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 A293112 Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; square array A(n,k), n>=0, k>=0, read by antidiagonals. 13
 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 5, 2, 0, 1, 1, 2, 6, 10, 3, 0, 1, 1, 2, 6, 14, 23, 4, 0, 1, 1, 2, 6, 15, 39, 51, 5, 0, 1, 1, 2, 6, 15, 44, 104, 111, 6, 0, 1, 1, 2, 6, 15, 45, 129, 284, 243, 8, 0, 1, 1, 2, 6, 15, 45, 135, 386, 775, 530, 10, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 LINKS Alois P. Heinz, Antidiagonals n = 0..80, flattened FORMULA G.f. of column k: Product_{j>=1} (1+x^j)^A182172(j,k). A(n,k) = Sum_{j=0..k} A293113(n,j). EXAMPLE Square array A(n,k) begins:   1, 1,   1,   1,   1,   1,   1,   1, ...   0, 1,   1,   1,   1,   1,   1,   1, ...   0, 1,   2,   2,   2,   2,   2,   2, ...   0, 2,   5,   6,   6,   6,   6,   6, ...   0, 2,  10,  14,  15,  15,  15,  15, ...   0, 3,  23,  39,  44,  45,  45,  45, ...   0, 4,  51, 104, 129, 135, 136, 136, ...   0, 5, 111, 284, 386, 422, 429, 430, ... MAPLE h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]     n, 0, g(n-i, i, [l[], i])))))     end: b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(b(n-i*j, i-1, k)*binomial(g(i, k, []), j), j=0..n/i)))     end: A:= (n, k)-> b(n\$2, k): seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]] n, 0, g[n - i, i, Append[l, i]]]]]]; b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*Binomial[g[i, k, {}], j], {j, 0, n/i}]]]; A[n_, k_] := b[n, n, k]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *) CROSSREFS Columns k=0-10 give: A000007, A000009, A293741, A293742, A293743, A293744, A293745, A293746, A293747, A293748, A293749. Main diagonal gives A293114. Cf. A182172, A293108, A293113. Sequence in context: A255636 A292085 A262163 * A306910 A112185 A192062 Adjacent sequences:  A293109 A293110 A293111 * A293113 A293114 A293115 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 30 2017 STATUS approved

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Last modified May 9 20:41 EDT 2021. Contains 343746 sequences. (Running on oeis4.)