function is an alternating sum of
969:hover over the image to move the ruler.
5203:"Infinite wedge and random partitions"
5637:. New York: Perennial-HarperCollins.
4459:of a partition is the largest number
2875:in 1748 and later was generalized as
582:, and as Young diagrams, named after
7:
5739:Partition and composition calculator
5524:. Vol. 4, Tables of partitions.
159:, and the two distinct compositions
5619:Collected Papers of Hans Rademacher
5585:Elementary Methods in Number Theory
4684:, where it is used to describe the
4022:{\displaystyle p(N,M;n)-p(N,M-1;n)}
3820:denote the number of partitions of
3264:in which the largest part has size
1505:that accurately approximate it and
5555:. Oxford Mathematical Monographs.
4400:
4113:
4073:
3050:
3004:
2958:
2097:of its generating function is the
1734:
1623:
1444:
1407:
1386:
1340:
1228:. The values of this function for
975:Partition function (number theory)
25:
5482:10.1090/S0002-9947-1939-0000410-9
4722:Robinson–Schensted correspondence
3232:By taking conjugates, the number
293:or in the even more compact form
4565:
4559:
4553:
4548:
4542:
4537:
4532:
4526:
4521:
4516:
4510:
4505:
4500:
4495:
3775:and the number of partitions of
2773:
2768:
2762:
2757:
2752:
2746:
2741:
2736:
2731:
2725:
2720:
2715:
2710:
2705:
2699:
2694:
2689:
2684:
2679:
2668:
2663:
2658:
2652:
2647:
2642:
2637:
2632:
2627:
2622:
2616:
2611:
2606:
2601:
2596:
2591:
2586:
2581:
2576:
2559:
2553:
2547:
2542:
2537:
2527:
2522:
2517:
2512:
2507:
2455:
2449:
2443:
2438:
2432:
2427:
2422:
2416:
2411:
2406:
2400:
2395:
2390:
2385:
2375:
2369:
2364:
2359:
2353:
2348:
2343:
2338:
2332:
2327:
2322:
2317:
2312:
2307:
921:
915:
910:
905:
900:
894:
889:
884:
879:
874:
858:
800:
794:
788:
782:
773:
767:
761:
756:
747:
742:
736:
731:
722:
716:
711:
706:
697:
692:
687:
682:
669:
663:
658:
653:
647:
642:
637:
632:
626:
621:
616:
611:
606:
601:
167:represent the same partition as
151:The order-dependent composition
5840:from the original on 2021-12-11
5825:Grime, James (April 28, 2016).
5779:from the original on 2021-02-24
5384:. World Scientific Publishing.
5070:, p. 826, 24.2.2 eq. II(A)
4874:Journal of Combinatorial Theory
3626:, all of whose parts belong to
178:. The number of partitions of
5806:Integer::Partition Perl module
5766:Lectures on Integer Partitions
5682:Pacific Journal of Mathematics
5316:. Cambridge University Press.
5297:. Cambridge University Press.
5058:, p. 825, 24.2.2 eq. I(B)
4781:Stars and bars (combinatorics)
4614:{\displaystyle \lambda _{k}-k}
4376:
4358:
4282:
4263:
4239:
4220:
4194:
4175:
4016:
3992:
3983:
3965:
3942:
3912:
3903:
3879:
3870:
3852:
3679:
3659:
3530:
3524:
3461:
3455:
3418:
3412:
3028:
3009:
2972:
2966:
2212:
2200:
2191:
2179:
2170:
2158:
2149:
2137:
2128:
2122:
2073:
2061:
2008:
1996:
1966:
1954:
1924:
1918:
1755:
1749:
1693:
1687:
1659:
1653:
1620:
1536:
1530:
1517:of its argument., as follows:
1469:
1449:
1354:
1348:
1265:{\displaystyle n=0,1,2,\dots }
1047:
1041:
998:
992:
260:The seven partitions of 5 are
1:
5522:Royal Society of Math. Tables
5344:Graduate Texts in Mathematics
4956:American Mathematical Monthly
3797:Gaussian binomial coefficient
3228:Triangle of partition numbers
2871:). This result was proved by
2838:partition with distinct parts
2832:1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
2805:Odd parts and distinct parts
961:Using Euler's method to find
5068:Abramowitz & Stegun 1964
5056:Abramowitz & Stegun 1964
5026:Abramowitz & Stegun 1964
4839:Kostant's partition function
4645:{\displaystyle \lambda _{k}}
3828:parts, each of size at most
1629:{\displaystyle n\to \infty }
5727:Encyclopedia of Mathematics
5201:Okounkov, A. (2001-04-01).
4736:and representation theory.
4686:irreducible representations
3955:obtained by observing that
939:group representation theory
250:group representation theory
5879:
5827:"Partitions - Numberphile"
5424:. (6th ed.). Oxford:
4897:10.1016/j.jcta.2010.03.006
4663:
4448:
3794:
3707:This can be used to solve
3391:One recovers the function
3225:
2678:
2574:
972:
843:
29:
5658:Enumerative Combinatorics
5179:10.1155/S1073792800000532
5160:Okounkov, Andrei (2000).
5081:Enumerative Combinatorics
4029:counts the partitions of
3286:satisfies the recurrence
3107:pentagonal number theorem
2103:pentagonal number theorem
240:, including the study of
155:is the same partition as
5790:Counting with partitions
5676:Whiteman, A. L. (1956).
5583:Nathanson, M.B. (2000).
5295:The Theory of Partitions
4950:Alder, Henry L. (1969).
4807:Ewens's sampling formula
4797:Multiplicative partition
4749:, a different notion of
4726:Tracy–Widom distribution
4625:parts with largest part
2285:will be divisible by 5.
2109:powers of its argument.
1086:has the five partitions
36:Infinitary combinatorics
5633:The Music of the Primes
5557:Oxford University Press
5426:Oxford University Press
5013:Hardy & Wright 2008
5001:Hardy & Wright 2008
4989:Hardy & Wright 2008
4938:Hardy & Wright 2008
4827:Smallest-parts function
4659:
4467:parts of size at least
3189:− 22) − ...
