Lattice paths in regions with the catalan property

Abstract

If α = { α 0, α 1,…, α n } and β = { β 0, β 1,…, β n } are two non-decreasing sets of integers such that α 0 = 0 < β 0, α n < β n = n, and α i < i < β i for 1 ⩽ i ⩽ n − 1, let L denote the set of lattice points ( p, q) such that 0 ⩽ p ⩽ n and α p ⩽ q ⩽ β p . We determine all such regions L with the property that the number of lattice paths from (0, 0) to ( p, p) in L is the Catalan number (p + 2) −1( 2p+2 p+1 ) for 0 ⩽ p ⩽ n.