Dun Qiu ; Jeffrey B. Remmel - Quadrant marked mesh patterns in 123-avoiding permutations

dmtcs:3297 - Discrete Mathematics & Theoretical Computer Science, August 3, 2018, Vol. 19 no. 2, Permutation Patterns 2016 - https://doi.org/10.23638/DMTCS-19-2-12
Quadrant marked mesh patterns in 123-avoiding permutations

Authors: Dun Qiu ; Jeffrey B. Remmel

    Given a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $\sigma_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $\sigma$ if there are at least $a$ points to the right of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $b$ points to the left of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $c$ points to the left of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$, and at least $d$ points to the right of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$. Kitaev, Remmel, and Tiefenbruck systematically studied the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 132-avoiding permutations. The operation of reverse and complement on permutations allow one to translate their results to find the distribution of the number of $\mathrm{MMP}(a,b,c,d)$ matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this paper, we study the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 123-avoiding permutations. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions.


    Volume: Vol. 19 no. 2, Permutation Patterns 2016
    Section: Permutation Patterns
    Published on: August 3, 2018
    Accepted on: August 3, 2018
    Submitted on: May 2, 2017
    Keywords: Mathematics - Combinatorics

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