Permutation patterns and statistics

Details Permutation patterns and statistics Permutation patterns and statistics

2011-09-23

arxiv.org/abs/1109.4976v1

Mathematics

Combinatorics

32 pages, 5 figures

Scientific paper

Let S_n denote the symmetric group of all permutations of the set {1, 2, ...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of Pi in the sense of pattern avoidance. One of the celebrated notions in pattern theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if #Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage proposed studying a q-analogue of this concept defined as follows. Suppose st:S->N is a permutation statistic where N represents the nonnegative integers. Consider the corresponding generating function, F_n^{st}(Pi;q) = sum_{sigma in Av_n(Pi)} q^{st sigma}, and call Pi,Pi' st-Wilf equivalent if F_n^{st}(Pi;q)=F_n^{st}(Pi';q) for all n\ge0. We present the first in-depth study of this concept for the inv and maj statistics. In particular, we determine all inv- and maj-Wilf equivalences for any Pi containd in S_3. This leads us to consider various q-analogues of the Catalan numbers, Fibonacci numbers, triangular numbers, and powers of two. Our proof techniques use lattice paths, integer partitions, and Foata's fundamental bijection. We also answer a question about Mahonian pairs raised in the Sagan-Savage article.

Affiliated with

Also associated with

No associations

Rating

Permutation patterns and statistics does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Permutation patterns and statistics, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Permutation patterns and statistics will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-723827