This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A064428 #78 Mar 29 2024 11:43:57
%S A064428 1,0,1,2,3,4,6,8,12,16,23,30,42,54,73,94,124,158,206,260,334,420,532,
%T A064428 664,835,1034,1288,1588,1962,2404,2953,3598,4392,5328,6466,7808,9432,
%U A064428 11338,13632,16326,19544,23316,27806,33054,39273,46534,55096,65076,76808
%N A064428 Number of partitions of n with nonnegative crank.
%C A064428 For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).
%C A064428 From _Gus Wiseman_, Mar 30 2021 and May 21 2022: (Start)
%C A064428 Also the number of even-length compositions of n with alternating parts strictly decreasing, or properly 2-colored partitions (proper = no equal parts of the same color) with the same number of parts of each color, or ordered pairs of strict partitions of the same length with total n. The odd-length case is A001522, and there are a total of A000041 compositions with alternating parts strictly decreasing (see A342528 for a bijective proof). The a(2) = 1 through a(7) = 8 ordered pairs of strict partitions of the same length are:
%C A064428 (1)(1) (1)(2) (1)(3) (1)(4) (1)(5) (1)(6)
%C A064428 (2)(1) (2)(2) (2)(3) (2)(4) (2)(5)
%C A064428 (3)(1) (3)(2) (3)(3) (3)(4)
%C A064428 (4)(1) (4)(2) (4)(3)
%C A064428 (5)(1) (5)(2)
%C A064428 (21)(21) (6)(1)
%C A064428 (21)(31)
%C A064428 (31)(21)
%C A064428 Conjecture: Also the number of integer partitions y of n without a fixed point y(i) = i, ranked by A352826. This is stated at A238394, but Resta tells me he may not have had a proof. The a(2) = 1 through a(7) = 8 partitions without a fixed point are:
%C A064428 (2) (3) (4) (5) (6) (7)
%C A064428 (21) (31) (41) (33) (43)
%C A064428 (211) (311) (51) (61)
%C A064428 (2111) (411) (331)
%C A064428 (3111) (511)
%C A064428 (21111) (4111)
%C A064428 (31111)
%C A064428 (211111)
%C A064428 The version for permutations is A000166, complement A002467.
%C A064428 The version for compositions is A238351.
%C A064428 This is column k = 0 of A352833.
%C A064428 A238352 counts reversed partitions by fixed points, rank statistic A352822.
%C A064428 A238394 counts reversed partitions without a fixed point, ranked by A352830.
%C A064428 A238395 counts reversed partitions with a fixed point, ranked by A352872. (End)
%C A064428 The above conjecture is true. See Section 4 of the Blecher-Knopfmacher paper in the Links section. - _Jeremy Lovejoy_, Sep 26 2022
%D A064428 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (i).
%D A064428 G. E. Andrews, B. C. Berndt, Ramanujan's Lost Notebook Part I, Springer, see p. 169 Entry 6.7.1.
%H A064428 Vaclav Kotesovec, <a href="/A064428/b064428.txt">Table of n, a(n) for n = 0..10000</a>
%H A064428 George E. Andrews and David Newman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Andrews/andrews5.html">The Minimal Excludant in Integer Partitions</a>, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
%H A064428 Cody Armond and Oliver T. Dasbach, <a href="https://arxiv.org/abs/1106.3948">Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial</a>, arXiv:1106.3948 [math.GT], 2011.
%H A064428 Cristina Ballantine and Mircea Merca, <a href="https://arxiv.org/abs/1710.05960">Bisected theta series, least r-gaps in partitions, and polygonal numbers</a>, arXiv:1710.05960 [math.CO], 2017.
%H A064428 Rupam Barman and Ajit Singh, <a href="https://arxiv.org/abs/2009.11602">On Mex-related partition functions of Andrews and Newman</a>, arXiv:2009.11602 [math.NT], 2020.
%H A064428 Aubrey Blecher and Arnold Knopfmacher, <a href="http://doi.org/10.1007/s11139-022-00551-x">Fixed points and matching points in partitions</a>, Ramanujan J. 58 (2022), 23-41.
%H A064428 Brian Hopkins, James A. Sellers, and Ae Ja Yee, <a href="https://arxiv.org/abs/2108.09414">Combinatorial Perspectives on the Crank and Mex Partition Statistics</a>, arXiv:2108.09414 [math.CO], 2021.
