A369336 Number of n X n Fishburn matrices with entries in the set {0,1,...,n}.

1, 1, 12, 2052, 5684480, 305416893750, 391129148721673152, 14286237711414132094989064, 17309880507327972883933887341789184, 792117985317303404452447777723478865406570410, 1534214120588806182890487155420702132205591283310000000000

Offset

0,3

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..35

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

Example

a(0) = 1: [].
a(1) = 1: [1].
a(2) = 12:
  [10] [10] [20] [20]  [11] [11] [21] [21]  [12] [12] [22] [22]
  [ 1] [ 2] [ 1] [ 2]  [ 1] [ 2] [ 1] [ 2]  [ 1] [ 2] [ 1] [ 2].

Maple

a:= n-> coeff(series(add(x^j*mul(((n+1)^i-1)/(1+x*
    ((n+1)^i-1)), i=1..j), j=0..n), x, n+1), x, n):
seq(a(n), n=0..10);

Crossrefs

Cf. A000007, A005321, A289314, A289315.

Main diagonal of A369415.

Sequence in context: A323816 A208252 A204622 * A004823 A342964 A009063

Adjacent sequences: A369333 A369334 A369335 * A369337 A369338 A369339

Keyword

nonn

Author

Alois P. Heinz, Jan 20 2024

Status

approved