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A289314 Number of n X n Fishburn matrices with entries in the set {0,1,2}. 7
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
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 - ...))))))).
a(n) ~ c * 3^(n*(n+1)/2), where c = QPochhammer(1/3)^2 = 0.313741223174946734265526469975707962872482170305592991802056615373429729... - Vaclav Kotesovec, Aug 31 2023, updated Mar 17 2024
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
Sequence in context: A009610 A012546 A193204 * A091504 A098137 A214005
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jul 03 2017
STATUS
approved

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Last modified July 16 08:10 EDT 2024. Contains 374345 sequences. (Running on oeis4.)