OFFSET
0,4
COMMENTS
The Heinz numbers of these partitions are given by A325792.
The number of submultisets of an integer partition is the product of its multiplicities, each plus one.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
EXAMPLE
The 12 = 11 + 1 submultisets of the partition (4331) are: (), (1), (3), (4), (31), (33), (41), (43), (331), (431), (433), (4331), so (4331) is counted under a(11).
The a(5) = 3 through a(11) = 12 partitions:
221 111111 421 3311 22221 1111111111 4322
311 2221 11111111 51111 4331
11111 4111 111111111 4421
1111111 5411
6221
6311
7211
33311
44111
222221
611111
11111111111
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[#])-1==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, r = Quotient[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 11 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 25 2019
STATUS
approved