Linear $k$-Chord Diagrams

We generalize the notion of linear chord diagrams to the case of matched sets of size $k$, which we call $k$-chord diagrams. We provide formal generating functions and recurrence relations enumerating these $k$-chord diagrams by the number of short chords, where the latter is defined as all members of the matched set being adjacent, and is the generalization of a short chord or loop in a linear chord diagram. We also enumerate $k$-chord diagrams by the number of connected components built from short chords and provide the associated generating functions in this case. We show that the distributions of short chords and connected components are asymptotically Poisson, and provide the associated means. Finally, we provide recurrence relations enumerating non-crossing $k$-chord diagrams by the number of short chords, generalising the Narayana numbers, and establish asymptotic normality, providing the associated means and variances. Applications to generalized games of memory are also discussed.