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A logical limit law for $231$-avoiding permutations
- Source :
- Discrete Mathematics & Theoretical Computer Science, vol. 26:1, Permutation Patterns 2023, Special issues (April 2, 2024) dmtcs:11751
- Publication Year :
- 2022
-
Abstract
- We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $\Psi$, in the language of two total orders, the probability $p_{n,\Psi}$ that a uniform random 231-avoiding permutation of size $n$ satisfies $\Psi$ admits a limit as $n$ is large. Moreover, we establish two further results about the behavior and value of $p_{n,\Psi}$: (i) it is either bounded away from $0$, or decays exponentially fast; (ii) the set of possible limits is dense in $[0,1]$. Our tools come mainly from analytic combinatorics and singularity analysis.<br />Comment: 15 pages; version 3 is the final version, ready for publication in DMTCS
Details
- Database :
- arXiv
- Journal :
- Discrete Mathematics & Theoretical Computer Science, vol. 26:1, Permutation Patterns 2023, Special issues (April 2, 2024) dmtcs:11751
- Publication Type :
- Report
- Accession number :
- edsarx.2210.05537
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.46298/dmtcs.11751