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{\bf Adam M. Goyt and David Mathisen}
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{\bf Permutation Statistics and $q$-Fibonacci Numbers}
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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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