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A371788
Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.
5
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 8, 4, 1, 0, 2, 19, 24, 6, 1, 0, 2, 47, 95, 49, 9, 1, 0, 6, 105, 363, 297, 93, 12, 1, 0, 12, 248, 1292, 1660, 753, 158, 16, 1, 0, 11, 563, 4649, 8409, 5591, 1653, 250, 20, 1, 0, 2, 1414, 15976, 41264, 38074, 15590, 3249, 380, 25, 1
OFFSET
0,8
EXAMPLE
The set partition {{1,3},{2},{4}} has two distinct block-sums {2,4} so is counted under T(4,2).
Triangle begins:
1
0 1
0 1 1
0 2 2 1
0 2 8 4 1
0 2 19 24 6 1
0 2 47 95 49 9 1
0 6 105 363 297 93 12 1
0 12 248 1292 1660 753 158 16 1
0 11 563 4649 8409 5591 1653 250 20 1
0 2 1414 15976 41264 38074 15590 3249 380 25 1
Row n = 4 counts the following set partitions:
. {{1,4},{2,3}} {{1},{2,3,4}} {{1},{2},{3,4}} {{1},{2},{3},{4}}
{{1,2,3,4}} {{1,2},{3},{4}} {{1},{2,3},{4}}
{{1,2},{3,4}} {{1},{2,4},{3}}
{{1,3},{2},{4}} {{1,4},{2},{3}}
{{1,3},{2,4}}
{{1,2,3},{4}}
{{1,2,4},{3}}
{{1,3,4},{2}}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Select[sps[Range[n]], Length[Union[Total/@#]]==k&]], {n, 0, 5}, {k, 0, n}]
CROSSREFS
Row sums are A000110.
Column k = 1 is A035470.
A version for integer partitions is A116608.
For block lengths instead of sums we have A208437.
A008277 counts set partitions by length.
A275780 counts set partitions with distinct block-sums.
A371737 counts quanimous strict partitions, non-strict A321452.
A371789 counts non-quanimous sets, differences A371790.
Sequence in context: A363733 A062135 A190182 * A068926 A276770 A258291
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 16 2024
STATUS
approved