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A325335
Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.
3
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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 01 2019
STATUS
approved