OFFSET
0,2
COMMENTS
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.
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.
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.
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.
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
FORMULA
O.g.f.: Sum_{n >= 0} ( Product_{i = 1..n} ((1 + x)/(1 - x))^i - 1 ).
The o.g.f. has several alternative forms:
Sum_{n >= 0} ( Product_{i = 1..n} ( 1 - ((1 - x)/(1 + x))^(2*i-1) ) );
Sum_{n >= 0} ((1 - x)/(1 + x))^(n+1) * ( Product_{i = 1..n} 1 - ((1 - x)/(1 + x))^(2*i) );
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 ) ).
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 ).
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
EXAMPLE
a(2) = 10: The ten generalized row-Fishburn matrices of size 2 are
(+-2),
/+-1 0\ and /0 +-1\
| | | |
\0 +-1/ \0 +-1/.
MAPLE
G:= add(mul( ((1 + x)/(1 - x))^i - 1, i=1..n), n=0..20):
S:= series(G, x, 21):
seq(coeff(S, x, j), j=0..20);
# Peter Bala, Jul 24 2017
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Peter Bala, Jul 02 2017
STATUS
approved