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.

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

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