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We study q-analogues of k-Fibonacci numbers that arise from weighted tilings of an $n\times1$ board with tiles of length at most k. The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics. We use these q-analogues to produce q-analogues of identities involving k-Fibonacci numbers. This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations. In this paper we give general q-analogues of k-Fibonacci identities for arbitrary weights that depend only on lengths and locations of tiles. We then determine weights for specific permutation or set partition statistics and use these specific weights and the general identities to produce specific identities.

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Permutation Statistics and $q$-Fibonacci Numbers

The Electronic Journal of Combinatorics ◽

10.37236/190 ◽

2009 ◽

Vol 16 (1) ◽

Cited By ~ 2

Author(s):

Adam M. Goyt ◽

David Mathisen

Keyword(s):

Fibonacci Numbers ◽

Permutation Statistics ◽

Set Partition ◽

Set Partitions ◽

Restricted Permutations ◽

Mahonian Statistics ◽

Partition Statistics

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

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