Combinatorial enumeration of lattice paths by flaws with respect to a linear boundary of rational slope

Federico Firoozi, Jonathan Jedwab, Amarpreet Rattan

Published 2024-06-13Version 1

Let $a,b$ be fixed positive coprime integers. For a positive integer $g$, write $N_k(g)$ for the set of lattice paths from the startpoint $(0,0)$ to the endpoint $(ga,gb)$ with steps restricted to $\{(1,0), (0,1)\}$, having exactly $k$ flaws (lattice points lying above the linear boundary). We wish to determine $|N_k(g)|$. The enumeration of lattice paths with respect to a linear boundary while accounting for flaws has a long and rich history, dating back to the 1949 results of Chung and Feller. The only previously known values of $|N_k(g)|$ are the extremal cases $k = 0$ and $k = g(a+b)-1$, determined by Bizley in 1954. Our main result is that a certain subset of $N_k(g)$ is in bijection with $N_{k+1}(g)$. One consequence is that the value $|N_k(g)|$ is constant over each successive set of $a+b$ values of $k$. This in turn allows us to derive a recursion for $|N_k(g)|$ whose base case is given by Bizley's result for $k=0$. We solve this recursion to obtain a closed form expression for $|N_k(g)|$ for all $k$ and $g$. Our methods are purely combinatorial.