A325766 Number of integer partitions of n covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 4, 5, 4, 6, 7, 8, 6, 12, 11, 19, 16, 22, 22, 25, 32, 38, 45, 45, 51, 53, 71, 69, 85, 92, 118, 125, 147, 149, 184, 187, 230, 254, 290, 317, 372, 397, 449, 502, 544, 616, 680, 758, 841, 930, 1042, 1151, 1262

Offset: 0

Author

Gus Wiseman, May 19 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A325767.

Examples

			The initial terms count the following partitions:
   1: (1)
   4: (2,1,1)
   5: (2,2,1)
   6: (2,2,1,1)
   7: (3,2,1,1)
   8: (3,2,1,1,1)
   9: (3,2,2,1,1)
  10: (3,2,2,1,1,1)
  11: (3,3,2,2,1)
  11: (3,3,2,1,1,1)
  11: (3,2,2,2,1,1)
  12: (4,3,2,1,1,1)
  13: (4,3,2,2,1,1)
  13: (4,3,2,1,1,1,1)
  13: (3,3,3,2,1,1)
  13: (3,3,2,2,2,1)
  13: (3,3,2,2,1,1,1)
  14: (4,3,2,2,1,1,1)
  14: (3,3,3,2,2,1)
  14: (3,3,3,2,1,1,1)
  14: (3,3,2,2,2,1,1)
		

Crossrefs

Cf. A000009 (partitions covering an initial interval), A055932, A114639, A114640, A290689, A324753, A325702, A325706, A325707, A325708, A325767.

Programs

Mathematica

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap]
Table[Length[Select[IntegerPartitions[n],Range[Length[Union[#]]]==Union[#]&&submultQ[Sort[Length/@Split[#]],Sort[#]]&]],{n,0,30}]