A327712 - OEIS

%I #16 Jul 17 2023 14:48:40

%S 1,1,1,3,3,9,29,57,135,615,2635,6273,25151,82623,525281,2941047,

%T 9100709,38766777,205155713,902705793,7714938567,52987356783,

%U 204844103977,1042657233471,5520661314689,38159472253821,211945677298567,2404720648663335,19773733727088813

%N 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 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 Alois P. Heinz, <a href="/A327712/b327712.txt">Table of n, a(n) for n = 0..706</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%p with(combinat):

%p a:= n-> add(multinomial(n-nops(p), map(x-> x-1, p)[], 0), p=map(h->

%p     permute(h)[], select(l-> nops(l)=nops({l[]}), partition(n)))):

%p seq(a(n), n=0..28);

%p # second Maple program:

%p a:= proc(m) option remember; local b; b:=

%p       proc(n, i, j) option remember; `if`(i*(i+1)/2>=n,

%p        `if`(n=0, (m-j)!*j!, b(n, i-1, j)+

%p         b(n-i, min(n-i, i-1), j+1)/(i-1)!), 0)

%p       end: b(m$2, 0):

%p     end:

%p seq(a(n), n=0..28);

%t 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 a /@ Range[0, 28] (* _Jean-François Alcover_, May 10 2020, after 2nd Maple program *)

%Y Cf. A026898, A326493, A327711, A364281.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Sep 22 2019