3109:gives a recurrence for
2243:Ramanujan's congruences
1117:{\displaystyle 1+1+1+1}
18:Partition of an integer
5695:10.2140/pjm.1956.6.159
5543:which is in Whiteman.)
5469:Trans. Amer. Math. Soc
4812:Faà di Bruno's formula
4646:
4615:
4445:Rank and Durfee square
4436:
4354:
4295:
4262:
4219:
4174:
4037:parts of size at most
4023:
3949:
3766:
3723:into 1 or 2 parts) is
3709:change-making problems
3698:
3609:
3576:
3471:
3444:
3096:
3054:
3008:
2962:
2470:4 + 3 + 3 + 2 + 1 + 1
2279:
2259:
2225:
2095:multiplicative inverse
2080:
2079:{\displaystyle s(m,k)}
2045:
1893:
1738:
1666:
1630:
1604:
1499:closed-form expression
1488:
1448:
1411:
1390:
1344:
1308:
1266:
1222:
1202:
1176:
1150:
1118:
1080:
1060:
1059:{\displaystyle p(4)=5}
1025:
1005:
970:
580:Norman Macleod Ferrers
564:
537:
380:is the number of 1's,
365:
94:, is a way of writing
68:
53:
4833:Goldbach's conjecture
4771:Integer factorization
4682:representation theory
4654:Ramanujan congruences
4647:
4616:
4437:
4331:
4296:
4242:
4199:
4148:
4024:
3950:
3767:
3699:
3610:
3556:
3472:
3424:
3097:
3034:
2988:
2942:
2898:) are (starting with
2829:3 + 1 + 1 + 1 + 1 + 1
2289:Restricted partitions
2280:
2260:
2226:
2081:
2046:
1894:
1718:
1667:
1631:
1605:
1503:asymptotic expansions
1489:
1428:
1391:
1370:
1324:
1309:
1267:
1223:
1203:
1177:
1151:
1149:{\displaystyle 1+1+2}
1119:
1081:
1061:
1026:
1006:
960:
565:
563:{\displaystyle 1^{5}}
538:
366:
242:symmetric polynomials
59:
48:
5093:Hardy, G.H. (1920).
4756:Crank of a partition
4629:
4592:
4328:
4062:
3959:
3846:
3730:
3640:
3495:
3406:
3339:with initial values
3218:and is 0 otherwise.
3181:− 15) −
2939:
2269:
2249:
2116:
2055:
1905:
1681:
1665:{\displaystyle p(n)}
1647:
1614:
1524:
1511:exponential function
1507:recurrence relations
1321:
1298:
1232:
1212:
1186:
1160:
1128:
1090:
1070:
1066:because the integer
1035:
1015:
1004:{\displaystyle p(n)}
986:
547:
405:
304:
5653:Stanley, Richard P.
5207:Selecta Mathematica
5135:Romik, Dan (2015).
4822:Newton's identities
4747:Rank of a partition
4670:There is a natural
4621:for a partition of
4586:rank of a partition
4303:generating function
3632:generating function
3618:More generally, if
3388:are not both zero.
3157:− 5) −
3149:− 2) −
2922:generating function
2235:Srinivasa Ramanujan
1292:generating function
1201:{\displaystyle 2+2}
1175:{\displaystyle 1+3}
935:symmetric functions
5863:Integer partitions
5796:Integer partitions
5746:Weisstein, Eric W.
5627:Sautoy, Marcus Du.
5314:Integer Partitions
5291:Andrews, George E.
5263:Abramowitz, Milton
5229:10.1007/PL00001398
5099:. Clarendon Press.
5039:Andrews, George E.
4776:Partition of a set
4706:differential poset
4642:
4611:
4432:
4291:
4019:
3945:
3762:
3694:
3658:
3605:
3513:
3467:
3092:
2877:Glaisher's theorem
2788:5 + 5 + 4 + 3 + 2
2275:
2255:
2239:modular arithmetic
2221:
2076:
2041:
1976:
1889:
1662:
1626:
1600:
1484:
1304:
1262:
1218:
1198:
1172:
1146:
1114:
1076:
1056:
1021:
1001:
981:partition function
971:
953:Partition function
560:
533:
361:
217:is a partition of
184:partition function
83:of a non-negative
69:
64:with largest part
54:
32:Partition of a set
5762:Wilf, Herbert S.
5579:(See section I.1)
5549:Macdonald, Ian G.
5435:978-0-19-921986-5
5414:D. R. Heath-Brown
5146:978-1-107-42882-9
5112:, pp. 33–34.
