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A326322 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) = sum of the k-th powers of multinomials M(n; mu), where mu ranges over all compositions of n.

%I A326322 #37 Jan 28 2021 14:42:59
%S A326322 1,1,1,1,1,2,1,1,3,4,1,1,5,13,8,1,1,9,55,75,16,1,1,17,271,1077,541,32,
%T A326322 1,1,33,1459,19353,32951,4683,64,1,1,65,8263,395793,2699251,1451723,
%U A326322 47293,128,1,1,129,48115,8718945,262131251,650553183,87054773,545835,256
%N A326322 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) = sum of the k-th powers of multinomials M(n; mu), where mu ranges over all compositions of n.
%C A326322 For k>=1, A(n,k) is the number of k-tuples (p_1,p_2,...,p_k) of ordered set partitions of [n] such that the sequence of block lengths in each ordered partition p_i is identical.  Equivalently, A(n,k) is the number of chains from s to t where [s,t] is any n-interval in the binomial poset B_k = B*B*...*B (k times),  B is the lattice of all finite subsets of {1,2,...} ordered by inclusion and * is the Segre product.  See Stanley reference. - _Geoffrey Critzer_, Dec 16 2020
%D A326322 R. P. Stanley, Enumerative Combinatorics, Vol. I, second edition, Example 3.18.3d page 322.
%H A326322 Alois P. Heinz, <a href="/A326322/b326322.txt">Antidiagonals n = 0..60, flattened</a>
%H A326322 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%F A326322 Let E_k(x) = Sum_{n>=0} x^n/n!^k.  Then 1/(2-E_k(x)) = Sum_{n>=0} A(n,k)*x^n/n!^k. - _Geoffrey Critzer_, Dec 16 2020
%e A326322 A(2,2) = M(2; 2)^2 + M(2; 1,1)^2 = 1 + 4 = 5.
%e A326322 Square array A(n,k) begins:
%e A326322    1,   1,     1,       1,         1,           1, ...
%e A326322    1,   1,     1,       1,         1,           1, ...
%e A326322    2,   3,     5,       9,        17,          33, ...
%e A326322    4,  13,    55,     271,      1459,        8263, ...
%e A326322    8,  75,  1077,   19353,    395793,     8718945, ...
%e A326322   16, 541, 32951, 2699251, 262131251, 28076306251, ...
%p A326322 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A326322       add(b(n-i, k)/i!^k, i=1..n))
%p A326322     end:
%p A326322 A:= (n, k)-> n!^k*b(n, k):
%p A326322 seq(seq(A(n, d-n), n=0..d), d=0..12);
%p A326322 # second Maple program:
%p A326322 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A326322       add(binomial(n, j)^k*A(j, k), j=0..n-1))
%p A326322     end:
%p A326322 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A326322 A[n_, k_] := A[n, k] = If[n==0, 1, Sum[Binomial[n, j]^k A[j, k], {j, 0, n-1}]];
%t A326322 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten  (* _Jean-François Alcover_, Dec 03 2020, after 2nd Maple program *)
%Y A326322 Columns k=0-2 give: A011782, A000670, A102221.
%Y A326322 Rows n=0+1, 2 give A000012, A000051.
%Y A326322 Main diagonal gives A326321.
%Y A326322 Cf. A183610.
%K A326322 nonn,tabl
%O A326322 0,6
%A A326322 _Alois P. Heinz_, Sep 11 2019