|
%I #20 Aug 31 2023 05:16:54
%S 1,2,10,74,722,8786,128218,2182554,42456226,929093538,22590839466,
%T 604225121258,17630145814898,557285515817970,18970857530674554,
%U 691929648113663802,26919562120779248962,1112769248605003393858,48704349211392743606602
%N The number of upper-triangular matrices with integer entries whose absolute sum is equal to n and such that each row contains a nonzero entry.
%C A row-Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and such that each row contains a nonzero entry. See A158691.
%C Here we consider generalized row-Fishburn matrices where we allow the row_Fishburn matrices to have positive and negative nonzero entries. We define the size of a generalized row-Fishburn matrix to be the absolute sum of the matrix entries. This sequence gives the number of generalized row-Fishburn matrices of size n.
%C Alternatively, this sequence gives the number of 2-colored row-Fishburn matrices of size n, that is, ordinary row-Fishburn matrices of size n where each nonzero entry in the matrix can have one of two different colors.
%C More generally, if F(x) = Sum_{n >= 0} ( Product_{i = 1..n} (1 + x)^i - 1 ) is the o.g.f. for primitive row-Fishburn matrices A179525 (i.e., row-Fishburn matrices with entries restricted to the set {0,1}) and C(x) := c_1*x + c_2*x^2 + ..., where c_i is a sequence of nonnegative integers, then the composition F(C(x)) is the o.g.f. for colored row-Fishburn matrices where entry i in the matrix can have one of c_i different colors: c_i = 0 for some i means i does not appear as an entry in the Fishburn matrix. This result is an application of Lemma 2.2.22 of Goulden and Jackson.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.
%H Seiichi Manyama, <a href="/A289313/b289313.txt">Table of n, a(n) for n = 0..200</a>
%H Hsien-Kuei Hwang, Emma Yu Jin, <a href="https://arxiv.org/abs/1911.06690">Asymptotics and statistics on Fishburn matrices and their generalizations</a>, arXiv:1911.06690 [math.CO], 2019.
%F O.g.f.: Sum_{n >= 0} ( Product_{i = 1..n} ((1 + x)/(1 - x))^i - 1 ).
%F The o.g.f. has several alternative forms:
%F Sum_{n >= 0} ( Product_{i = 1..n} ( 1 - ((1 - x)/(1 + x))^(2*i-1) ) );
%F Sum_{n >= 0} ((1 - x)/(1 + x))^(n+1) * ( Product_{i = 1..n} 1 - ((1 - x)/(1 + x))^(2*i) );
%F 1/2*( 1 + Sum_{n >= 0} ((1 + x)/(1 - x))^((n+1)*(n+2)/2) * Product_{i = 1..n} ( 1 - ((1 - x)/(1 + x))^i ) ).
%F Conjectural g.f.: Sum_{n >= 0} ((1 + x)/(1 - x))^((n+1)*(2*n+1)) * Product_{i = 1..2*n} ( ((1 - x)/(1 + x))^i - 1 ).
%F a(n) ~ 2^(3*n+2) * 3^(n+1) * n^(n + 1/2) / (exp(n) * Pi^(2*n + 3/2)). - _Vaclav Kotesovec_, Aug 31 2023
%e a(2) = 10: The ten generalized row-Fishburn matrices of size 2 are
%e (+-2),
%e /+-1 0\ and /0 +-1\
%e | | | |
%e \0 +-1/ \0 +-1/.
%p G:= add(mul( ((1 + x)/(1 - x))^i - 1, i=1..n),n=0..20):
%p S:= series(G,x,21):
%p seq(coeff(S,x,j),j=0..20);
%p # - _Peter Bala_, Jul 24 2017
%Y Cf. A022493, A158691, A179525, A289312.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Jul 02 2017
| 