login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A064410 Number of partitions of n with zero crank. 29
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 (list; graph; refs; listen; history; text; internal format)
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.
A257989 gives the crank of the partition with Heinz number n.
Sequence in context: A280954 A339244 A197122 * A304178 A266776 A363214
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Sep 29 2001
EXTENSIONS
More terms from Reiner Martin, Dec 26 2001
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)