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

Lattice paths called $\mathrm{\ell }$-Schröder paths are introduced. They are paths on the upper half-plane consisting of $\mathrm{\ell }+2$ types of steps: $(i,\mathrm{\ell }-i)$ for $i=0,\dots ,\mathrm{\ell }$, 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 $\mathrm{\ell }$-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 $\mathrm{\ell }$-Schröder paths can be factorized in closed forms.