Neal Madras ; Justin M. Troyka - Bounded affine permutations I. Pattern avoidance and enumeration

dmtcs:6178 - Discrete Mathematics & Theoretical Computer Science, March 29, 2021, vol. 22 no. 2, Permutation Patterns 2019 - https://doi.org/10.46298/dmtcs.6178
Bounded affine permutations I. Pattern avoidance and enumeration

Authors: Neal Madras ; Justin M. Troyka

We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $\tau$ is one of the finite increasing oscillations, then every $\tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size $n$. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations.

Volume: vol. 22 no. 2, Permutation Patterns 2019
Section: Special issues
Published on: March 29, 2021
Accepted on: March 13, 2021
Submitted on: March 3, 2020
Keywords: Mathematics - Combinatorics,Mathematics - Probability,05A05 (primary), 05A16, 60C05