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A327712 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all compositions of n into distinct parts (k is a composition length).

%I A327712 #16 Jul 17 2023 14:48:40
%S A327712 1,1,1,3,3,9,29,57,135,615,2635,6273,25151,82623,525281,2941047,
%T A327712 9100709,38766777,205155713,902705793,7714938567,52987356783,
%U A327712 204844103977,1042657233471,5520661314689,38159472253821,211945677298567,2404720648663335,19773733727088813
%N A327712 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all compositions of n into distinct parts (k is a composition length).
%C A327712 Number of partitions of [n] with distinct block sizes such that each block contains exactly one block size as an element. a(5) = 9: 12345, 1235|4, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 1|2345.
%H A327712 Alois P. Heinz, <a href="/A327712/b327712.txt">Table of n, a(n) for n = 0..706</a>
%H A327712 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H A327712 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A327712 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A327712 with(combinat):
%p A327712 a:= n-> add(multinomial(n-nops(p), map(x-> x-1, p)[], 0), p=map(h->
%p A327712     permute(h)[], select(l-> nops(l)=nops({l[]}), partition(n)))):
%p A327712 seq(a(n), n=0..28);
%p A327712 # second Maple program:
%p A327712 a:= proc(m) option remember; local b; b:=
%p A327712       proc(n, i, j) option remember; `if`(i*(i+1)/2>=n,
%p A327712        `if`(n=0, (m-j)!*j!, b(n, i-1, j)+
%p A327712         b(n-i, min(n-i, i-1), j+1)/(i-1)!), 0)
%p A327712       end: b(m$2, 0):
%p A327712     end:
%p A327712 seq(a(n), n=0..28);
%t A327712 a[m_] := a[m] = Module[{b}, b[n_, i_, j_] := b[n, i, j] = If[i(i + 1)/2 >= n, If[n == 0, (m - j)! j!, b[n, i - 1, j] + b[n - i, Min[n - i, i - 1], j + 1]/(i - 1)!], 0]; b[m, m, 0]];
%t A327712 a /@ Range[0, 28] (* _Jean-François Alcover_, May 10 2020, after 2nd Maple program *)
%Y A327712 Cf. A026898, A326493, A327711, A364281.
%K A327712 nonn
%O A327712 0,4
%A A327712 _Alois P. Heinz_, Sep 22 2019