A367684 - OEIS

A367684

Number of integer partitions of n whose multiset multiplicity kernel is a submultiset.

1, 1, 2, 2, 4, 5, 8, 10, 14, 17, 25, 30, 39, 51, 66, 79, 102, 125, 154, 191, 233, 284, 347, 420, 499, 614, 726, 867, 1031, 1233, 1437, 1726, 2002, 2375, 2770, 3271, 3760, 4455, 5123, 5994, 6904, 8064, 9199, 10753, 12241, 14202, 16189, 18704, 21194, 24504

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OFFSET

0,3

COMMENTS

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

LINKS

Table of n, a(n) for n=0..49.

EXAMPLE

The a(1) = 1 through a(7) = 10 partitions:

(1) (2) (3) (4) (5) (6) (7)

(11) (111) (22) (221) (33) (322)

(211) (311) (222) (331)

(1111) (2111) (411) (511)

(11111) (2211) (2221)

(3111) (4111)

(21111) (22111)

(111111) (31111)

(211111)

(1111111)

MATHEMATICA

submultQ[cap_, fat_]:=And@@Function[i, Count[fat, i]>=Count[cap, i]]/@Union[List@@cap];

mmk[q_List]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];

Table[Length[Select[IntegerPartitions[n], submultQ[mmk[#], #]&]], {n, 0, 15}]

CROSSREFS

The case of strict partitions is A000012.

Includes all partitions with distinct multiplicities A098859, ranks A130091.

These partitions have ranks A367685.

A000041 counts integer partitions, strict A000009.

A072233 counts partitions by number of parts.

A091602 counts partitions by greatest multiplicity, least A243978.

A116608 counts partitions by number of distinct parts.

A116861 counts partitions by sum of distinct parts.

Cf. A000837, A032741, A071625, A109297, A114640, A367579, A367580, A367581.

Sequence in context: A053097 A331443 A367411 * A035946 A303939 A326446

Adjacent sequences: A367681 A367682 A367683 * A367685 A367686 A367687

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 30 2023

STATUS

approved