A325828 Number of integer partitions of n having exactly n + 1 submultisets.

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

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

Cf. A002033, A098859, A126796, A188431, A325694, A325792, A325793, A325830, A325831, A325832, A325833, A325834, A325836.

Sequence in context: A368198 A108466 A211110 * A200780 A338899 A362071

Adjacent sequences: A325825 A325826 A325827 * A325829 A325830 A325831

Keyword

nonn

Author

Gus Wiseman, May 25 2019

Status

approved