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A353698
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Number of integer partitions of n whose product equals their length.
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4
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0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 3, 0, 3, 2, 2, 1, 5, 0, 1, 2, 5, 1, 4, 0, 3, 3, 2, 1, 4, 2, 3, 2, 2, 0, 5, 1, 4, 2, 2, 3, 6, 1, 2, 2, 5, 1, 4, 0, 4, 3, 3, 1, 6, 2, 3, 4, 4, 2, 4, 1, 4, 2, 3, 1, 8, 2, 4, 2, 4, 2, 5, 2, 4, 2
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OFFSET
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0,10
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LINKS
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EXAMPLE
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The a(n) partitions for selected n (A..H = 10..17):
n=9: n=21: n=27: n=33:
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51111 B1111111111 E1111111111111 H1111111111111111
321111 72111111111111 921111111111111111 B211111111111111111111
531111111111111 54111111111111111111 831111111111111111111111
4221111111111111 5511111111111111111111111
333111111111111111111111111
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Times@@#==Length[#]&]], {n, 0, 30}]
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PROG
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(PARI) a(r, m=r, p=1, k=0) = {(p==k+r) + sum(m=2, min(m, (k+r)\p), self()(r-m, min(m, r-m), p*m, k+1))} \\ Andrew Howroyd, Jan 02 2023
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CROSSREFS
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The LHS (product of parts) is counted by A339095, rank statistic A003963.
These partitions are ranked by A353699.
A266477 counts partitions by product of multiplicities, rank stat A005361.
A353504 counts partitions w/ product less than product of multiplicities.
A353505 counts partitions w/ product greater than product of multiplicities.
A353506 counts partitions w/ prod equal to prod of mults, ranked by A353503.
Cf. A000041, A002033, A098859, A114640, A181819, A225485, A266499, A319000, A325280, A353398, A353507.
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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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