A353505 - OEIS

%I #5 May 20 2022 08:51:38

%S 0,0,1,2,3,5,7,11,17,24,35,47,66,89,121,162,214,276,362,464,599,763,

%T 971,1219,1537,1918,2393,2966,3668,4512,5549,6784,8287,10076,12238,

%U 14807,17898,21556,25931,31094,37243,44486,53075,63158,75069,89025,105447,124636

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

%e The a(0) = 0 through a(7) = 11 partitions:

%e   .  .  (2)  (3)   (4)   (5)    (6)    (7)

%e              (21)  (22)  (32)   (33)   (43)

%e                    (31)  (41)   (42)   (52)

%e                          (221)  (51)   (61)

%e                          (311)  (222)  (322)

%e                                 (321)  (331)

%e                                 (411)  (421)

%e                                        (511)

%e                                        (2221)

%e                                        (3211)

%e                                        (4111)

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

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

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

%Y The version for less instead of greater is A353504.

%Y The version for equality is A353506, ranked by A353503.

%Y A124010 gives prime signature, sorted A118914.

%Y A181819 gives prime shadow, with an inverse A181821.

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

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

%K nonn

%O 0,4

%A _Gus Wiseman_, May 19 2022