OFFSET
0,6
COMMENTS
The number of submultisets of a partition is the product of its multiplicities, each plus one.
The Heinz numbers of these partitions are given by A325694.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
EXAMPLE
The a(3) = 1 through a(13) = 15 partitions (empty columns not shown):
(3) (22) (32) (322) (432) (3322) (32222) (4432)
(41) (331) (531) (4411) (71111) (5332)
(511) (621) (5422)
(3222) (5521)
(6111) (6322)
(6331)
(6511)
(7411)
(8221)
(8311)
(9211)
(33322)
(55111)
(322222)
(811111)
MAPLE
b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
`if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
(w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
end:
a:= n-> b(n$2, n-1):
seq(a(n), n=0..80); # Alois P. Heinz, Aug 17 2019
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])==n-1&]], {n, 0, 30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = p/(j + 1);
Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := b[n, n, n-1];
a /@ Range[0, 80] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 29 2019
STATUS
approved