A292795 Number A(n,k) of sets 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, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 7, 2, 0, 1, 1, 3, 13, 18, 3, 0, 1, 1, 3, 13, 36, 42, 4, 0, 1, 1, 3, 13, 60, 122, 110, 5, 0, 1, 1, 3, 13, 60, 206, 433, 250, 6, 0, 1, 1, 3, 13, 60, 326, 865, 1356, 627, 8, 0, 1, 1, 3, 13, 60, 326, 1345, 3408, 4449, 1439, 10, 0

Offset

0,13

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Formula

G.f. of column k: Product_{j>=1} (1+x^j)^A226873(j,k).

A(n,k) = Sum_{j=0..n} A319498(n,j).

Example

A(2,3) = 3: {aa}, {ab}, {ba}.
A(3,2) = 7: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}.
A(3,3) = 13: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,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, 1,   3,    3,     3,     3,     3,     3,      3, ...
  0, 2,   7,   13,    13,    13,    13,    13,     13, ...
  0, 2,  18,   36,    60,    60,    60,    60,     60, ...
  0, 3,  42,  122,   206,   326,   326,   326,    326, ...
  0, 4, 110,  433,   865,  1345,  2065,  2065,   2065, ...
  0, 5, 250, 1356,  3408,  6228,  9468, 14508,  14508, ...
  0, 6, 627, 4449, 15025, 29845, 51325, 76525, 116845, ...

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)):
h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))
    end:
A:= (n, k)-> h(n$2, min(n, k)):
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]];
h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[h[n - i*j, i - 1, k]*Binomial[g[i, k], j], {j, 0, n/i}]]];
A[n_, k_] := h[n, n, Min[n, k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten(* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000007, A000009, A340409, A340410, A340411, A340412, A340413, A340414, A340415, A340416, A340417.

Rows n=0-1 give: A000012, A057427.

Main diagonal gives A292796.

Cf. A226873, A292712, A292804, A319498.

Sequence in context: A290216 A293202 A280265 * A295028 A294201 A079618

Adjacent sequences: A292792 A292793 A292794 * A292796 A292797 A292798

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 23 2017

Status

approved