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Abstract: In a recent paper, Goyt and Sagan studied distributions of certain setpartition statistics over pattern restricted sets of set partitions that werecounted by the Fibonacci numbers. Their study produced a class of $q$-Fibonaccinumbers, which they related to $q$-Fibonacci numbers studied by Carlitz andCigler. In this paper we will study the distributions of some Mahonianstatistics over pattern restricted sets of permutations. We will give bijectiveproofs connecting some of our $q$-Fibonacci numbers to those of Carlitz,Cigler, Goyt and Sagan. We encode these permutations as words and use a weightto produce bijective proofs of $q$-Fibonacci identities. Finally, we study thedistribution of some of these statistics on pattern restricted permutationsthat West showed were counted by even Fibonacci numbers.



Author: Adam M. Goyt, David Mathisen

Source: https://arxiv.org/



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