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A226874 Number T(n,k) of n-length words w over a k-ary alphabet {a1, a2, ..., ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows. 18
1, 0, 1, 0, 1, 2, 0, 1, 3, 6, 0, 1, 10, 12, 24, 0, 1, 15, 50, 60, 120, 0, 1, 41, 180, 300, 360, 720, 0, 1, 63, 497, 1260, 2100, 2520, 5040, 0, 1, 162, 1484, 6496, 10080, 16800, 20160, 40320, 0, 1, 255, 5154, 20916, 58464, 90720, 151200, 181440, 362880 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

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

FORMULA

T(n,k) = A226873(n,k) - A226873(n,k-1) for k > 0, T(n,0) = A226873(n,0).

EXAMPLE

T(4,2) = 10: aaab, aaba, aabb, abaa, abab, abba, baaa, baab, baba, bbaa.

T(4,3) = 12: aabc, aacb, abac, abca, acab, acba, baac, baca, bcaa, caab, caba, cbaa.

T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa.

Triangle T(n,k) begins:

  1;

  0,  1;

  0,  1,   2;

  0,  1,   3,    6;

  0,  1,  10,   12,   24;

  0,  1,  15,   50,   60,   120;

  0,  1,  41,  180,  300,   360,   720;

  0,  1,  63,  497, 1260,  2100,  2520,  5040;

  0,  1, 162, 1484, 6496, 10080, 16800, 20160, 40320;

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:

T:= (n, k)-> `if`(n*k=0, `if`(n=k, 1, 0), n!*b(n, 1, k)):

seq(seq(T(n, k), k=0..n), n=0..12);

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}]]; t[n_, k_] := If[n*k == 0, If[n == k, 1, 0], n!*b[n, 1, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Dec 13 2013, translated from Maple *)

PROG

(PARI)

T(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}

{my(t=T(10)); for(n=1, #t, for(k=0, n-1, print1(polcoeff(t[n], k), ", ")); print)} \\ Andrew Howroyd, Dec 20 2017

CROSSREFS

Columns k=0-10 give: A000007, A057427, A226881, A226882, A226883, A226884, A226885, A226886, A226887, A226888, A226889.

Main diagonal gives: A000142.

Row sums give: A005651.

T(2n,n) gives A318796.

Cf. A131632, A285824.

The sub-triangle T(n,k), for n >= k >= 1, is A292222.

Sequence in context: A195772 A062104 A257783 * A267901 A276561 A153506

Adjacent sequences:  A226871 A226872 A226873 * A226875 A226876 A226877

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 21 2013

STATUS

approved

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Last modified January 17 19:58 EST 2019. Contains 319251 sequences. (Running on oeis4.)