Set partition statistics and [formula omitted]-Fibonacci numbers

Abstract

We consider the set partition statistics l s and r b introduced by Wachs and White and investigate their distribution over set partitions that avoid 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 ( l s , r b ) give rise to p , q -analogues of identities.