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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.12789 |
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| _version_ | 1866911007070224384 |
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| author | Beluhov, Nikolai |
| author_facet | Beluhov, Nikolai |
| contents | Let $W_d(n)$ be the number of $2n$-step walks in $\mathbb{Z}^d$ which begin and end at the origin. We study the exponent of $2$ in the prime factorisation of this number; i.e., $w_d(n) = ν_2(W_d(n))$. We show that, for each $d$, there is a relationship between $w_d(n)$ and the number $s_2(n)$ of $1$s in the binary expansion of $n$. For example, $w_d(n) = s_2(n)$ if $d$ is odd and $w_d(n) = 2s_2(n)$ if $ν_2(d) = 1$; while $w_d(n) \ge 3s_2(n)$ if $ν_2(d) = 2$. The pattern changes further when $ν_2(d) \ge 3$. However, for each $d$, we give the best analogous estimate of $w_d(n)$ together with a description of all $n$ where equality is attained. The methods we develop apply to a wider range of problems as well, and so might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12789 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Powers of 2 in High-Dimensional Lattice Walks Beluhov, Nikolai Combinatorics 05A05, 05A10 Let $W_d(n)$ be the number of $2n$-step walks in $\mathbb{Z}^d$ which begin and end at the origin. We study the exponent of $2$ in the prime factorisation of this number; i.e., $w_d(n) = ν_2(W_d(n))$. We show that, for each $d$, there is a relationship between $w_d(n)$ and the number $s_2(n)$ of $1$s in the binary expansion of $n$. For example, $w_d(n) = s_2(n)$ if $d$ is odd and $w_d(n) = 2s_2(n)$ if $ν_2(d) = 1$; while $w_d(n) \ge 3s_2(n)$ if $ν_2(d) = 2$. The pattern changes further when $ν_2(d) \ge 3$. However, for each $d$, we give the best analogous estimate of $w_d(n)$ together with a description of all $n$ where equality is attained. The methods we develop apply to a wider range of problems as well, and so might be of independent interest. |
| title | Powers of 2 in High-Dimensional Lattice Walks |
| topic | Combinatorics 05A05, 05A10 |
| url | https://arxiv.org/abs/2506.12789 |