A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.

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

Links

Table of n, a(n) for n=0..65.

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.

Cf. A007837, A038041, A327899, A336137, A336138, A365663, A365661, A365925.

Sequence in context: A363733 A062135 A190182 * A068926 A276770 A258291

Adjacent sequences: A371785 A371786 A371787 * A371789 A371790 A371791

Keyword

nonn,tabl

Author

Gus Wiseman, Apr 16 2024

Status

approved