Set partition statistics and q-Fibonacci numbers

2007/07/18 by Adam M. Goyt, Goyt, Adam, Bruce E. Sagan +1 · 1 citation
Computer Science · Mathematics · #05A18 #05A19 #05A30 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.0707.2781

openalex publication_date 2007/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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, Π_n(13/2), and those avoiding both 13/2 and 123, Π_n(13/2,123). We show that the distribution over Π_n(13/2) enumerates certain integer partitions, and the distribution over Π_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.

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