A369415 Number A(n,k) of n X n Fishburn matrices with entries in the set {0,1,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 12, 10, 0, 1, 4, 36, 264, 122, 0, 1, 5, 80, 2052, 19632, 3346, 0, 1, 6, 150, 9280, 505764, 4606752, 196082, 0, 1, 7, 252, 30750, 5684480, 511718148, 3311447232, 23869210, 0, 1, 8, 392, 83160, 39378750, 17672135680, 2088275673636, 7202118117504, 5939193962, 0
Offset: 0
Examples
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, ... ...
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
Crossrefs
Programs
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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);
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)).
Comments