5079:Richard Stanley,
5024:Notation follows
4712:Random partitions
4573:
4572:
4415:
4286:
4128:
4088:
3746:
3643:
3600:
3498:
3248:of partitions of
3214:for some integer
3202:is (−1) if
3087:
2802:
2801:
2567:
2566:
2474:
2473:
2278:{\displaystyle n}
2258:{\displaystyle n}
2107:pentagonal number
1930:
1873:
1866:
1845:
1833:
1811:
1810:
1808:
1781:
1763:
1716:
1713:
1674:convergent series
1593:
1592:
1562:
1559:
1307:{\displaystyle p}
1221:{\displaystyle 4}
1079:{\displaystyle 4}
1024:{\displaystyle n}
929:
928:
837:
836:
282:1 + 1 + 1 + 1 + 1
114:.) For example,
104:positive integers
92:integer partition
90:, also called an
40:Partition problem
16:(Redirected from
5870:
5849:
5847:
5845:
5831:
5786:
5785:
5784:
5778:
5771:
5759:
5758:
5735:
5707:
5697:
5672:
5648:
5636:
5622:
5615:Rademacher, Hans
5610:
5578:
5525:
5502:
5484:
5456:
5455:
5395:
5369:
5341:
5327:
5308:
5286:
5249:
5248:
5222:
5198:
5192:
5191:
5181:
5157:
5151:
5150:
5132:
5126:
5119:
5113:
5107:
5101:
5100:
5090:
5084:
5077:
5071:
5065:
5059:
5053:
5047:
5046:
5035:
5029:
5022:
5016:
5010:
5004:
4998:
4992:
4986:
4980:
4979:
4947:
4941:
4935:
4929:
4923:
4917:
4915:
4890:
4881:(8): 1218–1230,
4868:
4862:
4856:
4734:Riemann surfaces
4690:symmetric groups
4651:
4649:
4648:
4643:
4641:
4640:
4620:
4618:
4617:
4612:
4604:
4603:
4569:
4563:
4557:
4552:
4546:
4541:
4536:
4530:
4525:
4520:
4514:
4509:
4504:
4499:
4492:
4491:
4441:
4439:
4438:
4433:
4428:
4427:
4422:
4421:
4420:
4411:
4399:
4388:
4387:
4353:
4345:
4324:by the equality
4323:
4300:
4298:
4297:
4292:
4287:
4285:
4281:
4280:
4261:
4256:
4238:
4237:
4218:
4213:
4197:
4193:
4192:
4173:
4162:
4146:
4141:
4140:
4135:
4134:
4133:
4124:
4112:
4101:
4100:
4095:
4094:
4093:
4084:
4072:
4054:
4050:
4040:
4036:
4032:
4028:
4026:
4025:
4020:
3954:
3952:
3951:
3946:
3841:
3831:
3827:
3823:
3819:
3771:
3769:
3768:
3763:
3758:
3754:
3747:
3739:
3703:
3701:
3700:
3695:
3690:
3689:
3677:
3676:
3657:
3614:
3612:
3611:
3606:
3601:
3599:
3598:
3597:
3578:
3575:
3570:
3555:
3554:
3542:
3541:
3523:
3522:
3512:
3476:
3474:
3473:
3468:
3454:
3453:
3443:
3438:
3387:
3381:
3375:
3364:
3348:
3335:
3285:
3270:. The function
3269:
3263:
3253:
3247:
3101:
3099:
3098:
3093:
3088:
3086:
3085:
3084:
3056:
3053:
3048:
3027:
3026:
3007:
3002:
2984:
2983:
2961:
2956:
2911:
2777:
2772:
2766:
2761:
2756:
2750:
2745:
2740:
2735:
2729:
2724:
2719:
2714:
2709:
2703:
2698:
2693:
2688:
2683:
2672:
2667:
2662:
2656:
2651:
2646:
2641:
2636:
2631:
2626:
2620:
2615:
2610:
2605:
2600:
2595:
2590:
2585:
2580:
2572:
2571:
2563:
2557:
2551:
2546:
2541:
2534: ↔
2531:
2526:
2521:
2516:
2511:
2504:
2503:
2459:
2453:
2447:
2442:
2436:
2431:
2426:
2420:
2415:
2410:
2404:
2399:
2394:
2389:
2379:
2373:
2368:
2363:
2357:
2352:
2347:
2342:
2336:
2331:
2326:
2321:
2316:
2311:
2304:
2303:
2284:
2282:
2281:
2276:
2264:
2262:
2261:
2256:
2230:
2228:
2227:
2222:
2085:
2083:
2082:
2077:
2050:
2048:
2047:
2042:
2037:
2036:
2035:
2031:
2027:
1975:
1917:
1916:
1898:
1896:
1895:
1890:
1888:
1884:
1883:
1879:
1875:
1874:
1872:
1868:
1867:
1859:
1846:
1838:
1836:
1834:
1826:
1812:
1809:
1801:
1793:
1789:
1782:
1780:
1769:
1764:
1759:
1748:
1747:
1737:
1732:
1717:
1715:
1714:
1709:
1700:
1671:
1669:
1668:
1663:
1635:
1633:
1632:
1627:
1609:
1607:
1606:
1601:
1599:
1595:
1594:
1588:
1580:
1579:
1563:
1561:
1560:
1555:
1543:
1493:
1491:
1490:
1485:
1480:
1479:
1467:
1466:
1447:
1442:
1424:
1423:
1410:
1405:
1389:
1384:
1366:
1365:
1343:
1338:
1313:
1311:
1310:
1305:
1281:
1271:
1269:
1268:
1263:
1227:
1225:
1224:
1219:
1207:
1205:
1204:
1199:
1181:
1179:
1178:
1173:
1155:
1153:
1152:
1147:
1123:
1121:
1120:
1115:
1085:
1083:
1082:
1077:
1065:
1063:
1062:
1057:
1031:. For instance,
1030:
1028:
1027:
1022:
1010:
1008:
1007:
1002:
925:
919:
914:
909:
904:
898:
893:
888:
883:
878:
871:
870:
862:
804:
798:
792:
786:
777:
771:
765:
760:
751:
746:
740:
735:
726:
720:
715:
710:
701:
696:
691:
686:
679:
678:
673:
667:
662:
657:
651:
646:
641:
636:
630:
625:
620:
615:
610:
605:
588:English notation
569:
567:
566:
561:
559:
558:
542:
540:
539:
534:
532:
531:
522:
521:
509:
508:
499:
498:
486:
485:
476:
475:
463:
462:
453:
452:
440:
439:
430:
429:
417:
416:
400:
388:
379:
370:
368:
367:
362:
357:
356:
355:
354:
340:
339:
338:
337:
323:
322:
321:
320:
296:
292:
230:Ferrers diagrams
220:
216:
212:
202:
195:
182:is given by the
181:
170:
166:
162:
158:
154:
144:
139:
134:
129:
124:
117:
97:
89:
67:
63:
21:
5878:
5877:
5873:
5872:
5871:
5869:
5868:
5867:
5853:
5852:
5843:
5841:
5829:
5824:
5782:
5780:
5776:
5769:
5763:
5744:
5743:
5720:
5717:
5675:
5669:
5651:
5645:
5625:
5613:
5599:
5589:Springer-Verlag
5582:
5567:
5547:
5536:
5519:
5511:
5459:
5436:
5420:. Foreword by
5418:J. H. Silverman
5400:
5399:
5392:
5376:
5358:
5348:Springer-Verlag
5332:Apostol, Tom M.
