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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.09590 |
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| _version_ | 1866911058222907392 |
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| author | Firoozi, Federico Jedwab, Jonathan Rattan, Amarpreet |
| author_facet | Firoozi, Federico Jedwab, Jonathan Rattan, Amarpreet |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09590 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Combinatorial enumeration of lattice paths by flaws with respect to a linear boundary of rational slope Firoozi, Federico Jedwab, Jonathan Rattan, Amarpreet Combinatorics 05A15 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. |
| title | Combinatorial enumeration of lattice paths by flaws with respect to a linear boundary of rational slope |
| topic | Combinatorics 05A15 |
| url | https://arxiv.org/abs/2406.09590 |