OFFSET
0,8
COMMENTS
Number of upper triangular n X n {0,1,...,k}-matrices with no zero rows or columns.
LINKS
Alois P. Heinz, Antidiagonals n = 0..53, flattened
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; arXiv preprint, arXiv:1106.2261 [math.CO], 2011.
Wikipedia, Peter C. Fishburn
FORMULA
A(n,k) = [x^n] Sum_{j=0..n} x^j * Product_{i=1..j} ((k+1)^i-1)/(1+x*((k+1)^i-1)).
EXAMPLE
A(2,3) = 3*3*4 = 36:
[10] [10] [10] [20] [20] [20] [30] [30] [30]
[ 1] [ 2] [ 3] [ 1] [ 2] [ 3] [ 1] [ 2] [ 3]
.
[11] [11] [11] [21] [21] [21] [31] [31] [31]
[ 1] [ 2] [ 3] [ 1] [ 2] [ 3] [ 1] [ 2] [ 3]
.
[12] [12] [12] [22] [22] [22] [32] [32] [32]
[ 1] [ 2] [ 3] [ 1] [ 2] [ 3] [ 1] [ 2] [ 3]
.
[13] [13] [13] [23] [23] [23] [33] [33] [33]
[ 1] [ 2] [ 3] [ 1] [ 2] [ 3] [ 1] [ 2] [ 3]
.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 2, 12, 36, 80, 150, ...
0, 10, 264, 2052, 9280, 30750, ...
0, 122, 19632, 505764, 5684480, 39378750, ...
0, 3346, 4606752, 511718148, 17672135680, 305416893750, ...
...
MAPLE
A:= (n, k)-> coeff(series(add(x^j*mul(((k+1)^i-1)/(1+x*
((k+1)^i-1)), i=1..j), j=0..n), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..10);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jan 22 2024
STATUS
approved