Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài:Permutation Statistics and q-Fibonacci Numbers. | Permutation Statistics and q-Fibonacci Numbers Adam M. Goyt and David Mathisent Mathematics Department Minnesota State University Moorhead Moorhead MN 56562 Submitted Apr 2 2009 Accepted Aug 2 2009 Published Aug 7 2009 Mathematics Subject Classification 05A19 Abstract 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. 1 Introduction We will study the distribution of two Mahonian statistics inv and maj over sets of pattern restricted permutations. In particular we will study the distributions of these statistics over pattern-restricted sets which are counted by the Fibonacci numbers. These distributions will give us q-Fibonacci numbers which are related to the q-Fibonacci numbers of Carlitz 2 Cigler 3 4 and Goyt and Sagan 6 . Let the nth Fibonacci number be Fn where Fn Fn-1 Fn-2 and F0 1 and F1 1. Let n 1 2 . n and k n k k 1 . n . We will call two integer sequences a1 a2 . .ak and blb2 . .bk are order isomorphic if ai aj whenever bi bj. Let Sn be the set of permutations of n and suppose n p1p2 . .pm G Sm and Ơ q1q2 . qn G Sn. We say that Ơ contains the pattern n if there is a subsequence ơ qix qi2 . .qim of Ơ email goytadam@ temail mathisda@ THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R101 1 which .
