Generalized Schröder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials

Abstract

Lattice paths called ℓ-Schröder paths are introduced. They are paths on the upper half-plane consisting of $\ell +2$ types of steps: $(i,\ell -i)$ for $i=0,\dots ,\ell$, and $(1,-1)$. Those paths generalize Schröder paths and some variants, such as m-Schröder paths by Yang and Jiang and Motzkin-Schröder paths by Kim and Stanton. We show that ℓ-Schröder paths arise naturally from a combinatorial interpretation of the moments of generalized Laurent bi-orthogonal polynomials introduced by Wang, Chang, and Yue. We also show that some generating functions of non-intersecting ℓ-Schröder paths can be factorized in closed forms.