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A325832 Number of integer partitions of n whose number of submultisets is greater than or equal to n.

%I A325832 #11 May 16 2021 12:24:44
%S A325832 1,1,2,2,3,4,6,8,13,16,22,35,50,58,85,120,162,199,267,347,462,592,773,
%T A325832 1006,1293,1504,1929,2455,3081,3859,4815,5953,7363,8737,10743,13193,
%U A325832 16102,19241,23413,28344,34260,40911,49197,58917,70515,84055,100070,118914
%N A325832 Number of integer partitions of n whose number of submultisets is greater than or equal to n.
%C A325832 The number of submultisets of a partition is the product of its multiplicities, each plus one.
%C A325832 The Heinz numbers of these partitions are given by A325796.
%H A325832 Alois P. Heinz, <a href="/A325832/b325832.txt">Table of n, a(n) for n = 0..500</a>
%F A325832 a(n) = A000041(n) - A325833(n).
%F A325832 For n even, a(n) = A325831(n) + A325830(n/2); for n odd, a(n) = A325831(n).
%e A325832 The a(1) = 1 through a(8) = 13 partitions:
%e A325832   (1)  (2)   (21)   (31)    (221)    (321)     (421)      (431)
%e A325832        (11)  (111)  (211)   (311)    (411)     (2221)     (521)
%e A325832                     (1111)  (2111)   (2211)    (3211)     (3221)
%e A325832                             (11111)  (3111)    (4111)     (3311)
%e A325832                                      (21111)   (22111)    (4211)
%e A325832                                      (111111)  (31111)    (5111)
%e A325832                                                (211111)   (22211)
%e A325832                                                (1111111)  (32111)
%e A325832                                                           (41111)
%e A325832                                                           (221111)
%e A325832                                                           (311111)
%e A325832                                                           (2111111)
%e A325832                                                           (11111111)
%p A325832 b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
%p A325832       `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
%p A325832       (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
%p A325832     end:
%p A325832 a:= n-> combinat[numbpart](n)-add(b(n$2, k), k=0..n-1):
%p A325832 seq(a(n), n=0..55);  # _Alois P. Heinz_, Aug 17 2019
%t A325832 Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])>=n&]],{n,0,30}]
%t A325832 (* Second program: *)
%t A325832 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, Function [w, b[w, Min[w, i - 1], p/(j + 1)]][n - i*j], 0], {j, 0, n/i}]];
%t A325832 a[n_] := PartitionsP[n] - Sum[b[n, n, k], {k, 0, n - 1}];
%t A325832 Table[a[n], {n, 0, 55}] (* _Jean-François Alcover_, May 16 2021, after _Alois P. Heinz_ *)
%Y A325832 Cf. A002033, A098859, A108917, A126796, A325694, A325792, A325796, A325828, A325830, A325831, A325833, A325834, A325836.
%K A325832 nonn
%O A325832 0,3
%A A325832 _Gus Wiseman_, May 25 2019