A289315 Number of n X n Fishburn matrices with entries in the set {0,1,2,3}.

1, 3, 36, 2052, 505764, 511718148, 2088275673636, 34176650317115652, 2239082850356711468964, 586908388119824949146284548, 615402202729113953115253336166436, 2581165458211746544190089033131172341252, 43304685245392697816407075492352986194550240164

Offset

0,2

Comments

A Fishburn matrix is defined to be an upper-triangular matrix with nonnegative integer entries such that each row and column contains a nonzero entry. See A005321 for primitive Fishburn matrices of dimension n, that is, Fishburn matrices of dimension n with entries in the set {0,1}.

The present sequence has an alternative description as the number of primitive Fishburn matrices of dimension n where each entry equal to 1 can have one of three different colors. Cf. A289314.

Links

Robert Israel, Table of n, a(n) for n = 0..57

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.

Formula

O.g.f.: A(x) = Sum_{n >= 0} x^n Product_{i = 1..n} (4^i - 1)/(1 + x*(4^i - 1)) = 1 + 3*x + 36*x^2 + 2052*x^3 + ... (use Jelínek, Theorem 2.1 with v = w = x = y = 3).

Two conjectural continued fractions for the o.g.f.:

A(x) = 1/(1 - 3*x/(1 - 9*x/(1 - 60*x/(1 - 225*x/(1 - 1008*x/(1 - 3969*x/(1 - ... - 4^(n-1)*(4^n - 1)*x/(1 - (4^n - 1)^2*x/(1 - ...))))))))) and

A(x) = 1 + 3*x/(1 - 12*x/(1 - 45*x/(1 - 240*x/(1 - 945*x/(1 - ... - 4^n*(4^n - 1)*x/(1 - (4^(n+1) - 1)*(4^n - 1)*x/(1 - ...))))))).

a(n) ~ c * 2^(n*(n+1)), where c = QPochhammer(1/4)^2 = 0.474083940023743191581900099468175063640311684514259231... - Vaclav Kotesovec, Aug 31 2023, updated Mar 17 2024

Example

a(2) = 36: The 36 2 X 2 Fishburn matrices with entries 0, 1, 2 or 3 are
/1 0\ /1 0\ /2 0\ /2 0\
\0 1/ \0 2/ \0 1/ \0 2/
/1 0\ /3 0\ /3 0\
\0 3/ \0 1/ \0 3/
/2 0\ /3 0\
\0 3/ \0 2/
/1 2\ /1 1\ /1 2\ /2 1\ /2 2\ /2 1\ /2 2\
\0 1/ \0 2/ \0 2/ \0 1/ \0 1/ \0 2/ \0 2/
/1 1\ /1 3\ /1 1\ /1 3\ /3 1\ /3 3\ /3 1\
\0 1/ \0 1/ \0 3/ \0 3/ \0 1/ \0 1/ \0 3/
/2 3\ /2 2\ /2 3\ /3 2\ /3 3\ /3 2\ /3 3\
\0 2/ \0 3/ \0 3/ \0 2/ \0 2/ \0 3/ \0 3/
/1 2\ /1 3\ /2 3\ /2 1\ /3 1\ /3 2\.
\0 3/ \0 2/ \0 1/ \0 3/ \0 2/ \0 1/
Alternatively, the 36 3-colored primitive Fishburn matrices of dimension 2 (using 1_j, j = 1,2,3 to denote the three different colored versions of 1) are
/1_j  0\ (9 possibilities)
\ 0 1_j/
and
/1_j 1_j\ (27 possibilities).
\ 0  1_j/

Maple

G:= add(x^n*mul((4^i-1)/(1+x*(4^i-1)), i=1..n), n=0..40):
S:= series(G, x, 41):
seq(coeff(S, x, j), j=0..40); # Robert Israel, Jul 10 2017

Mathematica

Table[SeriesCoefficient[Sum[x^j*Product[(4^i - 1)/(1 + x (4^i - 1)), {i, j}], {j, 0, n}], {x, 0, n}], {n, 0, 12}] (* Michael De Vlieger, Jul 10 2017, after Maple by Robert Israel *)

Crossrefs

Cf. A005321, A022493, A138265, A289314.

Sequence in context: A300770 A272660 A193302 * A102579 A368076 A136393

Adjacent sequences: A289312 A289313 A289314 * A289316 A289317 A289318

Keyword

nonn,easy

Author

Peter Bala, Jul 03 2017

Status

approved