Event Type:

Number Theory Seminar

Date/Time:

Tuesday, October 28, 2014 - 16:00 to 17:00

Location:

Graf 307

Guest Speaker:

Institution:

Lewis and Clark

Abstract:

For any permutations $\sigma \in S_n$ and $\pi \in S_k$, we say that $\sigma$ contains the pattern $\pi$ if $\sigma$ has a subsequence that is order isomorphic to $\pi$. In this case, we refer to $\pi$ as a pattern of length $k.$ We may endow a pattern $\pi=\pi_1\pi_2\cdots\pi_k$ with proximity restrictions, that is, requiring certain elements of the pattern to correspond to adjacent elements in the permutation $\sigma$. If the elements $\pi_i,\pi_{i+1}$ of the pattern $\pi$ are required to correspond to adjacent elements of $\sigma$, then we place an underscore beneath $\pi_i,\pi_{i+1}$. For example, $\sigma=45213$ contains the patterns $312$ and $3\underline{12}$. When $\sigma$ contains no subsequence order isomorphic to $\pi$ then we say that $\sigma$ avoids $\pi$. A pattern where every element receives an underscore is called a consecutive pattern.

In 2012, Sagan and Savage introduced the notion of statistical Wilf equivalence for sets of permutations that avoid particular permutation patterns. In this talk, we will consider Wilf equivalence of the inversion statistic on sets of permutations that avoid consecutive patterns. We say that two sets of permutation patterns $\Pi$ and $\Pi'$ are $\inv$-Wilf equivalent if the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of $\Pi$ is equal to the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of $\Pi'$. We will discuss the use of Fibonacci tableaux as a model to study the inversion statistic on permutations that avoid $\Pi$ where $\Pi$ is a set of three or more consecutive permutation patterns. We will give combinatorial arguments for the distribution of the inversion statistic for $\Pi$ a subset of 4 or 5 consecutive permutation patterns and for all but one of the cases where $\Pi$ is a subset of three consecutive permutation patterns.

Host: