The generalized Schröder theory (Q2571296)

\noindent An \(m\)-Schröder path of order \(n\) is a lattice path from \((0,0)\) to \((mn,n)\), in the integer lattice, that never goes below the diagonal \(y=\frac xm\) and permits only steps of the form \((0,1)\), \((1,0)\) and \((1,1)\). The author proves: Theorem 2.1. The number of \(m\)-Schröder paths of order \(n\) with \(d(1,1)\) steps, denoted by \(S_{n,d}^m\), is equal to \[ \frac1{mn-d+1}{{mn+n-d}\choose{mn-d,n-d,d}}. \] \vskip.2cm \noindent The author goes on to prove some other results on \(m\)-Schröder paths and \(m\)-Schröder words. The author introduces the \((q,t)\)-\(m\)-Schröder polynomial through \(m\)-parking functions and relates it to the \(m\)-shuffle conjecture of Haglund, Loehr, Remmel and Ulyanov.