Naiomi T. Cameron ; Kendra Killpatrick - Statistics on Linear Chord Diagrams

dmtcs:5211 - Discrete Mathematics & Theoretical Computer Science, January 23, 2020, Vol. 21 no. 2, Permutation Patters 2018 - https://doi.org/10.23638/DMTCS-21-2-11
Statistics on Linear Chord DiagramsArticle

Authors: Naiomi T. Cameron ; Kendra Killpatrick

    Linear chord diagrams are partitions of $\left[2n\right]$ into $n$ blocks of size two called chords. We refer to a block of the form $\{i,i+1\}$ as a short chord. In this paper, we study the distribution of the number of short chords on the set of linear chord diagrams, as a generalization of the Narayana distribution obtained when restricted to the set of noncrossing linear chord diagrams. We provide a combinatorial proof that this distribution is unimodal and has an expected value of one. We also study the number of pairs $(i,i+1)$ where $i$ is the minimal element of a chord and $i+1$ is the maximal element of a chord. We show that the distribution of this statistic on linear chord diagrams corresponds to the second-order Eulerian triangle and is log-concave.


    Volume: Vol. 21 no. 2, Permutation Patters 2018
    Published on: January 23, 2020
    Accepted on: January 7, 2020
    Submitted on: February 26, 2019
    Keywords: Mathematics - Combinatorics

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