%H A064428 Mbavhalelo Mulokwe and Konstantinos Zoubos, <a href="https://arxiv.org/abs/2403.08531">Free fermions, neutrality and modular transformations</a>, arXiv:2403.08531 [hep-th], 2024.
%F A064428 a(n) = (A000041(n) + A064410(n)) / 2, n>1. - _Michael Somos_, Jul 28 2003
%F A064428 G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1-x^k). - _Michael Somos_, Jul 28 2003
%F A064428 G.f.: Sum_{i>=0} x^(i*(i+1)) / (Product_{j=1..i} 1-x^j )^2. - _Jon Perry_, Jul 18 2004
%F A064428 a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - _Vaclav Kotesovec_, Sep 26 2016
%F A064428 G.f.: (Sum_{i>=0} x^i / (Product_{j=1..i} 1-x^j)^2 ) * (Product_{k>0} 1-x^k). - _Li Han_, May 23 2020
%F A064428 a(n) = A000041(n) - A001522(n). - _Gus Wiseman_, Mar 30 2021
%F A064428 a(n) = A064410(n) + A001522(n). - _Gus Wiseman_, May 21 2022
%e A064428 G.f. = 1 + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 12*x^8 + 16*x^9 + 23*x^10 + ... - _Michael Somos_, Jan 15 2018
%e A064428 From _Gus Wiseman_, May 21 2022: (Start)
%e A064428 The a(0) = 1 through a(8) = 12 partitions with nonnegative crank:
%e A064428 () . (2) (3) (4) (5) (6) (7) (8)
%e A064428 (21) (22) (32) (33) (43) (44)
%e A064428 (31) (41) (42) (52) (53)
%e A064428 (221) (51) (61) (62)
%e A064428 (222) (322) (71)
%e A064428 (321) (331) (332)
%e A064428 (421) (422)
%e A064428 (2221) (431)
%e A064428 (521)
%e A064428 (2222)
%e A064428 (3221)
%e A064428 (3311)
%e A064428 (End)
%t A064428 a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^(k (k + 1)/2) , {k, 0, (Sqrt[1 + 8 n] - 1)/2}] / QPochhammer[ x], {x, 0, n}]]; (* _Michael Somos_, Jan 15 2018 *)
%t A064428 a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k (k + 1)) / QPochhammer[ x, x, k]^2 , {k, 0, (Sqrt[1 + 4 n] - 1)/2}], {x, 0, n}]]; (* _Michael Somos_, Jan 15 2018 *)
%t A064428 ck[y_]:=With[{w=Count[y,1]},If[w==0,If[y=={},0,Max@@y],Count[y,_?(#>w&)]-w]];Table[Length[Select[IntegerPartitions[n],ck[#]>=0&]],{n,0,30}] (* _Gus Wiseman_, Mar 30 2021 *)
%t A064428 ici[q_]:=And@@Table[q[[i]]>q[[i+2]],{i,Length[q]-2}];
%t A064428 Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], EvenQ@*Length],ici]],{n,0,15}] (* _Gus Wiseman_, Mar 30 2021 *)
%o A064428 (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(1 + 8*n) -1)\2, (-1)^k * x^((k+k^2)/2)) / eta( x + x * O(x^n)), n))}; /* _Michael Somos_, Jul 28 2003 */
%Y A064428 These are the row-sums of the right (or left) half of A064391, inclusive.
%Y A064428 The case of crank 0 is A064410, ranked by A342192.
%Y A064428 The strict case is A352828.
%Y A064428 These partitions are ranked by A352873.
%Y A064428 A000700 = self-conjugate partitions, ranked by A088902, complement A330644.
%Y A064428 A001522 counts partitions with positive crank, ranked by A352874.
%Y A064428 A034008 counts even-length compositions.
%Y A064428 A115720 and A115994 count partitions by their Durfee square.
%Y A064428 A224958 counts compositions w/ alternating parts unequal (even: A342532).
%Y A064428 A257989 gives the crank of the partition with Heinz number n.
%Y A064428 A342527 counts compositions w/ alternating parts equal (even: A065608).
%Y A064428 A342528 = compositions w/ alternating parts weakly decr. (even: A114921).
%Y A064428 Cf. A000041, A008292, A062968, A118199, A188674, A325547, A325548.
%K A064428 nonn
%O A064428 0,4
%A A064428 _Vladeta Jovovic_, Sep 30 2001