Set partition statistics and qq-Fibonacci numbers

We consider the set partition statistics lsls and rbrb 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 13/213/2, Πn(13/2)Πn(13/2), and those avoiding both 13/2 and 123, Πn(13/2,123)Πn(13/2,123). We show that the distribution over Πn(13/2)Πn(13/2) enumerates certain integer partitions, and the distribution over Πn(13/2,123)Πn(13/2,123) gives qq-Fibonacci numbers. These qq-Fibonacci numbers are closely related to qq-Fibonacci numbers studied by Carlitz and by Cigler. We provide combinatorial proofs that these qq-Fibonacci numbers satisfy qq-analogues of many Fibonacci identities. Finally, we indicate how p,qp,q-Fibonacci numbers arising from the bistatistic (ls,rb)(ls,rb) give rise to p,qp,q-analogues of identities.