Planar networks and total positivity of Riordan arrays

In 2015, Chen, Liang and Wang provided several sufficient conditions for the total positivity of Riordan arrays and asked for combinatorial proofs of these results. In this paper, we present such proofs by constructing suitable planar networks with non-negative weights and applying the Lindström-Gessel-Viennot lemma. Moreover, we slightly generalize one of the results and give more totally positive Riordan arrays.

💡 Research Summary

The paper addresses a problem raised by Chen, Liang, and Wang in 2015 concerning sufficient conditions for the total positivity (TP) of Riordan arrays. While the original proofs relied on algebraic manipulations of recurrence relations and determinant calculations, the authors of the present work provide purely combinatorial proofs by constructing appropriate planar networks with non‑negative edge weights and invoking the Lindström‑Gessel‑Viennot (LGV) lemma.

A Riordan array (R(g,f)) is defined by two formal power series (g(x)=\sum_{n\ge0}g_nx^n) and (f(x)=\sum_{n\ge0}f_nx^n) (with (g_0\neq0), (f_0=0), (f_1\neq0)). Its ((n,k))-entry is the coefficient of (x^n) in (g(x)f(x)^k). Equivalently, the array can be described by an A‑sequence ((a_n){n\ge0}) and a Z‑sequence ((z_n){n\ge0}) through the recurrences
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