5330:
5324:
5311:
5305:
5289:
5283:
5261:
5258:
5253:
5252:
5200:
5199:
5195:
5159:
5158:
5154:
5147:
5134:
5133:
5129:
5120:
5116:
5108:
5104:
5092:
5091:
5087:
5078:
5074:
5066:
5062:
5054:
5050:
5037:
5036:
5032:
5023:
5019:
5011:
5007:
4999:
4995:
4987:
4983:
4968:10.2307/2317861
4949:
4948:
4944:
4936:
4932:
4924:
4920:
4870:
4869:
4865:
4857:
4853:
4848:
4843:
4786:Plane partition
4761:Dominance order
4742:
4718:symmetric group
4714:
4699:
4677:Young's lattice
4668:
4666:Young's lattice
4662:
4660:Young's lattice
4632:
4627:
4626:
4595:
4590:
4589:
4564:
4558:
4547:
4531:
4515:
4453:
4447:
4401:
4394:
4392:
4379:
4326:
4325:
4306:
4272:
4229:
4198:
4184:
4147:
4114:
4107:
4105:
4074:
4067:
4065:
4060:
4059:
4052:
4042:
4038:
4034:
4030:
3957:
3956:
3844:
3843:
3833:
3829:
3825:
3821:
3802:
3799:
3793:
3737:
3733:
3728:
3727:
3711:(where the set
3678:
3668:
3638:
3637:
3589:
3582:
3546:
3533:
3514:
3493:
3492:
3445:
3404:
3403:
3383:
3377:
3366:
3358:
3350:
3346:
3340:
3329:
3311:
3298:
3290:
3279:
3271:
3265:
3259:
3249:
3241:
3233:
3230:
3224:
3201:
3132:
3067:
3060:
3018:
2975:
2937:
2936:
2932:) is given by
2907:
2807:
2798:self-conjugate
2767:
2751:
2730:
2704:
2657:
2621:
2558:
2552:
2495:Proof (outline)
2454:
2448:
2437:
2421:
2405:
2374:
2358:
2337:
2299:
2291:
2267:
2266:
2247:
2246:
2241:, now known as
2114:
2113:
2053:
2052:
1992:
1988:
1977:
1908:
1903:
1902:
1851:
1847:
1823:
1819:
1783:
1773:
1739:
1704:
1679:
1678:
1645:
1644:
1641:Hans Rademacher
1612:
1611:
1581:
1570:
1547:
1522:
1521:
1468:
1458:
1412:
1357:
1319:
1318:
1296:
1295:
1277:
1230:
1229:
1210:
1209:
1184:
1183:
1158:
1157:
1126:
1125:
1088:
1087:
1068:
1067:
1033:
1032:
1013:
1012:
984:
983:
977:
955:
920:
899:
848:
842:
799:
793:
787:
772:
766:
741:
721:
668:
652:
631:
596:
594:Ferrers diagram
576:
550:
545:
544:
523:
513:
500:
490:
477:
467:
454:
444:
431:
421:
408:
403:
402:
398:
390:
387:
381:
378:
372:
346:
341:
329:
324:
312:
307:
302:
301:
294:
290:
258:
246:symmetric group
218:
214:
204:
203:. The notation
197:
186:
179:
168:
164:
160:
156:
152:
142:
137:
132:
127:
122:
115:
95:
87:
65:
61:
43:
28:
23:
22:
15:
12:
11:
5:
5876:
5874:
5866:
5865:
5855:
5854:
5851:
5850:
5822:
5817:
5812:
5803:
5793:
5787:
5760:
5741:
5736:
5716:
5715:External links
5713:
5712:
5711:
5688:(1): 159–176.
5673:
5667:
5649:
5643:
5623:
5611:
5597:
5580:
5565:
5545:
5532:
5517:
5507:
5457:
5434:
5397:
5390:
5374:
5356:
5328:
5322:
5309:
5303:
5287:
5281:
5257:
5254:
5251:
5250:
5193:
5152:
5145:
5127:
5114:
5102:
5085:
5072:
5060:
5048:
5030:
5017:
5015:, p. 365.
5005:
5003:, p. 368.
4993:
4991:, p. 362.
4981:
4962:(7): 733–746.
4942:
4940:, p. 380.
4930:
4918:
4863:
4861:, p. 199.
