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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. 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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 REFERENCES R. P. Stanley, Enumerative Combinatorics, Vol. I, second edition, Example 3.18.3d page 322. LINKS Alois P. Heinz, Antidiagonals n = 0..60, flattened Wikipedia, Multinomial coefficients FORMULA 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 EXAMPLE 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, ... MAPLE 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); MATHEMATICA 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 *) CROSSREFS Columns k=0-2 give: A011782, A000670, A102221. Rows n=0+1, 2 give A000012, A000051. Main diagonal gives A326321. Cf. A183610. Sequence in context: A320251 A210341 A160449 * A089940 A331598 A123974 Adjacent sequences: A326319 A326320 A326321 * A326323 A326324 A326325 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 11 2019 STATUS approved

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Last modified September 19 09:19 EDT 2024. Contains 376007 sequences. (Running on oeis4.)