The Narayana distribution

Abstract

Abstract The Narayana distribution is N n,k =1/n( n k−1 )( n k ) for 1⩽k⩽n. This paper concerns structures counted by the Narayana distribution and bijective relationships between them. Here the statistic counting pairs of ascent steps on Catalan paths is prominently considered. Defining the nth Narayana polynomial as Nn(w)=∑1⩽k⩽nNn,kwk, for n⩾1, the paper gives a combinatorial proof of a three term recurrence for these polynomials. It examines the Schroder numbers and the Kirkman numbers and establishes a sequence of bijections linking dissections of polygons to large Schroder paths.