Statistics on Linear Chord Diagrams
Discrete mathematics & theoretical computer science, Permutation Patters 2018, Tome 21 (2019) no. 2
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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},
publisher = {mathdoc},
volume = {21},
number = {2},
year = {2019},
doi = {10.23638/DMTCS-21-2-11},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-2-11/}
}
TY - JOUR AU - Cameron, Naiomi T. AU - Killpatrick, Kendra TI - Statistics on Linear Chord Diagrams JO - Discrete mathematics & theoretical computer science PY - 2019 VL - 21 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-2-11/ DO - 10.23638/DMTCS-21-2-11 LA - en ID - DMTCS_2019_21_2_a10 ER -
%0 Journal Article %A Cameron, Naiomi T. %A Killpatrick, Kendra %T Statistics on Linear Chord Diagrams %J Discrete mathematics & theoretical computer science %D 2019 %V 21 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-2-11/ %R 10.23638/DMTCS-21-2-11 %G en %F DMTCS_2019_21_2_a10
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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