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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 30 2023
STATUS
approved