Set partition statistics and q-Fibonacci numbers

TL;DR

This paper explores the distribution of set partition statistics over pattern-avoiding partitions, revealing connections to q-Fibonacci numbers and providing combinatorial proofs of related identities.

Contribution

It introduces new connections between set partition statistics, pattern avoidance, and q-Fibonacci numbers, including combinatorial proofs of their identities.

Findings

Distribution over pattern-avoiding partitions enumerates integer partitions.

Distribution over certain pattern-avoiding partitions yields q-Fibonacci numbers.

Provides combinatorial proofs of q-Fibonacci identities.

Abstract

We consider the set partition statistics ls and rb introduced by Wachs and White and investigate their distribution over set partitions avoiding certain patterns. In particular, we consider those set partitions avoiding the pattern 13/2, $\Pi_n(13/2)$, and those avoiding both 13/2 and 123, $\Pi_n(13/2,123)$. We show that the distribution over $\Pi_n(13/2)$ enumerates certain integer partitions, and the distribution over $\Pi_n(13/2,123)$ gives q-Fibonacci numbers. These q-Fibonacci numbers are closely related to q-Fibonacci numbers studied by Carlitz and by Cigler. We provide combinatorial proofs that these q-Fibonacci numbers satisfy q-analogues of many Fibonacci identities. Finally, we indicate how p,q-Fibonacci numbers arising from the bistatistic (ls, rb) give rise to p,q-analogues of identities.