Linear Chord Diagrams with Long Chords

A linear chord diagram of size n is a partition of the set {1, 2, . . . , 2n} into sets of size two, called chords. From a table showing the number of linear chord diagrams of degree n such that every chord has length at least k, we observe that if we proceed far enough along the diagonals, they are given by a geometric sequence. We prove that this holds for all diagonals, and identify when the effect starts.