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

0,6

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.

Links

Table of n, a(n) for n=0..58.

Example

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)

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}]

Crossrefs

Column k = 4 of A325336.

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

Sequence in context: A303974 A153868 A341147 * A262952 A245367 A304740

Adjacent sequences: A325332 A325333 A325334 * A325336 A325337 A325338

Keyword

nonn

Author

Gus Wiseman, May 01 2019

Status

approved