cp's OEIS Frontend

A367682 Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.

%I A367682 #6 Nov 30 2023 23:16:49
%S A367682 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 A367682 151,185,212,256,286,348,394,479,543,642,740,888,994,1176,1350,1589,
%U A367682 1789,2109,2371,2786,3144,3653,4126,4811,5385,6213
%N A367682 Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.
%C A367682 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 A367682 The a(1) = 1 through a(10) = 10 partitions:
%e A367682   (1)  .  (21)  (22)   (41)   (51)    (61)   (71)     (81)    (91)
%e A367682                 (31)   (221)  (321)   (421)  (431)    (333)   (541)
%e A367682                 (211)         (3111)         (521)    (531)   (631)
%e A367682                                              (3221)   (621)   (721)
%e A367682                                              (41111)  (4221)  (3322)
%e A367682                                                               (3331)
%e A367682                                                               (4321)
%e A367682                                                               (5221)
%e A367682                                                               (322111)
%e A367682                                                               (511111)
%t A367682 mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t A367682 Table[Length[Select[IntegerPartitions[n], Sort[Length/@Split[#]]==mmk[#]&]], {n,0,15}]
%Y A367682 The case of strict partitions is A025147, ranks A039956.
%Y A367682 The case of distinct multiplicities is A114640, ranks A109297.
%Y A367682 These partitions have ranks A367683.
%Y A367682 A000041 counts integer partitions, strict A000009.
%Y A367682 A072233 counts partitions by number of parts.
%Y A367682 A091602 counts partitions by greatest multiplicity, least A243978.
%Y A367682 A116608 counts partitions by number of distinct parts.
%Y A367682 Cf. A000837, A071625, A072774, A082090, A116861, A367579, A367580, A367581.
%K A367682 nonn
%O A367682 0,5
%A A367682 _Gus Wiseman_, Nov 30 2023