A064410 - OEIS

A064410

Number of partitions of n with zero crank.

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, 166, 206, 243, 302, 354, 435, 513, 622, 733, 887, 1039, 1249, 1467, 1750, 2049, 2438, 2847, 3371, 3934, 4634, 5398, 6343, 7367, 8626, 10009, 11677, 13521, 15737, 18184

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OFFSET

1,8

COMMENTS

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).

LINKS

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Brian Hopkins and James A. Sellers, On Blecher and Knopfmacher's Fixed Points for Integer Partitions, arXiv:2305.05096 [math.CO], 2023. Mentions this sequence.

Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020. Mentions this sequence.

FORMULA

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

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi / (3 * 2^(9/2) * n^(3/2)). - Vaclav Kotesovec, May 06 2018

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

EXAMPLE

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.

From Gus Wiseman, Apr 02 2021: (Start)

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

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

3311 4311 4411 5411 5511 6511 6611

5311 6311 6411 7411 7511

33211 43211 7311 8311 8411

44211 54211 9311

53211 63211 55211

332211 432211 64211

73211

442211

532211

3322211

(End)

MATHEMATICA

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 *)

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 *)

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

Table[Length[Select[IntegerPartitions[n], ck[#]==0&]], {n, 0, 30}] (* Gus Wiseman, Apr 02 2021 *)

PROG

(Sage)

[[p.crank() for p in Partitions(n)].count(0) for n in (1..20)] # Peter Luschny, Sep 15 2014

CROSSREFS

The version for positive crank is A001522.

Central column of A064391.

The version for nonnegative crank is A064428.

The Heinz numbers of these partitions are A342192.

A003242 counts anti-run compositions.

A224958 counts compositions with alternating parts unequal.