A325833 Number of integer partitions of n whose number of submultisets is less than n.

0, 0, 0, 1, 2, 3, 5, 7, 9, 14, 20, 21, 27, 43, 50, 56, 69, 98, 118, 143, 165, 200, 229, 249, 282, 454, 507, 555, 637, 706, 789, 889, 986, 1406, 1567, 1690, 1875, 2396, 2602, 2841, 3078, 3672, 3977, 4344, 4660, 5079, 5488, 5840, 6296, 10424, 11306

Offset

0,5

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 A325797.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Formula

a(n) = A000041(n) - A325832(n).

For n even, a(n) = A325834(n) - A325830(n/2); for n odd, a(n) = A325834(n).

Example

The a(3) = 1 through a(9) = 14 partitions:
  (3)  (4)   (5)   (6)    (7)    (8)     (9)
       (22)  (32)  (33)   (43)   (44)    (54)
             (41)  (42)   (52)   (53)    (63)
                   (51)   (61)   (62)    (72)
                   (222)  (322)  (71)    (81)
                          (331)  (332)   (333)
                          (511)  (422)   (432)
                                 (611)   (441)
                                 (2222)  (522)
                                         (531)
                                         (621)
                                         (711)
                                         (3222)
                                         (6111)

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-> 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, r = p/(j + 1);
     Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := Sum[b[n, n, k], {k, 0, n - 1}];
a /@ Range[0, 55] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)

Crossrefs

Cf. A002033, A088880, A088881, A098859, A108917, A307699, A325694, A325792, A325797, A325828, A325830, A325831, A325832, A325834, A325836.

Sequence in context: A218614 A338360 A028378 * A143587 A096305 A039888

Adjacent sequences: A325830 A325831 A325832 * A325834 A325835 A325836

Keyword

nonn

Author

Gus Wiseman, May 29 2019

Status

approved