A064410 - OEIS

%I #36 May 10 2023 06:18:16

%S 0,0,1,1,1,1,1,2,2,4,4,7,7,11,12,17,19,27,30,41,48,62,73,95,110,140,

%T 166,206,243,302,354,435,513,622,733,887,1039,1249,1467,1750,2049,

%U 2438,2847,3371,3934,4634,5398,6343,7367,8626,10009,11677,13521,15737,18184

%N Number of partitions of n with zero 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).

%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A064410/b064410.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)

%H Brian Hopkins and James A. Sellers, <a href="https://arxiv.org/abs/2305.05096">On Blecher and Knopfmacher's Fixed Points for Integer Partitions</a>, arXiv:2305.05096 [math.CO], 2023. Mentions this sequence.

%H Brian Hopkins, James A. Sellers, and Dennis Stanton, <a href="https://arxiv.org/abs/2009.10873">Dyson's Crank and the Mex of Integer Partitions</a>, arXiv:2009.10873 [math.CO], 2020. Mentions this sequence.

%F a(n) = A000041(n)-2*A001522(n). a(n) = A064391(n, 0).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi / (3 * 2^(9/2) * n^(3/2)). - _Vaclav Kotesovec_, May 06 2018

%F a(n > 1) = A064428(n) - A001522(n), where A001522/A064428 count odd/even-length compositions with alternating parts strictly decreasing. - _Gus Wiseman_, Apr 02 2021

%e a(10)=4 because there are 4 partitions of 10 with zero crank: 1+1+2+3+3, 1+1+4+4, 1+1+3+5 and 1+9.

%e From _Gus Wiseman_, Apr 02 2021: (Start)

%e The a(3) = 1 through a(14) = 11 partitions (A..D = 10..13):

%e   21  31  41  51  61  71    81    91     A1     B1      C1      D1

%e                       3311  4311  4411   5411   5511    6511    6611

%e                                   5311   6311   6411    7411    7511

%e                                   33211  43211  7311    8311    8411

%e                                                 44211   54211   9311

%e                                                 53211   63211   55211

%e                                                 332211  432211  64211

%e                                                                 73211

%e                                                                 442211

%e                                                                 532211

%e                                                                 3322211

%e (End)

%t nmax = 60; Rest[CoefficientList[Series[x - 1 + Sum[(-1)^k*(x^(k*(k + 1)/2) - x^(k*(k - 1)/2)), {k, 1, nmax}] / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 26 2016 *)

%t Flatten[{0, Table[PartitionsP[n] - 2*Sum[(-1)^(j+1)*PartitionsP[n - j*((j+1)/2)], {j, 1, Floor[(Sqrt[8*n + 1] - 1)/2]}], {n, 2, 60}]}] (* _Vaclav Kotesovec_, Sep 26 2016 *)

%t ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];

%t Table[Length[Select[IntegerPartitions[n],ck[#]==0&]],{n,0,30}] (* _Gus Wiseman_, Apr 02 2021 *)

%o (Sage)

%o [[p.crank() for p in Partitions(n)].count(0) for n in (1..20)] # _Peter Luschny_, Sep 15 2014

%Y The version for positive crank is A001522.

%Y Central column of A064391.

%Y The version for nonnegative crank is A064428.

%Y The Heinz numbers of these partitions are A342192.

%Y A003242 counts anti-run compositions.

%Y A224958 counts compositions with alternating parts unequal.

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

%Y Cf. A000009, A000041, A000726, A027193, A034008, A062968, A114921, A342194, A342343, A342528.

%K nonn

%O 1,8

%A _Vladeta Jovovic_, Sep 29 2001

%E More terms from _Reiner Martin_, Dec 26 2001