cp's OEIS Frontend

A367684 Number of integer partitions of n whose multiset multiplicity kernel is a submultiset.

%I A367684 #7 Dec 01 2023 09:31:29
%S A367684 1,1,2,2,4,5,8,10,14,17,25,30,39,51,66,79,102,125,154,191,233,284,347,
%T A367684 420,499,614,726,867,1031,1233,1437,1726,2002,2375,2770,3271,3760,
%U A367684 4455,5123,5994,6904,8064,9199,10753,12241,14202,16189,18704,21194,24504
%N A367684 Number of integer partitions of n whose multiset multiplicity kernel is a submultiset.
%C A367684 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 A367684 The a(1) = 1 through a(7) = 10 partitions:
%e A367684   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A367684        (11)  (111)  (22)    (221)    (33)      (322)
%e A367684                     (211)   (311)    (222)     (331)
%e A367684                     (1111)  (2111)   (411)     (511)
%e A367684                             (11111)  (2211)    (2221)
%e A367684                                      (3111)    (4111)
%e A367684                                      (21111)   (22111)
%e A367684                                      (111111)  (31111)
%e A367684                                                (211111)
%e A367684                                                (1111111)
%t A367684 submultQ[cap_,fat_]:=And@@Function[i, Count[fat,i]>=Count[cap, i]]/@Union[List@@cap];
%t A367684 mmk[q_List]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t A367684 Table[Length[Select[IntegerPartitions[n], submultQ[mmk[#],#]&]], {n,0,15}]
%Y A367684 The case of strict partitions is A000012.
%Y A367684 Includes all partitions with distinct multiplicities A098859, ranks A130091.
%Y A367684 These partitions have ranks A367685.
%Y A367684 A000041 counts integer partitions, strict A000009.
%Y A367684 A072233 counts partitions by number of parts.
%Y A367684 A091602 counts partitions by greatest multiplicity, least A243978.
%Y A367684 A116608 counts partitions by number of distinct parts.
%Y A367684 A116861 counts partitions by sum of distinct parts.
%Y A367684 Cf. A000837, A032741, A071625, A109297, A114640, A367579, A367580, A367581.
%K A367684 nonn
%O A367684 0,3
%A A367684 _Gus Wiseman_, Nov 30 2023