Generalized Dyck paths

Abstract

It is well known (see [3, 6, 9, 10, 11]) that Dyck paths are in bijection with “Dyck words”, “ballot sequences”, “well formed sequences of parentheses”, “2-lines standard-tableaux”, “binary trees”, “ordered trees”; all these are counted by Catalan numbers. In the present text, we replace the north-east steps of a Dyck path by steps from an arbitrary finite multi-set l of vectors with integral coordinates in the plane. In order to study these generalized Dyck paths, called A-paths, we have to introduce many closely related families of paths. The corresponding (multi-variable) generating functions satisfy an intricate system of algebraic equations which leads to a polynomial equation satisfied by A . For example when l is {(1, 1)} (respectively {(1,2), (2,1); {(1,3), (3,1)}) this polynomial equation is of degree 2 (resp. 4, 8). More generally when l ={u 1, u 2,…, u m} where u j = ( r j , j) then A = A( u 1, u 2,…, u m ) satisfies a polynomial equation of degree 2 m with coefficients in Z [ u 1, u 2,…, u m ].