A367684 - OEIS

A367684

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

%I #7 Dec 01 2023 09:31:29

%S 1,1,2,2,4,5,8,10,14,17,25,30,39,51,66,79,102,125,154,191,233,284,347,

%T 420,499,614,726,867,1031,1233,1437,1726,2002,2375,2770,3271,3760,

%U 4455,5123,5994,6904,8064,9199,10753,12241,14202,16189,18704,21194,24504

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

%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(7) = 10 partitions:

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

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

%e (211) (311) (222) (331)

%e (1111) (2111) (411) (511)

%e (11111) (2211) (2221)

%e (3111) (4111)

%e (21111) (22111)

%e (111111) (31111)

%e (211111)

%e (1111111)

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

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

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

%Y The case of strict partitions is A000012.

%Y Includes all partitions with distinct multiplicities A098859, ranks A130091.

%Y These partitions have ranks A367685.

%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 A116861 counts partitions by sum of distinct parts.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Nov 30 2023