Enumeration and Distribution of Permutation Rows and Columns in Equi-$n$-Squares
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909991842086912 |
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| author | Pendleton, Andrew |
| author_facet | Pendleton, Andrew |
| contents | We introduce consecutive equi-$n$-squares, a variant of equi-$n$-squares in which at least one row or column forms a fixed permutation of $\{1,\dots,n\}$, taken for concreteness to be $(1,\dots,n)$. More generally, the enumeration and probabilistic arguments presented here extend to the occurrence of any prescribed permutation as a row or column of an equi-$n$-square. We derive exact and asymptotic formulas for the number of consecutive equi-$n$-squares, showing precisely how their proportion among all equi-$n$-squares rapidly approaches zero as $n\to\infty$.
We also analyze the distribution of consecutive equi-$n$-squares under uniform random sampling and explore connections to algebraic structures, interpreting equi-$n$-squares and consecutive equi-$n$-squares as Cayley tables. Finally, we supplement our theoretical results with Monte Carlo simulations for small values of $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumeration and Distribution of Permutation Rows and Columns in Equi-$n$-Squares Pendleton, Andrew Combinatorics We introduce consecutive equi-$n$-squares, a variant of equi-$n$-squares in which at least one row or column forms a fixed permutation of $\{1,\dots,n\}$, taken for concreteness to be $(1,\dots,n)$. More generally, the enumeration and probabilistic arguments presented here extend to the occurrence of any prescribed permutation as a row or column of an equi-$n$-square. We derive exact and asymptotic formulas for the number of consecutive equi-$n$-squares, showing precisely how their proportion among all equi-$n$-squares rapidly approaches zero as $n\to\infty$. We also analyze the distribution of consecutive equi-$n$-squares under uniform random sampling and explore connections to algebraic structures, interpreting equi-$n$-squares and consecutive equi-$n$-squares as Cayley tables. Finally, we supplement our theoretical results with Monte Carlo simulations for small values of $n$. |
| title | Enumeration and Distribution of Permutation Rows and Columns in Equi-$n$-Squares |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.11980 |