{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T19:42:58Z","timestamp":1767987778985,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $a,b$ be fixed positive coprime integers. For a positive integer $g$, write $W_k(g)$ 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 connecting the startpoint to the endpoint). We determine $|W_k(g)|$ for all $k$ and $g$. The enumeration of lattice paths with respect to a linear boundary while accounting for flaws has a long and rich history, dating back at least to the 1949 results of Chung and Feller. The only previously known values of $|W_k(g)|$ are the extremal cases $k = 0$ and $k = g(a+b)-1$, determined by Bizley in 1954. Our main combinatorial result is that a certain subset of $W_k(g)$ is in bijection with $W_{k+1}(g)$. One consequence is that the value $|W_k(g)|$ is constant over each successive set of $a+b$ values of $k$. This in turn allows us to derive a recursion for $|W_k(g)|$ whose base case is given by Bizley's result for $k=0$. We solve this recursion to obtain a closed form expression for $|W_k(g)|$ for all $k$ and $g$. Our methods are purely combinatorial.<\/jats:p>","DOI":"10.37236\/13115","type":"journal-article","created":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T14:32:30Z","timestamp":1767969150000},"source":"Crossref","is-referenced-by-count":0,"title":["Combinatorial Enumeration of Lattice Paths by Flaws with Respect to a Linear Boundary of Rational Slope"],"prefix":"10.37236","volume":"33","author":[{"given":"Federico","family":"Firoozi","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Jedwab","sequence":"additional","affiliation":[]},{"given":"Amarpreet","family":"Rattan","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,1,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T14:32:30Z","timestamp":1767969150000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,9]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/13115","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,9]]},"article-number":"P1.3"}}