proposed

approved

editing

proposed

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

allocated for Gus WisemanNumber 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

0,6

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.

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)

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}]

Column k = 4 of A325336.

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

allocated

nonn

Gus Wiseman, May 01 2019

approved

editing

allocated for Gus Wiseman

allocated

approved