Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6

On the Total Positivity of Delannoy-Like Triangles

Lili Mu
School of Mathematics
Liaoning Normal University
Dalian, 116029
PR China

Sai-nan Zheng
School of Mathematical Sciences
Dalian University of Technology
Dalian, 116024
PR China


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,k = dn-1,k-1 + edn-1,k + hdn-2,k-1, where e, h are both nonnegative and dn,k = 0 unless nk ≥ 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.

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(Concerned with sequences A001850 A007318 A008288 A026729 A033184 A132372.)

Received June 29 2016; revised version received December 20 2016. Published in Journal of Integer Sequences, December 26 2016.

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