A369336 - OEIS

%I #16 Jan 22 2024 17:52:01

%S 1,1,12,2052,5684480,305416893750,391129148721673152,

%T 14286237711414132094989064,17309880507327972883933887341789184,

%U 792117985317303404452447777723478865406570410,1534214120588806182890487155420702132205591283310000000000

%N Number of n X n Fishburn matrices with entries in the set {0,1,...,n}.

%C Number of upper triangular n X n {0,1,...,n}-matrices with no zero rows or columns.

%H Alois P. Heinz, <a href="/A369336/b369336.txt">Table of n, a(n) for n = 0..35</a>

%H 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Peter_C._Fishburn">Peter C. Fishburn</a>

%F a(n) = [x^n] Sum_{j=0..n} x^j * Product_{i=1..j} ((n+1)^i-1)/(1+x*((n+1)^i-1)).

%e a(0) = 1: [].

%e a(1) = 1: [1].

%e a(2) = 12:

%e   [10] [10] [20] [20]  [11] [11] [21] [21]  [12] [12] [22] [22]

%e   [ 1] [ 2] [ 1] [ 2]  [ 1] [ 2] [ 1] [ 2]  [ 1] [ 2] [ 1] [ 2].

%p a:= n-> coeff(series(add(x^j*mul(((n+1)^i-1)/(1+x*

%p     ((n+1)^i-1)), i=1..j), j=0..n), x, n+1), x, n):

%p seq(a(n), n=0..10);

%Y Cf. A000007, A005321, A289314, A289315.

%Y Main diagonal of A369415.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 20 2024