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A289314 Number of n X n Fishburn matrices with entries in the set {0,1,2}. 5
1, 2, 12, 264, 19632, 4606752, 3311447232, 7202118117504, 47151987852663552, 927337336972381327872, 54741643544083873448266752, 9696222929066933463021344262144, 5152757080697434799933013959862300672, 8215035458438940398186389046297459974152192 (list; graph; refs; listen; history; text; internal format)
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 the 1's may be colored either black or white.

LINKS

Robert Israel, Table of n, a(n) for n = 0..64

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} (3^i - 1)/(1 + x*(3^i - 1)) = 1 + 2*x + 12*x^2 + 264*x^3 + ... (use Jelínek, Theorem 2.1 with v = w = x = y = 2).

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

A(x) = 1/(1 - 2*x/(1 - 4*x/(1 - 24*x/(1 - 64*x/(1 - 234*x/(1 - 676*x/(1 - ... - 3^(n-1)*(3^n - 1)*x/(1 - (3^n - 1)^2*x/(1 - ...))))))))) and

A(x) = 1 + 2*x/(1 - 6*x/(1 - 16*x/(1 - 72*x/(1 - 208*x/(1 - ... - 3^n*(3^n - 1)*x/(1 - (3^(n+1) - 1)*(3^n - 1)*x/(1 - ...))))))).

EXAMPLE

a(2) = 12: The twelve 2 X 2 Fishburn matrices with entries 0, 1 or 2 are

/1 0\  /1 0\  /2 0\  /2 0\

\0 1/  \0 2/  \0 1/  \0 2/

/1 1\  /1 2\  /1 1\  /1 2\  /2 1\  /2 2\  /2 1\  /2 2\.

\0 1/  \0 1/  \0 2/  \0 2/  \0 1/  \0 1/  \0 2/  \0 2/

Alternatively, the twelve 2-colored primitive Fishburn matrices of dimension 2 (using +1 and -1 for the two different colored versions of 1) are

/+-1  0\ (4 possibilities)

\0  +-1/

   and

/+-1 +-1\ (8 possibilities).

\ 0  +-1/

MAPLE

N:= 20: # to get a(0)..a(N)

g:= add(x^n*mul((3^i-1)/(1+x*(3^i-1)), i=1..n), n=0..N):

S:= series(g, x, N+1):

seq(coeff(S, x, j), j=0..N); # Robert Israel, Jul 11 2017

MATHEMATICA

QP = QPochhammer; nmax = 14;

Sum[(-1)^n (1-x)^(-n-1) x^n QP[3, 3, n]/QP[x/(x-1), 3, n+1], {n, 0, nmax}] + O[x]^nmax // CoefficientList[#, x]& (* Jean-François Alcover, Sep 19 2018 *)

CROSSREFS

Cf. A005321, A022493, A138265, A289315.

Sequence in context: A009610 A012546 A193204 * A091504 A098137 A214005

Adjacent sequences:  A289311 A289312 A289313 * A289315 A289316 A289317

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Jul 03 2017

STATUS

approved

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Last modified August 17 23:13 EDT 2022. Contains 356204 sequences. (Running on oeis4.)