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 A292712 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that within each 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, 2, 0, 1, 1, 4, 3, 0, 1, 1, 4, 8, 5, 0, 1, 1, 4, 14, 25, 7, 0, 1, 1, 4, 14, 43, 53, 11, 0, 1, 1, 4, 14, 67, 139, 148, 15, 0, 1, 1, 4, 14, 67, 223, 495, 328, 22, 0, 1, 1, 4, 14, 67, 343, 951, 1544, 858, 30, 0, 1, 1, 4, 14, 67, 343, 1431, 3680, 5111, 1938, 42, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA G.f. of column k: Product_{j>=1} 1/(1-x^j)^A226873(j,k). A(n,n) = A(n,k) for all k >= n. A(n,k) = Sum_{j=0..n} A319495(n,j). EXAMPLE A(2,3) = 4: {aa}, {ab}, {ba}, {a,a}. A(3,2) = 8: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}. A(3,3) = 14: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}. Square array A(n,k) begins:   1,  1,   1,    1,     1,     1,     1,     1,      1, ...   0,  1,   1,    1,     1,     1,     1,     1,      1, ...   0,  2,   4,    4,     4,     4,     4,     4,      4, ...   0,  3,   8,   14,    14,    14,    14,    14,     14, ...   0,  5,  25,   43,    67,    67,    67,    67,     67, ...   0,  7,  53,  139,   223,   343,   343,   343,    343, ...   0, 11, 148,  495,   951,  1431,  2151,  2151,   2151, ...   0, 15, 328, 1544,  3680,  6620,  9860, 14900,  14900, ...   0, 22, 858, 5111, 16239, 31539, 53739, 78939, 119259, ... MAPLE b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,       add(b(n-j, j, t-1)/j!, j=i..n/t))     end: g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)): A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*       g(d, k), d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)     end: seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]]; g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]]; A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*g[d, k], {d, Divisors[j]}]* A[n - j, k], {j, 1, n}]/n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 07 2018, from Maple *) CROSSREFS Columns k=0-10 give: A000007, A000041, A292548, A292718, A292719, A292720, A292721, A292722, A292723, A292724, A292725. Rows n=0-1 give: A000012, A057427. Main diagonal gives A292713. Cf. A226873, A292795, A319495. Sequence in context: A049501 A102564 A215703 * A331571 A247504 A306800 Adjacent sequences:  A292709 A292710 A292711 * A292713 A292714 A292715 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 21 2017 STATUS approved

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Last modified September 23 21:27 EDT 2020. Contains 337315 sequences. (Running on oeis4.)