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

%I A325833 #12 May 12 2021 06:44:18
%S A325833 0,0,0,1,2,3,5,7,9,14,20,21,27,43,50,56,69,98,118,143,165,200,229,249,
%T A325833 282,454,507,555,637,706,789,889,986,1406,1567,1690,1875,2396,2602,
%U A325833 2841,3078,3672,3977,4344,4660,5079,5488,5840,6296,10424,11306
%N A325833 Number of integer partitions of n whose number of submultisets is less than n.
%C A325833 The number of submultisets of a partition is the product of its multiplicities, each plus one.
%C A325833 The Heinz numbers of these partitions are given by A325797.
%H A325833 Alois P. Heinz, <a href="/A325833/b325833.txt">Table of n, a(n) for n = 0..500</a>
%F A325833 a(n) = A000041(n) - A325832(n).
%F A325833 For n even, a(n) = A325834(n) - A325830(n/2); for n odd, a(n) = A325834(n).
%e A325833 The a(3) = 1 through a(9) = 14 partitions:
%e A325833   (3)  (4)   (5)   (6)    (7)    (8)     (9)
%e A325833        (22)  (32)  (33)   (43)   (44)    (54)
%e A325833              (41)  (42)   (52)   (53)    (63)
%e A325833                    (51)   (61)   (62)    (72)
%e A325833                    (222)  (322)  (71)    (81)
%e A325833                           (331)  (332)   (333)
%e A325833                           (511)  (422)   (432)
%e A325833                                  (611)   (441)
%e A325833                                  (2222)  (522)
%e A325833                                          (531)
%e A325833                                          (621)
%e A325833                                          (711)
%e A325833                                          (3222)
%e A325833                                          (6111)
%p A325833 b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
%p A325833       `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
%p A325833       (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
%p A325833     end:
%p A325833 a:= n-> add(b(n$2, k), k=0..n-1):
%p A325833 seq(a(n), n=0..55);  # _Alois P. Heinz_, Aug 17 2019
%t A325833 Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])<n&]],{n,0,30}]
%t A325833 (* Second program: *)
%t A325833 b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
%t A325833      If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = p/(j + 1);
%t A325833      Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
%t A325833 a[n_] := Sum[b[n, n, k], {k, 0, n - 1}];
%t A325833 a /@ Range[0, 55] (* _Jean-François Alcover_, May 12 2021, after _Alois P. Heinz_ *)
%Y A325833 Cf. A002033, A088880, A088881, A098859, A108917, A307699, A325694, A325792, A325797, A325828, A325830, A325831, A325832, A325834, A325836.
%K A325833 nonn
%O A325833 0,5
%A A325833 _Gus Wiseman_, May 29 2019