On the Total Positivity of Delannoy-Like Triangles
School of Mathematics
Liaoning Normal University
School of Mathematical Sciences
Dalian University of Technology
Define an infinite lower triangular matrix
D(e,h) = [dn,k]n,k ≥ 0 by the
recurrence d0,0 = d1,0 = d1,1 = 1,
dn-1,k-1 + edn-1,k +
where e, h are both nonnegative and dn,k = 0 unless n ≥ k ≥ 0. We call
D(e, h) the Delannoy-like triangle.
The entries dn,k count lattice
paths from (0, 0) to (n - k, k)
using the steps (0, 1), (1, 0) and (1, 1)
with assigned weights 1, e, and h. Some well-known combinatorial
triangles are such matrices, including the Pascal triangle D(1, 0), the
Fibonacci triangle D(0, 1), and the Delannoy triangle D(1, 1).
Futhermore, the Schröder triangle and Catalan triangle also arise as
inverses of Delannoy-like triangles. Here we investigate the total
positivity of Delannoy-like triangles. In addition, we show that each
row and diagonal of Delannoy-like triangles are all PF sequences.
Full version: pdf,
(Concerned with sequences
Received June 29 2016; revised version received December 20 2016.
Published in Journal of Integer Sequences, December 26 2016.
Journal of Integer Sequences home page