Abstract
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.
Highlights
Pattern-avoiding permutations have been studied actively in the combinatorics literature for the past four decades. (See Section 1.2 for definitions of terms we use.) Some sources on permutation patterns include: [5] for essential terminology, [9, Ch. 4] for a textbook introduction, and [30] for an in-depth survey of the literature
In a companion paper [24], we focus on the case of avoiding monotone decreasing patterns m(m − 1) · · · 321 in bounded affine permutations
We let Sn denote the set of permutations of size n, we let Sn denote the set of affine permutations of size n, and we let Sn// denote the set of bounded affine permutations of size n
Summary
Pattern-avoiding permutations have been studied actively in the combinatorics literature for the past four decades. (See Section 1.2 for definitions of terms we use.) Some sources on permutation patterns include: [5] for essential terminology, [9, Ch. 4] for a textbook introduction, and [30] for an in-depth survey of the literature. Regardless of whether S(R) is decomposable, we can enumerate the affine permutations in S(R) that are decomposable (assuming R is a set of indecomposable permutations) These are just the diagonal shifts of infinite sums of (ordinary) permutations that avoid R; the set of affine permutations that arise in this way is denoted ⊕S(R).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
