Combinatorial reciprocity for Riordan arrays

초록

In this paper, we develop a combinatorial reciprocity theory for Riordan arrays. Our focus is on calculating the negative territory using recurrence relations and investigating their combinatorial properties in connection with the original Riordan arrays. We take this negative part, flip it, and change signs on alternate diagonals. When this new matrix coincides with the original array, we call it a bogus-involution. These are our main focus. Noting the similarity between bogus-involutions and the previously defined pseudo-involutions, we also examine matrices that exhibit both properties. When these matrices have integer entries, as is often the case, we explore their combinatorial significance. © 2025 Elsevier Inc.

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