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A325836 Number of integer partitions of n having n - 1 different submultisets.

%I A325836 #13 May 12 2021 21:03:17
%S A325836 0,0,0,1,1,2,0,3,0,5,2,2,0,15,0,2,3,25,0,17,0,18,3,2,0,150,0,2,13,24,
%T A325836 0,43,0,351,3,2,2,383,0,2,3,341,0,60,0,37,51,2,0,3733,0,31,3,42,0,460,
%U A325836 1,633,3,2,0,1780,0,2,68,12460,0,87,0,55,3
%N A325836 Number of integer partitions of n having n - 1 different submultisets.
%C A325836 The number of submultisets of a partition is the product of its multiplicities, each plus one.
%C A325836 The Heinz numbers of these partitions are given by A325694.
%H A325836 Alois P. Heinz, <a href="/A325836/b325836.txt">Table of n, a(n) for n = 0..1000</a>
%e A325836 The a(3) = 1 through a(13) = 15 partitions (empty columns not shown):
%e A325836   (3)  (22)  (32)  (322)  (432)   (3322)  (32222)  (4432)
%e A325836              (41)  (331)  (531)   (4411)  (71111)  (5332)
%e A325836                    (511)  (621)                    (5422)
%e A325836                           (3222)                   (5521)
%e A325836                           (6111)                   (6322)
%e A325836                                                    (6331)
%e A325836                                                    (6511)
%e A325836                                                    (7411)
%e A325836                                                    (8221)
%e A325836                                                    (8311)
%e A325836                                                    (9211)
%e A325836                                                    (33322)
%e A325836                                                    (55111)
%e A325836                                                    (322222)
%e A325836                                                    (811111)
%p A325836 b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
%p A325836       `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
%p A325836       (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
%p A325836     end:
%p A325836 a:= n-> b(n$2,n-1):
%p A325836 seq(a(n), n=0..80);  # _Alois P. Heinz_, Aug 17 2019
%t A325836 Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==n-1&]],{n,0,30}]
%t A325836 (* Second program: *)
%t A325836 b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
%t A325836      If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = p/(j + 1);
%t A325836      Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
%t A325836 a[n_] := b[n, n, n-1];
%t A325836 a /@ Range[0, 80] (* _Jean-François Alcover_, May 12 2021, after _Alois P. Heinz_ *)
%Y A325836 Positions of zeros are A307699.
%Y A325836 Cf. A002033, A088880, A088881, A108917, A325694, A325768, A325792, A325798, A325828, A325830, A325833, A325835.
%K A325836 nonn
%O A325836 0,6
%A A325836 _Gus Wiseman_, May 29 2019