A369415 -id:A369415 - cp's OEIS frontend

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

Alois P. Heinz, Jan 20 2024

nonn

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

			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].

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

Cf. A000007, A005321, A289314, A289315.

Main diagonal of A369415.

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);

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

A369423 Number of 3 X 3 Fishburn matrices with entries in the set {0,1,...,n}.

Original entry on oeis.org

0, 10, 264, 2052, 9280, 30750, 83160, 194824, 410112, 794610, 1441000, 2475660, 4065984, 6428422, 9837240, 14634000, 21237760, 30155994, 41996232, 57478420, 77448000, 102889710, 134942104, 174912792, 224294400, 284781250, 358286760, 446961564, 553212352

Offset: 0

Alois P. Heinz, Jan 23 2024

Number of upper triangular 3 X 3 {0,1,...,n}-matrices with no zero rows or columns.

			a(1) = 10:
  [100] [110] [100] [110] [101] [111] [101] [111] [110] [111]
  [ 10] [ 10] [ 11] [ 11] [ 10] [ 10] [ 11] [ 11] [ 01] [ 01]
  [  1] [  1] [  1] [  1] [  1] [  1] [  1] [  1] [  1] [  1].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Row n=3 of A369415.

a:= n-> n^3*(n+1)*(n^2+3*n+1):
seq(a(n), n=0..28);

Table[n^3*(n + 1)*(n^2 + 3*n + 1), {n, 0, 50}] (* Paolo Xausa, Jun 09 2024 *)

a(n) = n^3*(n+1)*(n^2+3*n+1) = n^6 + 4*n^5 + 4*n^4 + n^3.

G.f.: 2*x*(4*x^4-55*x^3-207*x^2-97*x-5)/(x-1)^7.

cp's OEIS Frontend

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

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