A324572 Number of integer partitions of n whose multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in decreasing order.

1, 1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 3, 0, 2, 0, 4, 1, 2, 1, 4, 1, 3, 1, 5, 3, 5, 1, 6, 2, 6, 1, 7, 2, 7, 2, 11, 4, 8, 3, 11, 5, 10, 4, 13, 5, 11, 5, 16, 8, 14, 5, 19, 8, 18, 6, 22, 8, 22, 7, 26, 10, 25, 8, 33, 12, 29, 11, 36, 13, 34, 12, 40, 16, 41, 14, 47, 17, 45, 16, 55

Offset: 0

Author

Gus Wiseman, Mar 08 2019

Keywords

• nonn

Comments

These are a kind of self-describing partitions (cf. A001462, A304679).

The Heinz numbers of these partitions are given by A324571.

The case where the distinct parts are taken in increasing order is counted by A033461, with Heinz numbers given by A109298.

Examples

			The first 19 terms count the following integer partitions:
   1: (1)
   4: (22)
   4: (211)
   6: (3111)
   8: (41111)
   9: (333)
  10: (511111)
  10: (322111)
  12: (6111111)
  12: (4221111)
  12: (33222)
  14: (71111111)
  14: (52211111)
  16: (811111111)
  16: (622111111)
  16: (4444)
  16: (442222)
  17: (43331111)
  18: (9111111111)
  18: (7221111111)
  19: (533311111)
		

Crossrefs

Cf. A001156, A033461, A109298, A117144, A276078, A324524, A324571.

Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

Programs

Mathematica

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

Extensions

More terms from Alois P. Heinz, Mar 08 2019