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Every word has a shape determined by its image under the Robinson-Schensted-Knuth correspondence. We show that when a word w contains a separable (i.e., 3142- and 2413-avoiding) permutation sigma as a pattern, the shape of w contains the shape of sigma. As an application, we exhibit lower bounds for the lengths of supersequences of sets containing separable permutations.
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The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of permutation avoidance in cyclic permutations and characterized the avoidance classes for all single permutations of length 4. We continue this work. In particular, we establish a cyclic variant of the Erdos-Szekeres Theorem that any linear permutation of length mn+1 must contain either the increasing pattern of length m+1 or the decreasing pattern of length n+1. We then derive results about avoidance of multiple patterns of length 4. We also determine generating functions for the cyclic descent statistic on these classes. Finally, we end with various open questions and avenues for future research.
An alternating permutation of length $n$ is a permutation $pi=pi_1 pi_2 ... pi_n$ such that $pi_1 < pi_2 > pi_3 < pi_4 > ...$. Let $A_n$ denote set of alternating permutations of ${1,2,..., n}$, and let $A_n(sigma)$ be set of alternating permutations in $A_n$ that avoid a pattern $sigma$. Recently, Lewis used generating trees to enumerate $A_{2n}(1234)$, $A_{2n}(2143)$ and $A_{2n+1}(2143)$, and he posed several conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns. Some of these conjectures have been proved by Bona, Xu and Yan. In this paper, we prove the two relations $|A_{2n+1}(1243)|=|A_{2n+1}(2143)|$ and $|A_{2n}(4312)|=|A_{2n}(1234)|$ as conjectured by Lewis.
Babson and Steingr{i}msson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Recently, Amini investigated the equidistributions of these Mahonian statistics over sets of pattern avoiding permutations. Moreover, he posed several conjectures. In this paper, we construct a bijection from $S_n(213)$ to $S_n(231)$, which maps the statistic $(maj,stat)$ to the statistic $(stat,maj)$. This allows us to give solutions to some of Aminis conjectures.
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.
It is known that a sequence Pi_i of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Pi_i converges to 1/24. We show that there is a set S of 4-point permutations such that the sum of the pattern densities of the permutations from S in the permutations Pi_i converges to |S|/24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight.
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