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A064410
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Number of partitions of n with zero crank.
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29
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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
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1,8
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COMMENTS
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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).
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LINKS
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FORMULA
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a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi / (3 * 2^(9/2) * n^(3/2)). - Vaclav Kotesovec, May 06 2018
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EXAMPLE
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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.
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)
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MATHEMATICA
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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 *)
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PROG
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(Sage)
[[p.crank() for p in Partitions(n)].count(0) for n in (1..20)] # Peter Luschny, Sep 15 2014
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CROSSREFS
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The version for positive crank is A001522.
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.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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