A325335 - OEIS

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

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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]]];