|
|
A325828
|
|
Number of integer partitions of n having exactly n + 1 submultisets.
|
|
12
|
|
|
1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 12, 1, 3, 4, 21, 1, 14, 1, 18, 4, 3, 1, 116, 3, 3, 12, 25, 1, 40, 1, 271, 4, 3, 4, 325, 1, 3, 4, 295, 1, 56, 1, 36, 47, 3, 1, 3128, 4, 32, 4, 44, 1, 407, 4, 566, 4, 3, 1, 1598, 1, 3, 65, 10656, 5, 90, 1, 54, 4, 84, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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):
|
|
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];
|
|
CROSSREFS
|
Cf. A002033, A098859, A126796, A188431, A325694, A325792, A325793, A325830, A325831, A325832, A325833, A325834, A325836.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|