The [formula omitted]-Schröder paths and [formula omitted]-Schröder numbers

Abstract

In this paper, we introduce a variant of the m-Schröder paths, i.e. of those lattice paths from (0,0) to (mn,0) with up steps U=(1,1), down steps D(m)=(1,1−m) and horizontal-down steps H(m)=(2,2−m), which never go below the x-axis. The number of m-Schröder paths of length mn is called the nth m-Schröder number, while the number of m-Schröder paths of length mn with no horizontal-down steps ending on the x-axis is called nth small m-Schröder number. We present two types of generalization of Schröder matrices. The entries are interpreted in terms of the number of partial m-Schröder paths. Then we show that these matrices are Riordan arrays, and we give some properties from this result. In particular, we obtain that the nth m-Schröder number is twice of the nth small m-Schröder number for n≥1, and we show that the A-sequence of the m-Schröder matrix is the sequence of the (m−1)-Schröder numbers. Finally, we study other two m-Schröder matrices whose row sums are the m-Schröder numbers and small m-Schröder numbers respectively.