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Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.
4

%I #6 Nov 30 2023 23:16:49

%S 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,

%T 151,185,212,256,286,348,394,479,543,642,740,888,994,1176,1350,1589,

%U 1789,2109,2371,2786,3144,3653,4126,4811,5385,6213

%N Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.

%C 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.

%e The a(1) = 1 through a(10) = 10 partitions:

%e (1) . (21) (22) (41) (51) (61) (71) (81) (91)

%e (31) (221) (321) (421) (431) (333) (541)

%e (211) (3111) (521) (531) (631)

%e (3221) (621) (721)

%e (41111) (4221) (3322)

%e (3331)

%e (4321)

%e (5221)

%e (322111)

%e (511111)

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

%t Table[Length[Select[IntegerPartitions[n], Sort[Length/@Split[#]]==mmk[#]&]], {n,0,15}]

%Y The case of strict partitions is A025147, ranks A039956.

%Y The case of distinct multiplicities is A114640, ranks A109297.

%Y These partitions have ranks A367683.

%Y A000041 counts integer partitions, strict A000009.

%Y A072233 counts partitions by number of parts.

%Y A091602 counts partitions by greatest multiplicity, least A243978.

%Y A116608 counts partitions by number of distinct parts.

%Y Cf. A000837, A071625, A072774, A082090, A116861, A367579, A367580, A367581.

%K nonn

%O 0,5

%A _Gus Wiseman_, Nov 30 2023