Minimal Overlapping Patterns in Colored Permutations

Advances in Applied Mathematics

2018

Two permutations π and τ are c-Wilf equivalent if, for each n, the number of permutations in S n avoiding π as a consecutive pattern (i.e., in adjacent positions) is the same as the number of those avoiding τ . In addition, π and τ are strongly c-Wilf equivalent if, for each n and k, the number of permutations in S n containing k occurrences of π as a consecutive pattern is the same as for τ . In this paper we introduce a third, more restrictive equivalence relation, defining π and τ to be super-strongly c-Wilf equivalent if the above condition holds for any set of prescribed positions for the k occurrences. We show that, when restricted to non-overlapping permutations, these three equivalence relations coincide.We also give a necessary condition for two permutations to be strongly c-Wilf equivalent. Specifically, we show that if π, τ ∈ S m are strongly c-Wilf equivalent, then |π m − π 1 | = |τ m − τ 1 |.In the special case of non-overlapping permutations π and τ , this proves a weaker version of a conjecture of the second author stating that π and τ are c-Wilf equivalent if and only if π 1 = τ 1 and π m = τ m , up to trivial symmetries. Finally, we strengthen a recent result of Nakamura and Khoroshkin-Shapiro giving sufficient conditions for strong c-Wilf equivalence.

Section: Introduction and Summary Of Resultsmentioning

confidence: 60%

Wilf equivalence relations for consecutive patterns

Dwyer

Advances in Applied Mathematics

2018

“…We denote by N m the set of non-overlapping patterns in S m . These patterns have been considered recently by Duane and Remmel [11] and by Bóna [9], who shows that |N m |/|S m | > 0.364 for all m.…”

Section: Non-overlapping Patternsmentioning

confidence: 62%

“…For example, the patterns 132, 1243, 1342, 21534 and 34671285 are non-overlapping. Non-overlapping patterns have been studied by Duane and Remmel [11] and by Bóna [9], who shows that the proportion of non-overlapping patterns of any length m is at least 0.364. It is easy to see that 1 ∈ O σ if and only if σ is monotone.…”

Section: Consecutive Patternsmentioning

confidence: 99%

The most and the least avoided consecutive patterns

Proceedings of the London Mathematical Society

2012

We prove that the number of permutations avoiding the consecutive pattern 12… m, that is, containing no m adjacent entries in increasing order, is asymptotically larger than the number of permutations avoiding any other consecutive pattern of length m. This settles a conjecture of Elizalde and Noy from 2001. We also prove a recent conjecture of Nakamura stating that, at the other end of the spectrum, the number of permutations avoiding 12… (m−2)m(m−1) is asymptotically smaller than for any other pattern. Finally, we consider non‐overlapping patterns and obtain analogous results describing the most and least avoided ones. The techniques used include the cluster method of Goulden and Jackson, an interpretation of clusters as linear extensions of posets, and singularity analysis of generating functions.

“…Another important result from [55] is an expression of P σ (0, z) as the reciprocal of a power series whose coefficients are signed sums of permutations having occurrences of σ at prescribed positions determined by a certain set associated to σ . This power series is reminiscent of the cluster generating function R σ (−1, z), and in the case of non-overlapping patterns it actually coincides with it [25]. Subsequently, Liese and Remmel [54] gave explicit expressions for this power series in some cases where σ is a shuffle of an increasing sequence with another pattern.…”

Section: Symmetric Functions and Brick Tabloidsmentioning

confidence: 77%

A survey of consecutive patterns in permutations

The IMA Volumes in Mathematics and Its Applications

2016

A consecutive pattern in a permutation π is another permutation σ determined by the relative order of a subsequence of contiguous entries of π. Traditional notions such as descents, runs and peaks can be viewed as particular examples of consecutive patterns in permutations, but the systematic study of these patterns has flourished in the last 15 years, during which a variety of different techniques have been used. We survey some interesting developments in the subject, focusing on exact and asymptotic enumeration results, the classification of consecutive patterns into equivalence classes, and their applications to the study of one-dimensional dynamical systems. arXiv:1504.07265v2 [math.CO]