A367582 - OEIS

A367582

Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.

%I #6 Nov 29 2023 06:56:12

%S 1,0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,1,2,2,1,0,1,3,3,2,1,1,0,1,1,4,3,3,

%T 2,1,0,1,3,5,4,4,3,1,1,0,1,2,6,4,8,3,3,2,1,0,1,3,7,9,6,7,4,3,1,1,0,1,

%U 1,8,7,11,9,9,4,3,2,1

%N Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.

%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 Heinz numbers, it is represented by A367580.

%e Triangle begins:

%e 1

%e 0 1

%e 0 1 1

%e 0 1 1 1

%e 0 1 2 1 1

%e 0 1 1 2 2 1

%e 0 1 3 3 2 1 1

%e 0 1 1 4 3 3 2 1

%e 0 1 3 5 4 4 3 1 1

%e 0 1 2 6 4 8 3 3 2 1

%e 0 1 3 7 9 6 7 4 3 1 1

%e 0 1 1 8 7 11 9 9 4 3 2 1

%e 0 1 5 10 11 13 10 11 6 5 3 1 1

%e 0 1 1 10 11 17 14 18 10 9 4 3 2 1

%e 0 1 3 12 17 19 18 22 14 12 8 4 3 1 1

%e 0 1 3 12 15 27 19 31 19 19 10 9 5 3 2 1

%e 0 1 4 15 23 27 31 33 24 26 18 12 8 4 3 1 1

%e 0 1 1 14 20 35 33 48 32 37 25 20 11 10 4 3 2 1

%e Row n = 7 counts the following partitions:

%e (1111111) (61) (421) (52) (4111) (511) (7)

%e (2221) (331) (322) (43)

%e (22111) (31111) (3211)

%e (211111)

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

%t Table[Length[Select[IntegerPartitions[n], Total[mmk[#]]==k&]], {n,0,10}, {k,0,n}]

%Y Column k = 2 is A000005(n) - 1 = A032741(n).

%Y Row sums are A000041.

%Y The case of constant partitions is A051731, row sums A000005.

%Y The corresponding rank statistic is A367581, row sums of A367579.

%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, A056239, A066328, A071625, A072774, A082090, A367580.

%K nonn,tabl

%O 0,13

%A _Gus Wiseman_, Nov 28 2023