4850:
4849:
4847:
4844:
4842:
4841:
4836:
4829:
4824:
4819:
4817:Multipartition
4814:
4809:
4804:
4802:Twelvefold way
4799:
4794:
4788:
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4743:
4741:
4738:
4713:
4710:
4695:
4664:Main article:
4661:
4658:
4639:
4635:
4610:
4607:
4602:
4598:
4575:
4574:
4571:
4570:
4449:Main article:
4446:
4443:
4431:
4426:
4419:
4414:
4410:
4407:
4404:
4398:
4391:
4386:
4382:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4352:
4349:
4344:
4341:
4338:
4334:
4290:
4284:
4279:
4275:
4271:
4268:
4265:
4260:
4255:
4252:
4249:
4245:
4241:
4236:
4232:
4228:
4225:
4222:
4217:
4212:
4209:
4206:
4202:
4196:
4191:
4187:
4183:
4180:
4177:
4172:
4169:
4166:
4161:
4158:
4155:
4151:
4144:
4139:
4132:
4127:
4123:
4120:
4117:
4111:
4104:
4099:
4092:
4087:
4083:
4080:
4077:
4071:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3795:Main article:
3792:
3789:
3773:
3772:
3761:
3757:
3753:
3750:
3745:
3742:
3736:
3705:
3704:
3693:
3688:
3685:
3681:
3675:
3671:
3667:
3664:
3661:
3656:
3653:
3650:
3646:
3616:
3615:
3604:
3596:
3592:
3588:
3585:
3581:
3574:
3569:
3566:
3563:
3559:
3553:
3549:
3545:
3540:
3536:
3532:
3529:
3526:
3521:
3517:
3511:
3508:
3505:
3501:
3488:variable, is
3478:
3477:
3466:
3463:
3460:
3457:
3452:
3448:
3442:
3437:
3434:
3431:
3427:
3423:
3420:
3417:
3414:
3411:
3354:
3344:
3337:
3336:
3324:
3307:
3294:
3275:
3237:
3226:Main article:
3223:
3220:
3197:
3191:
3190:
3173:− 12) +
3128:
3103:
3102:
3091:
3083:
3080:
3077:
3074:
3070:
3066:
3063:
3059:
3052:
3047:
3044:
3041:
3037:
3033:
3030:
3025:
3021:
3017:
3014:
3011:
3006:
3001:
2998:
2995:
2991:
2987:
2982:
2978:
2974:
2971:
2968:
2965:
2960:
2955:
2952:
2949:
2945:
2918:
2917:
2873:Leonhard Euler
2861:
2860:
2857:
2854:
2851:
2848:
2845:
2834:
2833:
2830:
2827:
2824:
2821:
2818:
2806:
2803:
2800:
2799:
2796:
2794:
2790:
2789:
2786:
2783:
2779:
2778:
2677:
2674:
2565:
2564:
2535:
2532:
2483:self-conjugate
2472:
2471:
2468:
2465:
2464:6 + 4 + 3 + 1
2461:
2460:
2383:
2380:
2298:
2295:
2290:
2287:
2274:
2254:
2232:
2231:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2099:Euler function
2075:
2072:
2069:
2066:
2063:
2060:
2040:
2034:
2030:
2026:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1991:
1987:
1984:
1980:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1952:
1949:
1946:
1943:
1940:
1937:
1933:
1929:
1926:
1923:
1920:
1915:
1911:
1887:
1882:
1871:
1865:
1862:
1857:
1854:
1850:
1844:
1841:
1832:
1829:
1822:
1818:
1815:
1807:
1804:
1799:
1796:
1792:
1786:
1779:
1776:
1772:
1767:
1762:
1757:
1754:
1751:
1746:
1742:
1736:
1731:
1728:
1725:
1721:
1712:
1707:
1703:
1698:
1695:
1692:
1689:
1686:
1661:
1658:
1655:
1652:
1637:
1636:
1625:
1622:
1619:
1598:
1591:
1587:
1584:
1577:
1573:
1569:
1566:
1558:
1553:
1550:
1546:
1541:
1538:
1535:
1532:
1529:
1495:
1494:
1483:
1478:
1475:
1471:
1465:
1461:
1457:
1454:
1451:
1446:
1441:
1438:
1435:
1431:
1427:
1422:
1419:
1415:
1409:
1404:
1401:
1398:
1394:
1388:
1383:
1380:
1377:
1373:
1369:
1364:
1360:
1356:
1353:
1350:
1347:
1342:
1337:
1334:
1331:
1327:
1303:
1288:
1287:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1217:
1197:
1194:
1191:
1171:
1168:
1165:
1145:
1142:
1139:
1136:
1133:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1075:
1055:
1052:
1049:
1046:
1043:
1040:
1020:
1000:
997:
994:
991:
973:Main article:
954:
951:
943:Young tableaux
931:
930:
927:
926:
864:
863:
844:Main article:
841:
838:
835:
834:
833:1 + 1 + 1 + 1
831:
828:
825:
822:
819:
816:
813:
810:
806:
805:
780:
778:
754:
752:
729:
727:
704:
702:
595:
592:
575:
572:
557:
553:
530:
526:
520:
516:
512:
507:
503:
497:
493:
489:
484:
480:
474:
470:
466:
461:
457:
451:
447:
443:
438:
434:
428:
424:
420:
415:
411:
394:
385:
376:
360:
353:
349:
344:
336:
332:
327:
319:
315:
310:
284:
283:
280:
277:
274:
271:
268:
265:
257:
254:
226:Young diagrams
146:
145:
140:
135:
130:
125:
60:Partitions of
50:Young diagrams
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5875:
5864:
5861:
5860:
5858:
5839:
5835:
5828:
5823:
5821:
5818:
5816:
5813:
5811:
5807:
5804:
5801:
5798:entry in the
5797:
5794:
5791:
5788:
5775:
5768:
5767:
5761:
5756:
5755:
5750:
5747:
5742:
5740:
5737:
5733:
5729:
5728:
5723:
5719:
5718:
5714:
5710:
5705:
5701:
5696:
5691:
5687:
5683:
5679:
5674:
5670:
5668:0-521-56069-1
5664:
5660:
5659:
5654:
5650:
5646:
5644:9780066210704
5640:
5635:
5634:
5628:
5624:
5620:
5616:
5612:
5608:
5604:
5600:
5598:0-387-98912-9
5594:
5590:
5586:
5581:
5576:
5572:
5568:
5566:0-19-853530-9
5562:
5558:
5554:
5550:
5546:
5544:
5540:
5535:
5531:
5528:
5523:
5518:
5515:
5510:
5506:
5500:
5496:
5492:
5488:
5483:
5478:
5474:
5470:
5466:
5462:
5461:Lehmer, D. H.
5458:
5453:
5449:
5445:
5441:
5437:
5431:
5427:
5423:
5419:
5415:
5412:. Revised by
5411:
5407:
5406:Wright, E. M.
5403:
5398:
5393:
5391:981-02-4900-4
5387:
5383:
5379:
5375:
5372:
5367:
5363:
5359:
5357:0-387-97127-0
5353:
5349:
5345:
5340:
5339:
5333:
5329:
5325:
5323:0-521-60090-1
5319:
5315:
5310:
5306:
5304:0-521-63766-X
5300:
5296:
5292:
5288:
5284:
5282:0-486-61272-4
5278:
5274:
5273:
5268:
5267:Stegun, Irene
5264:
5260:
5259:
5255:
5246:
5242:
5238:
5234:
5230:
5226:
5221:
5216:
5212:
5208:
5204:
5197:
5194:
5189:
5185:
5180:
5175:
5171:
5167:
5163:
5156:
5153:
5148:
5142:
5138:
5131:
5128:
5124:
5118:
5115:
5111:
5106:
5103:
5098:
5097:
5089:
5086:
5082:
5076:
5073:
5069:
5064:
5061:
5057:
5052:
5049:
5044:
5043:Number Theory
5040:
5034:
5031:
5028:, p. 825
5027:
5021:
5018:
5014:
5009:
5006:
5002:
4997:
4994:
4990:
4985:
4982:
4977:
4973:
4969:
4965:
4961:
4957:
4953:
4946:
4943:
4939:
4934:
4931:
4928:, p. 69.
