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A326322
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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.
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7
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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, 1, 1, 33, 1459, 19353, 32951, 4683, 64, 1, 1, 65, 8263, 395793, 2699251, 1451723, 47293, 128, 1, 1, 129, 48115, 8718945, 262131251, 650553183, 87054773, 545835, 256
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OFFSET
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0,6
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COMMENTS
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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
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Vol. I, second edition, Example 3.18.3d page 322.
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LINKS
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FORMULA
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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
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EXAMPLE
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A(2,2) = M(2; 2)^2 + M(2; 1,1)^2 = 1 + 4 = 5.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
2, 3, 5, 9, 17, 33, ...
4, 13, 55, 271, 1459, 8263, ...
8, 75, 1077, 19353, 395793, 8718945, ...
16, 541, 32951, 2699251, 262131251, 28076306251, ...
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MAPLE
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b:= proc(n, k) option remember; `if`(n=0, 1,
add(b(n-i, k)/i!^k, i=1..n))
end:
A:= (n, k)-> n!^k*b(n, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
add(binomial(n, j)^k*A(j, k), j=0..n-1))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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A[n_, k_] := A[n, k] = If[n==0, 1, Sum[Binomial[n, j]^k A[j, k], {j, 0, n-1}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 03 2020, after 2nd Maple program *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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