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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.

%I A369415 #19 Jan 23 2024 11:45:19
%S A369415 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,
%T A369415 3346,0,1,6,150,9280,505764,4606752,196082,0,1,7,252,30750,5684480,
%U A369415 511718148,3311447232,23869210,0,1,8,392,83160,39378750,17672135680,2088275673636,7202118117504,5939193962,0
%N 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.
%C A369415 Number of upper triangular n X n {0,1,...,k}-matrices with no zero rows or columns.
%H A369415 Alois P. Heinz, <a href="/A369415/b369415.txt">Antidiagonals n = 0..53, flattened</a>
%H A369415 Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, <a href="https://arxiv.org/abs/2012.13570">Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow</a>, arXiv:2012.13570 [math.CO], 2020.
%H A369415 Vít Jelínek, <a href="http://dx.doi.org/10.1016/j.jcta.2011.11.010">Counting general and self-dual interval orders</a>, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; <a href="http://arxiv.org/abs/1106.2261">arXiv preprint</a>, arXiv:1106.2261 [math.CO], 2011.
%H A369415 Wikipedia, <a href="https://en.wikipedia.org/wiki/Peter_C._Fishburn">Peter C. Fishburn</a>
%F A369415 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)).
%e A369415 A(2,3) = 3*3*4 = 36:
%e A369415   [10] [10] [10]  [20] [20] [20]  [30] [30] [30]
%e A369415   [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
%e A369415 .
%e A369415   [11] [11] [11]  [21] [21] [21]  [31] [31] [31]
%e A369415   [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
%e A369415 .
%e A369415   [12] [12] [12]  [22] [22] [22]  [32] [32] [32]
%e A369415   [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
%e A369415 .
%e A369415   [13] [13] [13]  [23] [23] [23]  [33] [33] [33]
%e A369415   [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
%e A369415 .
%e A369415 Square array A(n,k) begins:
%e A369415   1,    1,       1,         1,           1,            1, ...
%e A369415   0,    1,       2,         3,           4,            5, ...
%e A369415   0,    2,      12,        36,          80,          150, ...
%e A369415   0,   10,     264,      2052,        9280,        30750, ...
%e A369415   0,  122,   19632,    505764,     5684480,     39378750, ...
%e A369415   0, 3346, 4606752, 511718148, 17672135680, 305416893750, ...
%e A369415   ...
%p A369415 A:= (n, k)-> coeff(series(add(x^j*mul(((k+1)^i-1)/(1+x*
%p A369415            ((k+1)^i-1)), i=1..j), j=0..n), x, n+1), x, n):
%p A369415 seq(seq(A(n, d-n), n=0..d), d=0..10);
%Y A369415 Columns k=0-3 give: A000007, A005321, A289314, A289315.
%Y A369415 Rows n=0-3 give: A000012, A001477, A011379, A369423.
%Y A369415 Main diagonal gives A369336.
%K A369415 nonn,tabl
%O A369415 0,8
%A A369415 _Alois P. Heinz_, Jan 22 2024