On 132-Avoiding Permutations with an Adjacency Constraint
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911619859087360 |
|---|---|
| author | Nadler, Nathaniel |
| author_facet | Nadler, Nathaniel |
| contents | We study permutations in $S_n$ that simultaneously avoid the pattern $132$ and satisfy the adjacency bound $|π_{i+1} - π_i| \leq m$ for all $i$, denoting their number by $A_n^{(m)}$. This combination of a global pattern restriction and a local bounded-difference condition produces a strong structural collapse: whereas unrestricted $132$-avoiding permutations are counted by the Catalan numbers with exponential growth rate $4$, the adjacency constraint forces the maximum element $n$ to occupy only positions in $\{1, 2, \ldots, m\} \cup \{n\}$. We give a complete solution for $m = 2$ by partitioning the class according to the position of the maximum element. This yields explicit recurrences and a rational generating function, from which we derive asymptotic growth of the form $A_n^{(2)} \sim C α^n$ with $α\approx 1.4656$. We conjecture that for each fixed $m$, the class admits a finite-state structural decomposition leading to linear recurrences with constant coefficients and rational generating functions, with growth constants increasing to $4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22135 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On 132-Avoiding Permutations with an Adjacency Constraint Nadler, Nathaniel Combinatorics 05A05 (Primary), 05A15, 05A16 (Secondary) We study permutations in $S_n$ that simultaneously avoid the pattern $132$ and satisfy the adjacency bound $|π_{i+1} - π_i| \leq m$ for all $i$, denoting their number by $A_n^{(m)}$. This combination of a global pattern restriction and a local bounded-difference condition produces a strong structural collapse: whereas unrestricted $132$-avoiding permutations are counted by the Catalan numbers with exponential growth rate $4$, the adjacency constraint forces the maximum element $n$ to occupy only positions in $\{1, 2, \ldots, m\} \cup \{n\}$. We give a complete solution for $m = 2$ by partitioning the class according to the position of the maximum element. This yields explicit recurrences and a rational generating function, from which we derive asymptotic growth of the form $A_n^{(2)} \sim C α^n$ with $α\approx 1.4656$. We conjecture that for each fixed $m$, the class admits a finite-state structural decomposition leading to linear recurrences with constant coefficients and rational generating functions, with growth constants increasing to $4$. |
| title | On 132-Avoiding Permutations with an Adjacency Constraint |
| topic | Combinatorics 05A05 (Primary), 05A15, 05A16 (Secondary) |
| url | https://arxiv.org/abs/2604.22135 |