A353505 Number of integer partitions of n whose product is greater than the product of their multiplicities.

0, 0, 1, 2, 3, 5, 7, 11, 17, 24, 35, 47, 66, 89, 121, 162, 214, 276, 362, 464, 599, 763, 971, 1219, 1537, 1918, 2393, 2966, 3668, 4512, 5549, 6784, 8287, 10076, 12238, 14807, 17898, 21556, 25931, 31094, 37243, 44486, 53075, 63158, 75069, 89025, 105447, 124636

Offset: 0

Author

Gus Wiseman, May 19 2022

Keywords

• nonn

Examples

			The a(0) = 0 through a(7) = 11 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)    (7)
             (21)  (22)  (32)   (33)   (43)
                   (31)  (41)   (42)   (52)
                         (221)  (51)   (61)
                         (311)  (222)  (322)
                                (321)  (331)
                                (411)  (421)
                                       (511)
                                       (2221)
                                       (3211)
                                       (4111)
		

Crossrefs

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

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

The version for less instead of greater is A353504.

The version for equality is A353506, ranked by A353503.

A124010 gives prime signature, sorted A118914.

A181819 gives prime shadow, with an inverse A181821.

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

Cf. A002033, A008284, A008619, A085629, A097318, A098859, A114640, A116608, A130091, A304678, A353394, A353507.

Programs

Mathematica

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