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Number of integer partitions of n whose number of submultisets is less than or equal to n.
10

%I #15 May 10 2021 07:41:05

%S 0,0,1,1,3,3,6,7,12,14,21,21,37,43,51,56,90,98,130,143,180,200,230,

%T 249,403,454,508,555,657,706,826,889,1295,1406,1568,1690,2194,2396,

%U 2603,2841,3387,3672,4024,4344,4693,5079,5489,5840,9731,10424,11336,12093

%N Number of integer partitions of n whose number of submultisets is less than or equal to n.

%C The number of submultisets of a partition is the product of its multiplicities, each plus one.

%C The Heinz numbers of these partitions are given by A325798.

%H Alois P. Heinz, <a href="/A325834/b325834.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = A000041(n) - A325831(n).

%F For n even, A325833(n) = a(n) - A325830(n/2); for n odd, A325833(n) = a(n).

%e The a(2) = 1 through a(9) = 14 partitions:

%e (2) (3) (4) (5) (6) (7) (8) (9)

%e (22) (32) (33) (43) (44) (54)

%e (31) (41) (42) (52) (53) (63)

%e (51) (61) (62) (72)

%e (222) (322) (71) (81)

%e (411) (331) (332) (333)

%e (511) (422) (432)

%e (431) (441)

%e (521) (522)

%e (611) (531)

%e (2222) (621)

%e (5111) (711)

%e (3222)

%e (6111)

%p b:= proc(n, i, p) option remember; `if`(n=0 or i=1,

%p `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,

%p (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))

%p end:

%p a:= n-> add(b(n$2, k), k=0..n):

%p seq(a(n), n=0..55); # _Alois P. Heinz_, Aug 17 2019

%t Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])<=n&]],{n,0,30}]

%t (* Second program: *)

%t 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], Quotient[p, j + 1]]][n - i*j], 0], {j, 0, n/i}]];

%t a[n_] := Sum[b[n, n, k], {k, 0, n}];

%t a /@ Range[0, 55] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)

%Y Cf. A002033, A088880, A088881, A108917, A126796, A307699, A325694, A325792, A325798, A325828, A325830, A325831, A325832, A325833, A325836.

%K nonn

%O 0,5

%A _Gus Wiseman_, May 29 2019