A289313 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.

1, 2, 10, 74, 722, 8786, 128218, 2182554, 42456226, 929093538, 22590839466, 604225121258, 17630145814898, 557285515817970, 18970857530674554, 691929648113663802, 26919562120779248962, 1112769248605003393858, 48704349211392743606602

Offset: 0

Author

Peter Bala, Jul 02 2017

Keywords

• nonn
• easy

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.

Examples

			a(2) = 10: The ten generalized row-Fishburn matrices of size 2 are
  (+-2),
  /+-1  0\ and  /0 +-1\
  |      |      |     |
  \0  +-1/      \0 +-1/.
		

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.

Crossrefs

Cf. A022493, A158691, A179525, A289312.

Programs

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

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