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A226873 Number A(n,k) of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w; square array A(n,k), n>=0, k>=0, read by antidiagonals. 27
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 4, 1, 0, 1, 1, 3, 10, 11, 1, 0, 1, 1, 3, 10, 23, 16, 1, 0, 1, 1, 3, 10, 47, 66, 42, 1, 0, 1, 1, 3, 10, 47, 126, 222, 64, 1, 0, 1, 1, 3, 10, 47, 246, 522, 561, 163, 1, 0, 1, 1, 3, 10, 47, 246, 882, 1821, 1647, 256, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

LINKS

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

FORMULA

A(n,k) = Sum_{i=0..min(n,k)} A226874(n,i).

EXAMPLE

A(4,3) = 23: aaaa, aaab, aaba, aabb, aabc, aacb, abaa, abab, abac, abba, abca, acab, acba, baaa, baab, baac, baba, baca, bbaa, bcaa, caab, caba, cbaa.

Square array A(n,k) begins:

  1, 1,  1,   1,    1,    1,    1,     1, ...

  0, 1,  1,   1,    1,    1,    1,     1, ...

  0, 1,  3,   3,    3,    3,    3,     3, ...

  0, 1,  4,  10,   10,   10,   10,    10, ...

  0, 1, 11,  23,   47,   47,   47,    47, ...

  0, 1, 16,  66,  126,  246,  246,   246, ...

  0, 1, 42, 222,  522,  882, 1602,  1602, ...

  0, 1, 64, 561, 1821, 3921, 6441, 11481, ...

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:

A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, 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}]]; a[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Dec 13 2013, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000007, A000012, A027306, A092255, A092429, A226875, A226876, A226877, A226878, A226879, A226880.

Main diagonal gives: A005651.

Cf. A131632, A182172.

Sequence in context: A181434 A294018 A192003 * A293960 A062719 A305161

Adjacent sequences:  A226870 A226871 A226872 * A226874 A226875 A226876

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 21 2013

STATUS

approved

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Last modified June 26 21:14 EDT 2022. Contains 354885 sequences. (Running on oeis4.)