4927:
4922:
4919:
4914:
4910:
4906:
4902:
4898:
4894:
4889:
4884:
4880:
4876:
4875:
4867:
4864:
4860:
4855:
4852:
4845:
4840:
4837:
4834:
4830:
4828:
4825:
4823:
4820:
4818:
4815:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4792:
4791:Polite number
4789:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4766:Factorization
4764:
4762:
4759:
4757:
4754:
4752:
4748:
4745:
4744:
4739:
4737:
4735:
4731:
4727:
4723:
4719:
4711:
4709:
4707:
4703:
4698:
4694:
4691:
4687:
4683:
4679:
4678:
4673:
4672:partial order
4667:
4657:
4655:
4637:
4633:
4624:
4608:
4605:
4600:
4596:
4587:
4582:
4580:
4568:
4562:
4556:
4551:
4545:
4540:
4535:
4529:
4524:
4519:
4513:
4508:
4503:
4498:
4494:
4493:
4490:
4489:
4488:
4486:
4485:Durfee square
4482:
4478:
4474:
4470:
4466:
4462:
4458:
4452:
4451:Durfee square
4444:
4442:
4429:
4424:
4412:
4408:
4405:
4402:
4389:
4384:
4380:
4373:
4370:
4367:
4364:
4361:
4355:
4350:
4347:
4342:
4339:
4336:
4332:
4321:
4317:
4313:
4309:
4304:
4288:
4277:
4273:
4269:
4266:
4258:
4253:
4250:
4247:
4243:
4234:
4230:
4226:
4223:
4215:
4210:
4207:
4204:
4200:
4189:
4185:
4181:
4178:
4170:
4167:
4164:
4159:
4156:
4153:
4149:
4142:
4137:
4125:
4121:
4118:
4115:
4102:
4097:
4085:
4081:
4078:
4075:
4056:
4051:into at most
4049:
4045:
4033:into exactly
4013:
4010:
4007:
4004:
4001:
3998:
3995:
3989:
3986:
3980:
3977:
3974:
3971:
3968:
3962:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3909:
3906:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3876:
3873:
3867:
3864:
3861:
3858:
3855:
3849:
3840:
3836:
3824:with at most
3817:
3813:
3809:
3805:
3798:
3790:
3788:
3786:
3782:
3778:
3759:
3755:
3751:
3748:
3743:
3740:
3734:
3726:
3725:
3724:
3722:
3718:
3714:
3710:
3691:
3686:
3683:
3673:
3669:
3665:
3662:
3654:
3651:
3648:
3644:
3636:
3635:
3634:
3633:
3629:
3625:
3621:
3602:
3594:
3590:
3586:
3583:
3579:
3572:
3567:
3564:
3561:
3557:
3551:
3547:
3543:
3538:
3534:
3527:
3519:
3515:
3509:
3506:
3503:
3499:
3491:
3490:
3489:
3487:
3483:
3464:
3458:
3450:
3446:
3440:
3435:
3432:
3429:
3425:
3421:
3415:
3409:
3402:
3401:
3400:
3398:
3394:
3389:
3386:
3380:
3373:
3370:≤ 0 or
3369:
3362:
3357:
3353:
3343:
3333:
3327:
3323:
3319:
3315:
3310:
3306:
3302:
3297:
3293:
3289:
3288:
3287:
3283:
3278:
3274:
3268:
3262:
3257:
3254:into exactly
3252:
3245:
3240:
3236:
3229:
3221:
3219:
3217:
3213:
3209:
3205:
3200:
3196:
3188:
3184:
3180:
3176:
3172:
3168:
3165:− 7) +
3164:
3160:
3156:
3152:
3148:
3144:
3141:− 1) +
3140:
3136:
3131:
3127:
3123:
3119:
3116:
3115:
3114:
3112:
3108:
3089:
3081:
3078:
3075:
3072:
3068:
3064:
3061:
3057:
3045:
3042:
3039:
3035:
3031:
3023:
3019:
3015:
3012:
2999:
2996:
2993:
2989:
2985:
2980:
2976:
2969:
2963:
2953:
2950:
2947:
2943:
2935:
2934:
2933:
2931:
2927:
2923:
2915:
2910:
2905:
2904:
2903:
2901:
2897:
2893:
2889:
2885:
2880:
2878:
2874:
2870:
2866:
2858:
2855:
2852:
2849:
2846:
2843:
2842:
2841:
2839:
2831:
2828:
2826:3 + 3 + 1 + 1
2825:
2823:5 + 1 + 1 + 1
2822:
2819:
2816:
2815:
2814:
2812:
2804:
2797:
2795:
2792:
2791:
2787:
2784:
2781:
2780:
2776:
2771:
2765:
2760:
2755:
2749:
2744:
2739:
2734:
2728:
2723:
2718:
2713:
2708:
2702:
2697:
2692:
2687:
2682:
2675:
2673:
2671:
2666:
2661:
2655:
2650:
2645:
2640:
2635:
2630:
2625:
2619:
2614:
2609:
2604:
2599:
2594:
2589:
2584:
2579:
2573:
2570:
2562:
2556:
2550:
2545:
2540:
2536:
2533:
2530:
2525:
2520:
2515:
2510:
2506:
2505:
2502:
2500:
2496:
2492:
2490:
2486:
2484:
2479:
2469:
2466:
2463:
2462:
2458:
2452:
2446:
2441:
2435:
2430:
2425:
2419:
2414:
2409:
2403:
2398:
2393:
2388:
2384:
2381:
2378:
2372:
2367:
2362:
2356:
2351:
2346:
2341:
2335:
2330:
2325:
2320:
2315:
2310:
2306:
2305:
2302:
2296:
2294:
2288:
2286:
2272:
2252:
2244:
2240:
2236:
2218:
2215:
2209:
2206:
2203:
2197:
2194:
2188:
2185:
2182:
2176:
2173:
2167:
2164:
2161:
2155:
2152:
2146:
2143:
2140:
2134:
2131:
2125:
2119:
2112:
2111:
2110:
2108:
2104:
2101:; by Euler's
2100:
2096:
2091:
2089:
2070:
2067:
2064:
2058:
2038:
2032:
2028:
2024:
2020:
2017:
2014:
2011:
2005:
2002:
1999:
1993:
1989:
1985:
1982:
1978:
1972:
1969:
1963:
1960:
1957:
1950:
1947:
1944:
1941:
1938:
1935:
1931:
1927:
1921:
1913:
1909:
1900:
1885:
1880:
1869:
1863:
1860:
1855:
1852:
1848:
1842:
1839:
1830:
1827:
1820:
1816:
1813:
1805:
1802:
1797:
1794:
1790:
1784:
1777:
1774:
1770:
1765:
1760:
1752:
1744:
1740:
1729:
1726:
1723:
1719:
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1690:
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1675:
1656:
1650:
