  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.

Combinatorics