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

%I A325828 #16 May 11 2021 06:14:25
%S A325828 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,
%T A325828 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,
%U A325828 566,4,3,1,1598,1,3,65,10656,5,90,1,54,4,84,1
%N A325828 Number of integer partitions of n having exactly n + 1 submultisets.
%C A325828 The Heinz numbers of these partitions are given by A325792.
%C A325828 The number of submultisets of an integer partition is the product of its multiplicities, each plus one.
%H A325828 Alois P. Heinz, <a href="/A325828/b325828.txt">Table of n, a(n) for n = 0..1000</a>
%e A325828 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).
%e A325828 The a(5) = 3 through a(11) = 12 partitions:
%e A325828   221    111111  421      3311      22221      1111111111  4322
%e A325828   311            2221     11111111  51111                  4331
%e A325828   11111          4111               111111111              4421
%e A325828                  1111111                                   5411
%e A325828                                                            6221
%e A325828                                                            6311
%e A325828                                                            7211
%e A325828                                                            33311
%e A325828                                                            44111
%e A325828                                                            222221
%e A325828                                                            611111
%e A325828                                                            11111111111
%p A325828 b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
%p A325828       `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
%p A325828       (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
%p A325828     end:
%p A325828 a:= n-> b(n$2,n+1):
%p A325828 seq(a(n), n=0..80);  # _Alois P. Heinz_, Aug 17 2019
%t A325828 Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])-1==n&]],{n,0,30}]
%t A325828 (* Second program: *)
%t A325828 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}]];
%t A325828 a[n_] := b[n, n, n+1];
%t A325828 a /@ Range[0, 80] (* _Jean-François Alcover_, May 11 2021, after _Alois P. Heinz_ *)
%Y A325828 Cf. A002033, A098859, A126796, A188431, A325694, A325792, A325793, A325830, A325831, A325832, A325833, A325834, A325836.
%K A325828 nonn
%O A325828 0,4
%A A325828 _Gus Wiseman_, May 25 2019