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1527:
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1512:
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1481:
1476:
1473:
1463:
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1439:
1436:
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1425:
1420:
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1402:
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1381:
1378:
1375:
1371:
1367:
1362:
1358:
1351:
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1335:
1332:
1329:
1325:
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1301:
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967:the SVG file,
964:
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892:
887:
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877:
873:
872:
869:
868:
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853:
852:Young diagram
847:
846:Young diagram
840:Young diagram
839:
832:
829:
826:
823:
820:
817:
814:
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808:
807:
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797:
791:
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614:
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573:
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342:
334:
330:
325:
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313:
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289:
281:
279:2 + 1 + 1 + 1
278:
275:
272:
269:
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262:
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253:
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247:
243:
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235:
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143:1 + 1 + 1 + 1
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119:
113:
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105:
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86:
82:
78:
77:combinatorics
74:
73:number theory
58:
51:
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41:
37:
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5842:. Retrieved
5781:, retrieved
5765:
5752:
5725:
5708:
5685:
5681:
5657:
5632:
5618:
5584:
5552:
5542:
5538:
5533:
5529:
5526:
5521:
5513:
5508:
5504:
5472:
5468:
5422:Andrew Wiles
5409:
5402:Hardy, G. H.
5381:
5378:Bóna, Miklós
5370:
5337:
5313:
5294:
5270:
5220:math/9907127
5213:(1): 57–81.
5210:
5206:
5196:
5172:(20): 1043.
5169:
5165:
5155:
5136:
5130:
5125:, p. 58
5123:Stanley 1999
5117:
5110:Andrews 1976
5105:
5095:
5088:
5080:
5075:
5063:
5051:
5042:
5033:
5020:
5008:
4996:
4984:
4959:
4955:
4945:
4933:
4926:Andrews 1976
4921:
4878:
4877:, Series A,
4872:
4866:
4859:Andrews 1976
4854:
4750:
4715:
4701:
4696:
4692:
4675:
4669:
4622:
4583:
4576:
4480:
4476:
4472:
4468:
4464:
4460:
4456:
4454:
4319:
4315:
4311:
4307:
4057:
4047:
4043:
3838:
3834:
3815:
3811:
3807:
3803:
3800:
3784:
3780:
3776:
3774:
3720:
3716:
3712:
3706:
3627:
3623:
3619:
3617:
3485:
3481:
3479:
3396:
3392:
3390:
3384:
3378:
3371:
3367:
3360:
3355:
3351:
3341:
3338:
3331:
3325:
3321:
3317:
3313:
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3304:
3300:
3295:
3291:
3281:
3276:
3272:
3266:
3260:
3255:
3250:
3243:
3238:
3234:
3231:
3215:
3211:
3207:
3203:
3198:
3194:
3192:
3186:
3182:
3178:
3174:
3170:
3166:
3162:
3158:
3154:
3150:
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3142:
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3125:
3121:
3117:
3110:
3104:
2929:
2925:
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2899:
2895:
2891:
2887:
2883:
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2868:
2864:
2862:
2837:
2835:
2810:
2808:
2575:
2568:
2498:
2494:
2493:
2488:
2487:
2482:
2477:
2475:
2300:
2292:
2233:
2092:
2088:Dedekind sum
1901:
1677:
1638:
1496:
1289:
978:
962:
932:
865:
851:
849:
675:
600:
597:
587:
584:Alfred Young
577:
395:
391:
382:
373:
299:
285:
259:
252:in general.
223:
209:
205:
198:
191:
187:
175:
173:
150:
147:
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80:
70:
5834:Brady Haran
5749:"Partition"
5722:"Partition"
5475:: 362–373.
5121:see, e.g.,
3787:+ 3) / 12.
1515:square root
244:and of the
234:mathematics
213:means that
112:composition
5783:2021-02-28
5704:0071.04004
5607:0953.11002
5575:0487.20007
5499:0022.20401
5452:1159.11001
5366:0697.10023
5256:References
3484:fixed and
3334:− 1)
2793:Dist. odd
2782:9 + 7 + 3
5754:MathWorld
5732:EMS Press
5408:(2008) .
5334:(1990) .
