A289317 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row and each column contains a nonzero entry.

1, 1, 1, 3, 7, 23, 84, 364, 1792, 9953, 61455, 417720, 3098515, 24902930, 215538825, 1998518430, 19761943208, 207571259703, 2307812703419, 27075591512866, 334263981931669

Offset: 0

Author

Peter Bala, Jul 25 2017

Keywords

• nonn
• easy

Comments

A Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and each row and each column contains a nonzero entry. See A022493. Here we are considering Fishburn matrices where the nonzero entries are all odd.

The g.f. for primitive Fishburn matrices (i.e., Fishburn matrices with entries restricted to the set {0,1}), is F(x) = Sum_{n>=0} Product_{k=1..n} ( 1 - 1/(1 + x)^k ). See A138265. Let C(x) = x/(1 - x^2) = x + x^3 + x^5 + x^7 + .... Then applying Lemma 2.2.22 of Goulden and Jackson gives the g.f. for this sequence as the composition F(C(x)).

Examples

			a(4) = 7: The Fishburn matrices of size 4 with odd nonzero entries are
/3 0\ /1 0\
\0 1/ \0 3/
/1 1 0\ /1 0 1\ /1 0 0\
|0 1 0| |0 1 0| |0 1 1|
\0 0 1/ \0 0 1/ \0 0 1/
/1 1 0\
|0 0 1|
\0 0 1/
/1 0 0 0\
|0 1 0 0|
|0 0 1 0|
\0 0 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, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A022493, A138265, A289312, A289313, A289314, A289315, A289316.

Programs

Maple

C:= x -> x/(1 - x^2):
G:= add(mul( 1 - 1/(1 + C(x))^k, k=1..n), n=0..20):
S:= series(G,x,21):
seq(coeff(S,x,j),j=0..20);

Formula

G.f.: A(x) = Sum_{n >= 0} Product_{k = 1..n} ( 1 - 1/(1 + x/(1 - x^2))^k ).

a(n) ~ 2^(n + 5/2) * 3^(n + 3/2) * n^(n+1) / (exp(n + Pi^2/12) * Pi^(2*n + 2)). - Vaclav Kotesovec, Aug 31 2023