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A367682
Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.
4
1, 1, 0, 1, 3, 2, 3, 2, 5, 5, 10, 9, 14, 14, 21, 20, 30, 36, 44, 50, 66, 75, 93, 106, 132, 151, 185, 212, 256, 286, 348, 394, 479, 543, 642, 740, 888, 994, 1176, 1350, 1589, 1789, 2109, 2371, 2786, 3144, 3653, 4126, 4811, 5385, 6213
OFFSET
0,5
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.
EXAMPLE
The a(1) = 1 through a(10) = 10 partitions:
(1) . (21) (22) (41) (51) (61) (71) (81) (91)
(31) (221) (321) (421) (431) (333) (541)
(211) (3111) (521) (531) (631)
(3221) (621) (721)
(41111) (4221) (3322)
(3331)
(4321)
(5221)
(322111)
(511111)
MATHEMATICA
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
Table[Length[Select[IntegerPartitions[n], Sort[Length/@Split[#]]==mmk[#]&]], {n, 0, 15}]
CROSSREFS
The case of strict partitions is A025147, ranks A039956.
The case of distinct multiplicities is A114640, ranks A109297.
These partitions have ranks A367683.
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.
Sequence in context: A258778 A112924 A267148 * A230406 A362688 A214254
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 30 2023
STATUS
approved