5245:119176413
5237:1420-9020
4888:0801.4928
4634:λ
4606:−
4597:λ
4333:∑
4270:−
4259:ℓ
4244:∏
4227:−
4201:∏
4182:−
4171:ℓ
4150:∏
4122:ℓ
4086:ℓ
4082:ℓ
4005:−
3987:−
3937:−
3919:−
3892:−
3684:−
3666:−
3652:∈
3645:∏
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3558:∏
3507:≥
3500:∑
3426:∑
3374:≤ 0
3079:−
3065:−
3051:∞
3036:∏
3005:∞
2990:∏
2959:∞
2944:∑
2859:4 + 3 + 1
2856:5 + 2 + 1
2811:odd parts
2478:conjugate
2219:⋯
2207:−
2195:−
2186:−
2174:−
2165:−
2144:−
2012:−
1983:π
1939:≤
1932:∑
1856:−
1828:π
1817:
1798:−
1766:⋅
1735:∞
1720:∑
1706:π
1639:In 1937,
1624:∞
1621:→
1576:π
1568:
1540:∼
1474:−
1456:−
1445:∞
1430:∏
1408:∞
1393:∑
1387:∞
1372:∏
1341:∞
1326:∑
1260:…
947:polyomino
827:2 + 1 + 1
359:⋯
291:(2, 2, 1)
276:2 + 2 + 1
273:3 + 1 + 1
169:2 + 1 + 1
165:1 + 1 + 2
161:1 + 2 + 1
138:2 + 1 + 1
81:partition
5857:Category
5838:Archived
5802:database
5800:FindStat
5774:archived
5655:(1999).
5629:(2003).
5617:(1974).
5551:(1979).
5463:(1939).
5380:(2002).
5293:(1976).
5269:(1964).
5188:14308256
5041:(1971).
4913:15392503
4740:See also
4730:Okounkov
4720:via the
4700:for all
3756:⌋
3735:⌊
3210:−
2902:(0)=1):
371:, where
256:Examples
108:summands
5830:(video)
5734:, 2001
5491:0000410
5444:2445243
4976:2317861
4905:2677686
4579:h-index
4055:parts.
3347:(0) = 1
2912:in the
2909:A000009
2086:is the
1672:by the
1513:of the
1282:in the
1279:A000041
248:and in
238:physics
201:(4) = 5
85:integer
5702:
5665:
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3630:, has
3193:where
2499:folded
1899:where
1208:, and
543:, and
295:(2, 1)
215:λ
206:λ
5844:5 May
5808:from
5777:(PDF)
5770:(PDF)
5241:S2CID
5215:arXiv
5184:S2CID
4972:JSTOR
4909:S2CID
4883:arXiv
4846:Notes
3399:) by
3363:) = 0
2853:5 + 3
2850:6 + 2
2847:7 + 1
2820:5 + 3
2817:7 + 1
2489:Claim
1272:are:
821:2 + 2
815:3 + 1
288:tuple
270:3 + 2
267:4 + 1
196:. So
157:3 + 1
153:1 + 3
133:2 + 2
128:3 + 1
98:as a
5846:2016
5810:CPAN
5663:ISBN
5639:ISBN
5593:ISBN
5561:ISBN
5430:ISBN
5416:and
5386:ISBN
5352:ISBN
5318:ISBN
5299:ISBN
5277:ISBN
5233:ISSN
5170:2000
5141:ISBN
4751:rank
4457:rank
4455:The
3382:and
3376:and
3349:and
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3105:The
2924:for
2920:The
2914:OEIS
2093:The
2051:and
1945:<
1814:sinh
1314:is
1290:The
1284:OEIS
979:The
937:and
236:and
176:part
163:and
79:, a
75:and
5700:Zbl
5690:doi
5603:Zbl
5571:Zbl
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5495:Zbl
5477:doi
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5362:Zbl
5225:doi
5174:doi
4964:doi
4893:doi
4879:117
4688:of
4305:of
3365:if
3206:= 3
3113::
1610:as
1565:exp
1497:No
1294:of
399:= 0
228:or
102:of
100:sum
71:In
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5751:.
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2928:(
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2068:,
2065:m
2062:(
2059:s
2039:.
2033:)
2029:k
2025:/
2021:m
2018:n
2015:2
2009:)
2006:k
2003:,
2000:m
1997:(
1994:s
1990:(
1986:i
1979:e
1973:1
1970:=
1967:)
1964:k
1961:,
1958:m
1955:(
1951:,
1948:k
1942:m
1936:0
1928:=
1925:)
1922:n
1919:(
1914:k
1910:A
1886:)
1881:]
1870:)
1861:1
1853:n
1849:(
1843:3
1840:2
1831:k
1821:[
1803:1
1795:n
1791:1
1785:(
1778:n
1775:d
1771:d
1761:k
1756:)
1753:n
1750:(
1745:k
1741:A
1730:1
1727:=
1724:k
1711:2
1702:1
1697:=
1694:)
1691:n
1688:(
1685:p
1660:)
1657:n
1654:(
1651:p
1618:n
1597:)
1590:3
1586:n
1583:2
1572:(
1557:3
1552:n
1549:4
1545:1
1537:)
1534:n
1531:(
1528:p
1482:.
1477:1
1470:)
1464:j
1460:q
1453:1
1450:(
1440:1
1437:=
1434:j
1426:=
1421:i
1418:j
1414:q
1403:0
1400:=
1397:i
1382:1
1379:=
1376:j
1368:=
1363:n
1359:q
1355:)
1352:n
1349:(
1346:p
1336:0
1333:=
1330:n
1302:p
1257:,
1254:2
1251:,
1248:1
1245:,
1242:0
1239:=
1236:n
1216:4
1196:2
1193:+
1190:2
1170:3
1167:+
1164:1
1144:2
1141:+
1138:1
1135:+
1132:1
1112:1
1109:+
1106:1
1103:+
1100:1
1097:+
1094:1
1074:4
1054:5
1051:=
1048:)
1045:4
1042:(
1039:p
1019:n
999:)
996:n
993:(
990:p
963:p
809:4
556:5
552:1
529:1
525:2
519:3
515:1
511:,
506:2
502:2
496:1
492:1
488:,
483:1
479:3
473:2
469:1
465:,
460:1
456:3
450:1
446:2
442:,
437:1
433:4
427:1
423:1
419:,
414:1
410:5
396:i
392:m
386:2
383:m
377:1
374:m
352:3
348:m
343:3
335:2
331:m
326:2
318:1
314:m
309:1
264:5
219:n
210:n
199:p
194:)
192:n
190:(
188:p
180:n
123:4
116:4
96:n
88:n
66:k
62:n
42:.
20:)
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