A325334 Number of integer partitions of n with adjusted frequency depth 3 whose parts cover an initial interval of positive integers.

0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 1, 1, 0, 4, 0, 0, 1, 0, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 3, 0, 0, 2, 0, 1, 1, 0, 0, 2, 1, 1, 1, 0, 0, 4, 0, 0, 2, 0, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 2, 0, 0, 3, 0, 1, 1, 0, 0, 4, 0, 0

Offset

0,7

Comments

The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

The Heinz numbers of these partitions are given by A325374.

Links

Table of n, a(n) for n=0..86.

Formula

a(n) = A007862(n) - 1.

Example

The first 30 terms count the following partitions:
   3: (21)
   6: (321)
   6: (2211)
   9: (222111)
  10: (4321)
  12: (332211)
  12: (22221111)
  15: (54321)
  15: (2222211111)
  18: (333222111)
  18: (222222111111)
  20: (44332211)
  21: (654321)
  21: (22222221111111)
  24: (333322221111)
  24: (2222222211111111)
  27: (222222222111111111)
  28: (7654321)
  30: (5544332211)
  30: (444333222111)
  30: (333332222211111)
  30: (22222222221111111111)

Mathematica

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
unifQ[m_]:=SameQ@@Length/@Split[m];
Table[Length[Select[IntegerPartitions[n], normQ[#]&&!SameQ@@#&&unifQ[#]&]], {n, 0, 30}]

Crossrefs

Column k = 3 of A325336.

Cf. A007862, A181819, A182850, A320348, A323014, A325245, A325280, A325326, A325335, A325374.

Sequence in context: A151843 A276422 A323069 * A280287 A147696 A001842

Adjacent sequences: A325331 A325332 A325333 * A325335 A325336 A325337

Keyword

nonn

Author

Gus Wiseman, May 01 2019

Status

approved