cp's OEIS Frontend

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

%I A289315 #26 Mar 17 2024 10:41:57
%S A289315 1,3,36,2052,505764,511718148,2088275673636,34176650317115652,
%T A289315 2239082850356711468964,586908388119824949146284548,
%U A289315 615402202729113953115253336166436,2581165458211746544190089033131172341252,43304685245392697816407075492352986194550240164
%N A289315 Number of n X n Fishburn matrices with entries in the set {0,1,2,3}.
%C A289315 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 A289315 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.
%H A289315 Robert Israel, <a href="/A289315/b289315.txt">Table of n, a(n) for n = 0..57</a>
%H A289315 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 A289315 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 A289315 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).
%F A289315 Two conjectural continued fractions for the o.g.f.:
%F A289315 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
%F A289315 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 - ...))))))).
%F A289315 a(n) ~ c * 2^(n*(n+1)), where c = QPochhammer(1/4)^2 = 0.474083940023743191581900099468175063640311684514259231... - _Vaclav Kotesovec_, Aug 31 2023, updated Mar 17 2024
%e A289315 a(2) = 36: The 36 2 X 2 Fishburn matrices with entries 0, 1, 2 or 3 are
%e A289315 /1 0\ /1 0\ /2 0\ /2 0\
%e A289315 \0 1/ \0 2/ \0 1/ \0 2/
%e A289315 /1 0\ /3 0\ /3 0\
%e A289315 \0 3/ \0 1/ \0 3/
%e A289315 /2 0\ /3 0\
%e A289315 \0 3/ \0 2/
%e A289315 /1 2\ /1 1\ /1 2\ /2 1\ /2 2\ /2 1\ /2 2\
%e A289315 \0 1/ \0 2/ \0 2/ \0 1/ \0 1/ \0 2/ \0 2/
%e A289315 /1 1\ /1 3\ /1 1\ /1 3\ /3 1\ /3 3\ /3 1\
%e A289315 \0 1/ \0 1/ \0 3/ \0 3/ \0 1/ \0 1/ \0 3/
%e A289315 /2 3\ /2 2\ /2 3\ /3 2\ /3 3\ /3 2\ /3 3\
%e A289315 \0 2/ \0 3/ \0 3/ \0 2/ \0 2/ \0 3/ \0 3/
%e A289315 /1 2\ /1 3\ /2 3\ /2 1\ /3 1\ /3 2\.
%e A289315 \0 3/ \0 2/ \0 1/ \0 3/ \0 2/ \0 1/
%e A289315 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
%e A289315 /1_j  0\ (9 possibilities)
%e A289315 \ 0 1_j/
%e A289315 and
%e A289315 /1_j 1_j\ (27 possibilities).
%e A289315 \ 0  1_j/
%p A289315 G:= add(x^n*mul((4^i-1)/(1+x*(4^i-1)), i=1..n),n=0..40):
%p A289315 S:= series(G,x,41):
%p A289315 seq(coeff(S,x,j),j=0..40); # _Robert Israel_, Jul 10 2017
%t A289315 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_ *)
%Y A289315 Cf. A005321, A022493, A138265, A289314.
%K A289315 nonn,easy
%O A289315 0,2
%A A289315 _Peter Bala_, Jul 03 2017