Geodesic
Statistics on Linear Chord Diagrams
Discrete mathematics & theoretical computer science, Permutation Patters 2018, Tome 21 (2019) no. 2 Cet article a éte moissonné depuis la source Episciences

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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. 
@article{DMTCS_2019_21_2_a10,
     author = {Cameron, Naiomi T. and Killpatrick, Kendra},
     title = {Statistics on {Linear} {Chord} {Diagrams}},
     journal = {Discrete mathematics & theoretical computer science},
     year = {2019},
     volume = {21},
     number = {2},
     doi = {10.23638/DMTCS-21-2-11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-2-11/}
}
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Cameron, Naiomi T.; Killpatrick, Kendra. Statistics on Linear Chord Diagrams. Discrete mathematics & theoretical computer science, Permutation Patters 2018, Tome 21 (2019) no. 2. doi: 10.23638/DMTCS-21-2-11

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