An identity for the central binomial coefficient

TL;DR

This paper derives a new triple sum identity for the central binomial coefficient by analyzing lattice path statistics and connects it to compositions studied by Bender et al., providing new insights and evaluations.

Contribution

It introduces a novel triple sum identity for the central binomial coefficient based on lattice path statistics and links it to existing composition counts.

Findings

Derived a triple sum identity for {2n} choose {n}

Connected the identity to irreducible pairs of compositions

Evaluated some of the sums explicitly

Abstract

We find the joint distribution of three simple statistics on lattice paths of n upsteps and n downsteps leading to a triple sum identity for the central binomial coefficient {2n}-choose-{n}. We explain why one of the constituent double sums counts the irreducible pairs of compositions considered by Bender et al., and we evaluate some of the other sums.