A353504 - OEIS

A353504

Number of integer partitions of n whose product is less than the product of their multiplicities.

0, 0, 1, 1, 1, 2, 2, 4, 4, 6, 6, 8, 9, 11, 12, 14, 14, 18, 21, 23, 26, 29, 29, 33, 36, 39, 40, 43, 44, 50, 53, 55, 59, 65, 69, 72, 78, 79, 81, 85, 92, 95, 97, 100, 103, 108, 109, 112, 118, 124, 129, 137, 139, 142, 149, 155, 159, 165, 166, 173, 178, 181, 187

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OFFSET

0,6

LINKS

Table of n, a(n) for n=0..62.

EXAMPLE

The a(2) = 1 through a(9) = 6 partitions:

11 111 1111 2111 21111 22111 221111 222111

11111 111111 31111 311111 411111

211111 2111111 2211111

1111111 11111111 3111111

21111111

111111111

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], Times@@#<Times@@Length/@Split[#]&]], {n, 0, 30}]

CROSSREFS

LHS (product of parts) is counted by A339095, ranked by A003963.

RHS (product of multiplicities) is counted by A266477, ranked by A005361.

The version for greater instead of less is A353505.

The version for equal instead of less is A353506, ranked by A353503.

A181819 gives prime shadow, with an inverse A181821.

A353398 counts partitions with the same product of multiplicities as of shadows, ranked by A353399.

Cf. A002033, A008284, A085629, A097318, A098859, A114640, A116608, A118914, A124010, A304678, A353394, A353500, A353507.

Sequence in context: A346663 A137501 A308767 * A345419 A285999 A005186

Adjacent sequences: A353501 A353502 A353503 * A353505 A353506 A353507

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 19 2022

STATUS

approved