|
|
A261835
|
|
Number A(n,k) of compositions of n into distinct parts where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.
|
|
13
|
|
|
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 16, 3, 0, 1, 5, 10, 46, 21, 5, 0, 1, 6, 15, 100, 75, 50, 11, 0, 1, 7, 21, 185, 195, 231, 205, 13, 0, 1, 8, 28, 308, 420, 736, 1414, 292, 19, 0, 1, 9, 36, 476, 798, 1876, 6032, 2376, 587, 27, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,8
|
|
COMMENTS
|
Also matrices with k rows of nonnegative integers with distinct positive column sums and total element sum n.
A(2,2) = 3: (matrices and corresponding marked compositions are given)
[1] [2] [0]
[1] [0] [2]
2ab, 2aa, 2bb.
|
|
LINKS
|
|
|
FORMULA
|
A(n,k) = Sum_{i=0..k} C(k,i) * A261836(n,k-i).
|
|
EXAMPLE
|
A(3,2) = 16: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a2aa, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1b2bb.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, ...
0, 1, 3, 6, 10, 15, 21, 28, ...
0, 3, 16, 46, 100, 185, 308, 476, ...
0, 3, 21, 75, 195, 420, 798, 1386, ...
0, 5, 50, 231, 736, 1876, 4116, 8106, ...
0, 11, 205, 1414, 6032, 19320, 51114, 117936, ...
0, 13, 292, 2376, 11712, 42610, 126288, 322764, ...
|
|
MAPLE
|
b:= proc(n, i, p, k) option remember;
`if`(i*(i+1)/2<n, 0, `if`(n=0, p!, b(n, i-1, p, k)+
`if`(i>n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))
end:
A:= (n, k)-> b(n$2, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
|
|
MATHEMATICA
|
b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2 < n, 0, If[n == 0, p!, b[n, i-1, p, k] + If[i>n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; A[n_, k_] := b[n, n, 0, k]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 16 2017, translated from Maple *)
|
|
CROSSREFS
|
Columns k=0-10 give: A000007, A032020, A261840, A261841, A261842, A261843, A261844, A261845, A261846, A261847, A261848.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|