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| Format: | Preprint |
| Published: |
2022
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| Online Access: | https://arxiv.org/abs/2208.08506 |
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| _version_ | 1866911850702045184 |
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| author | Sun, Nathan |
| author_facet | Sun, Nathan |
| contents | Multidimensional permutations, or $d$-permutations, are represented by their diagrams on $[n]^d$ such that there exists exactly one point per hyperplane $x_i$ that satisfies $x_i= j$ for $i \in [d]$ and $j \in [n]$. Bonichon and Morel previously enumerated $3$-permutations avoiding small patterns, and we extend their results by first proving four conjectures, which exhaustively enumerate $3$-permutations avoiding any two fixed patterns of size $3$. We further provide a enumerative result relating $3$-permutation avoidance classes with their respective recurrence relations. In particular, we show a recurrence relation for $3$-permutations avoiding the patterns $132$ and $213$, which contributes a new sequence to the OEIS database. We then extend our results to completely enumerate $3$-permutations avoiding three patterns of size $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_08506 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On $d$-permutations and Pattern Avoidance Classes Sun, Nathan Combinatorics 05A05 Multidimensional permutations, or $d$-permutations, are represented by their diagrams on $[n]^d$ such that there exists exactly one point per hyperplane $x_i$ that satisfies $x_i= j$ for $i \in [d]$ and $j \in [n]$. Bonichon and Morel previously enumerated $3$-permutations avoiding small patterns, and we extend their results by first proving four conjectures, which exhaustively enumerate $3$-permutations avoiding any two fixed patterns of size $3$. We further provide a enumerative result relating $3$-permutation avoidance classes with their respective recurrence relations. In particular, we show a recurrence relation for $3$-permutations avoiding the patterns $132$ and $213$, which contributes a new sequence to the OEIS database. We then extend our results to completely enumerate $3$-permutations avoiding three patterns of size $3$. |
| title | On $d$-permutations and Pattern Avoidance Classes |
| topic | Combinatorics 05A05 |
| url | https://arxiv.org/abs/2208.08506 |