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, 1, 0, 0, 5, 1, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 4

Offset: 0

Author

Gus Wiseman, May 01 2019

Keywords

• nonn

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.

Examples

			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)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..100000

Crossrefs

Column k = 3 of A325336.

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

Programs

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}]

PARI

A007862(n) = sumdiv(n, d, ispolygonal(d, 3));
A325334(n) = if(!n,n,A007862(n)-1); \\ Antti Karttunen, Jan 17 2025

Formula

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

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 17 2025