A320438 Irregular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with all block-sums equal to d, where d is the k-th divisor of n*(n+1)/2 that is >= n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 3, 7, 1, 1, 9, 1, 1, 1, 1, 43, 35, 1, 1, 102, 62, 1, 1, 1, 1, 68, 595, 1, 1, 17, 187, 871, 1480, 361, 1, 1, 2650, 657, 1, 1, 9294, 1, 1, 23728, 1, 1, 27763, 4110, 1, 1, 1850, 25035, 108516, 157991, 7636, 1, 1, 11421, 411474, 1

Offset

1,12

Links

Table of n, a(n) for n=1..68.

Example

Triangle begins:
    1
    1
    1    1
    1    1
    1    1
    1    1
    1    4    1
    1    3    7    1
    1    9    1
    1    1
    1   43   35    1
    1  102   62    1
    1    1
    1   68  595    1
    1   17  187  871 1480  361    1
    1 2650  657    1
Row 8 counts the following set partitions:
  {{18}{27}{36}{45}}  {{1236}{48}{57}}  {{12348}{567}}  {{12345678}}
                      {{138}{246}{57}}  {{12357}{468}}
                      {{156}{237}{48}}  {{12456}{378}}
                                        {{1278}{3456}}
                                        {{1368}{2457}}
                                        {{1458}{2367}}
                                        {{1467}{2358}}

Mathematica

spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsu[Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}];
Table[Length[spsu[Select[Subsets[Range[n]], Total[#]==d&], Range[n]]], {n, 12}, {d, Select[Divisors[n*(n+1)/2], #>=n&]}]

Crossrefs

Row lengths are A164978. Row sums are A035470.

Cf. A000110, A000258, A008277, A112956, A164977, A275714, A279375, A300335, A320423, A320424, A321455, A321469.

Sequence in context: A392664 A174834 A100642 * A255511 A014518 A316584

Adjacent sequences: A320435 A320436 A320437 * A320439 A320440 A320441

Keyword

nonn,tabf

Author

Gus Wiseman, Jan 08 2019

Extensions

More terms from Jinyuan Wang, Feb 27 2025

Name edited by Peter Munn, Mar 06 2025

Status

approved