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Number of partitions of n with nonnegative crank.
47

%I #78 Mar 29 2024 11:43:57

%S 1,0,1,2,3,4,6,8,12,16,23,30,42,54,73,94,124,158,206,260,334,420,532,

%T 664,835,1034,1288,1588,1962,2404,2953,3598,4392,5328,6466,7808,9432,

%U 11338,13632,16326,19544,23316,27806,33054,39273,46534,55096,65076,76808

%N Number of partitions of n with nonnegative crank.

%C 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 From _Gus Wiseman_, Mar 30 2021 and May 21 2022: (Start)

%C 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 (1)(1) (1)(2) (1)(3) (1)(4) (1)(5) (1)(6)

%C (2)(1) (2)(2) (2)(3) (2)(4) (2)(5)

%C (3)(1) (3)(2) (3)(3) (3)(4)

%C (4)(1) (4)(2) (4)(3)

%C (5)(1) (5)(2)

%C (21)(21) (6)(1)

%C (21)(31)

%C (31)(21)

%C 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 (2) (3) (4) (5) (6) (7)

%C (21) (31) (41) (33) (43)

%C (211) (311) (51) (61)

%C (2111) (411) (331)

%C (3111) (511)

%C (21111) (4111)

%C (31111)

%C (211111)

%C The version for permutations is A000166, complement A002467.

%C The version for compositions is A238351.

%C This is column k = 0 of A352833.

%C A238352 counts reversed partitions by fixed points, rank statistic A352822.

%C A238394 counts reversed partitions without a fixed point, ranked by A352830.

%C A238395 counts reversed partitions with a fixed point, ranked by A352872. (End)

%C The above conjecture is true. See Section 4 of the Blecher-Knopfmacher paper in the Links section. - _Jeremy Lovejoy_, Sep 26 2022

%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (i).

%D G. E. Andrews, B. C. Berndt, Ramanujan's Lost Notebook Part I, Springer, see p. 169 Entry 6.7.1.

%H Vaclav Kotesovec, <a href="/A064428/b064428.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 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 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 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 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 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 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 a(n) = (A000041(n) + A064410(n)) / 2, n>1. - _Michael Somos_, Jul 28 2003

%F G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1-x^k). - _Michael Somos_, Jul 28 2003

%F G.f.: Sum_{i>=0} x^(i*(i+1)) / (Product_{j=1..i} 1-x^j )^2. - _Jon Perry_, Jul 18 2004

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - _Vaclav Kotesovec_, Sep 26 2016

%F 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 a(n) = A000041(n) - A001522(n). - _Gus Wiseman_, Mar 30 2021

%F a(n) = A064410(n) + A001522(n). - _Gus Wiseman_, May 21 2022

%e 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 From _Gus Wiseman_, May 21 2022: (Start)

%e The a(0) = 1 through a(8) = 12 partitions with nonnegative crank:

%e () . (2) (3) (4) (5) (6) (7) (8)

%e (21) (22) (32) (33) (43) (44)

%e (31) (41) (42) (52) (53)

%e (221) (51) (61) (62)

%e (222) (322) (71)

%e (321) (331) (332)

%e (421) (422)

%e (2221) (431)

%e (521)

%e (2222)

%e (3221)

%e (3311)

%e (End)

%t 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 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 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 ici[q_]:=And@@Table[q[[i]]>q[[i+2]],{i,Length[q]-2}];

%t Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], EvenQ@*Length],ici]],{n,0,15}] (* _Gus Wiseman_, Mar 30 2021 *)

%o (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 These are the row-sums of the right (or left) half of A064391, inclusive.

%Y The case of crank 0 is A064410, ranked by A342192.

%Y The strict case is A352828.

%Y These partitions are ranked by A352873.

%Y A000700 = self-conjugate partitions, ranked by A088902, complement A330644.

%Y A001522 counts partitions with positive crank, ranked by A352874.

%Y A034008 counts even-length compositions.

%Y A115720 and A115994 count partitions by their Durfee square.

%Y A224958 counts compositions w/ alternating parts unequal (even: A342532).

%Y A257989 gives the crank of the partition with Heinz number n.

%Y A342527 counts compositions w/ alternating parts equal (even: A065608).

%Y A342528 = compositions w/ alternating parts weakly decr. (even: A114921).

%Y Cf. A000041, A008292, A062968, A118199, A188674, A325547, A325548.

%K nonn

%O 0,4

%A _Vladeta Jovovic_, Sep 30 2001