A325335 Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.

0, 0, 0, 0, 1, 2, 1, 3, 3, 3, 5, 8, 6, 13, 12, 14, 17, 22, 17, 28, 29, 30, 38, 50, 46, 67, 64, 75, 81, 104, 99, 127, 128, 150, 155, 201, 189, 236, 244, 293, 302, 363, 372, 437, 457, 548, 547, 638, 671, 754, 809, 922, 947, 1074, 1144, 1290, 1342, 1515, 1574

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 A325387.

Examples

			The a(4) = 1 through a(10) = 5 partitions:
  (211)  (221)   (21111)  (2221)    (22211)    (22221)     (222211)
         (2111)           (22111)   (221111)   (2211111)   (322111)
                          (211111)  (2111111)  (21111111)  (2221111)
                                                           (22111111)
                                                           (211111111)
		

Crossrefs

Column k = 4 of A325336.

Cf. A000009, A007862, A181819, A182850, A317081, A320348, A323014, A325280, A325326, A325334, A325387.

Programs

Mathematica

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==4&]],{n,0,30}]