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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.
11
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, 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, 1, 8, 7, 11, 9, 9, 4, 3, 2, 1
OFFSET
0,13
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 Heinz numbers, it is represented by A367580.
EXAMPLE
Triangle begins:
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 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 1 8 7 11 9 9 4 3 2 1
0 1 5 10 11 13 10 11 6 5 3 1 1
0 1 1 10 11 17 14 18 10 9 4 3 2 1
0 1 3 12 17 19 18 22 14 12 8 4 3 1 1
0 1 3 12 15 27 19 31 19 19 10 9 5 3 2 1
0 1 4 15 23 27 31 33 24 26 18 12 8 4 3 1 1
0 1 1 14 20 35 33 48 32 37 25 20 11 10 4 3 2 1
Row n = 7 counts the following partitions:
(1111111) (61) (421) (52) (4111) (511) (7)
(2221) (331) (322) (43)
(22111) (31111) (3211)
(211111)
MATHEMATICA
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
Table[Length[Select[IntegerPartitions[n], Total[mmk[#]]==k&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
Column k = 2 is A000005(n) - 1 = A032741(n).
Row sums are A000041.
The case of constant partitions is A051731, row sums A000005.
The corresponding rank statistic is A367581, row sums of A367579.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
A116861 counts partitions by sum of distinct parts.
Sequence in context: A213369 A020944 A287729 * A025897 A029421 A156749
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Nov 28 2023
STATUS
approved