A226878 -id:A226878 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Jun 21 2013

			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, ...

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

Columns k=0-10 give: A000007, A000012, A027306, A092255, A092429, A226875, A226876, A226877, A226878, A226879, A226880.
Main diagonal gives: A005651.
Cf. A131632, A182172.

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);

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-François Alcover, Dec 13 2013, translated from Maple *)

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

A292723 Number of multisets of nonempty words with a total of n letters over 8-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 4, 14, 67, 343, 2151, 14900, 119259, 692640, 4659774, 30077836, 209311030, 1433872892, 10689029713, 76772260527, 600293120366, 4142024767610, 30775147154084, 221566161988587, 1663421685691847, 12221938274124959, 93706886872251562, 696726353909296853

Offset: 0

Alois P. Heinz, Sep 21 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=8 of A292712.

Cf. A226878, A226873.

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:= proc(n) option remember; `if`(n=0, 1, add(add(d*d!*
      b(d, 0, 8), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

G.f.: Product_{j>=1} 1/(1-x^j)^A226878(j).

Euler transform of A226878.

A340415 Number of sets of nonempty words with a total of n letters over octonary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 3, 13, 60, 326, 2065, 14508, 116845, 676579, 4533285, 29337447, 204274255, 1401597565, 10464806200, 75242714351, 588938921227, 4060713617519, 30141138974325, 217182619165093, 1630762746458645, 11987353708674543, 91946531392941646, 683807822490949653

Offset: 0

Alois P. Heinz, Jan 06 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=8 of A292795.

Cf. A226878.

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-> h(n$2, min(n, 8)):
seq(a(n), n=0..32);

G.f.: Product_{j>=1} (1+x^j)^A226878(j).

cp's OEIS Frontend

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.

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A292723 Number of multisets of nonempty words with a total of n letters over 8-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

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A340415 Number of sets of nonempty words with a total of n letters over octonary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

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Maple

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