Abstract
In this paper, we look at Riordan arrays with nonnegative integer entries. In the set, if AT=B for a nonnegative integer matrix T, then we say A≤B. Our primary focus centers around six kinds of key arrays: Appell, Bell, two-Bell, Natural, Hankel, and Pseudo-involution. We examine relations among Hankel, Bell and two-Bell in general, while the other entries are treated case by case. We start by looking at individual cases related to the little and large Schröder numbers. Then we look at the families related to k-Catalan arrays, k-Schröder arrays, and k-Motzkin arrays. We often can give combinatorial interpretations. Along the way, several intriguing counterexamples occur.
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