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%I #28 Mar 17 2024 09:53:28
%S 1,2,12,264,19632,4606752,3311447232,7202118117504,47151987852663552,
%T 927337336972381327872,54741643544083873448266752,
%U 9696222929066933463021344262144,5152757080697434799933013959862300672,8215035458438940398186389046297459974152192
%N Number of n X n Fishburn matrices with entries in the set {0,1,2}.
%C 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}.
%C 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.
%H Robert Israel, <a href="/A289314/b289314.txt">Table of n, a(n) for n = 0..64</a>
%H Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, <a href="https://arxiv.org/abs/2012.13570">Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow</a>, arXiv:2012.13570 [math.CO], 2020.
%H Vít Jelínek, <a href="http://dx.doi.org/10.1016/j.jcta.2011.11.010">Counting general and self-dual interval orders</a>, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; <a href="http://arxiv.org/abs/1106.2261">arXiv preprint</a>, arXiv:1106.2261 [math.CO], 2011.
%F 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).
%F Two conjectural continued fractions for the o.g.f.:
%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
%F 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 - ...))))))).
%F 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
%e a(2) = 12: The twelve 2 X 2 Fishburn matrices with entries 0, 1 or 2 are
%e /1 0\ /1 0\ /2 0\ /2 0\
%e \0 1/ \0 2/ \0 1/ \0 2/
%e /1 1\ /1 2\ /1 1\ /1 2\ /2 1\ /2 2\ /2 1\ /2 2\.
%e \0 1/ \0 1/ \0 2/ \0 2/ \0 1/ \0 1/ \0 2/ \0 2/
%e Alternatively, the twelve 2-colored primitive Fishburn matrices of dimension 2 (using +1 and -1 for the two different colored versions of 1) are
%e /+-1 0\ (4 possibilities)
%e \0 +-1/
%e and
%e /+-1 +-1\ (8 possibilities).
%e \ 0 +-1/
%p N:= 20: # to get a(0)..a(N)
%p g:= add(x^n*mul((3^i-1)/(1+x*(3^i-1)),i=1..n),n=0..N):
%p S:= series(g,x,N+1):
%p seq(coeff(S,x,j),j=0..N); # _Robert Israel_, Jul 11 2017
%t QP = QPochhammer; nmax = 14;
%t 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 *)
%Y Cf. A005321, A022493, A138265, A289315.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Jul 03 2017
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