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A325836
Number of integer partitions of n having n - 1 different submultisets.
11
0, 0, 0, 1, 1, 2, 0, 3, 0, 5, 2, 2, 0, 15, 0, 2, 3, 25, 0, 17, 0, 18, 3, 2, 0, 150, 0, 2, 13, 24, 0, 43, 0, 351, 3, 2, 2, 383, 0, 2, 3, 341, 0, 60, 0, 37, 51, 2, 0, 3733, 0, 31, 3, 42, 0, 460, 1, 633, 3, 2, 0, 1780, 0, 2, 68, 12460, 0, 87, 0, 55, 3
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
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