OFFSET
0,3
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 A325796.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
EXAMPLE
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (21) (31) (221) (321) (421) (431)
(11) (111) (211) (311) (411) (2221) (521)
(1111) (2111) (2211) (3211) (3221)
(11111) (3111) (4111) (3311)
(21111) (22111) (4211)
(111111) (31111) (5111)
(211111) (22211)
(1111111) (32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
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-> combinat[numbpart](n)-add(b(n$2, k), k=0..n-1):
seq(a(n), n=0..55); # Alois P. Heinz, Aug 17 2019
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])>=n&]], {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, Function [w, b[w, Min[w, i - 1], p/(j + 1)]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := PartitionsP[n] - Sum[b[n, n, k], {k, 0, n - 1}];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 25 2019
STATUS
approved