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

dmtcs:5211 - Discrete Mathematics & Theoretical Computer Science, January 23, 2020, Vol. 21 no. 2, Permutation Patters 2018
Statistics on Linear Chord Diagrams

Authors: Cameron, Naiomi T. and Killpatrick, Kendra

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
Submitted on: February 26, 2019
Keywords: Mathematics - Combinatorics


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