Consider an $m\times n$ table $T$ and latices paths $\nu_1,\ldots,\nu_k$ in $T$ such that each step $\nu_{ i+1}-\nu_i=(1,1)$, $(1,0)$ or $(1,-1)$. The number of paths from the $(1,i)$-blank (resp. first column) to the $(s,t)$-blank is denoted by $\mathcal{ D}^i(s,t)$ (resp. $\mathcal{ D}(s,t)$). Also, the number of all paths form the first column to the las column is denoted by $\mathcal{ I}_m(n)$. We give explicit formulas for the numbers $\mathcal{ D}^1(s,t)$ and $\mathcal{ D}(s,t)$.
Categories: math.GM