Pattern Avoidance of Cyclic Permutations

Pattern avoidance of [4,k]-pairs in circular permutations

Cyclic pattern containment and avoidance

We consider two recent open problems stating that certain statistics on various sets of combinatorial objects are equidistributed. The first, posed by Anders Claesson and Svante Linusson, relates nestings in matchings on $\{1,2,\ldots,2n\}$ to occurrences of a certain pattern in permutations in $S_n$. The second, posed by Miles Jones and Jeffrey Remmel, relates occurrences of a large class of consecutive permutation patterns to occurrences of the same pattern in the cycles of permutations. We develop a general method that solves both of these problems and many more. We further employ the Garsia-Milne involution principle to obtain purely bijective proofs of these results. Nous considérons deux derniers problèmes ouverts indiquant que certaines statistiques sur les divers ensembles d'objets combinatoires sont équiréparties. La première, posée par Anders Claesson et Svante Linusson, concerne les imbrications dans des filtrages sur $\{1,2,\ldots,2n\}$ pour les occurrences d'un certain modèle de permutations dans $S_n$. La seconde, posée par Miles Jones et Jeffrey Remmel, concerne les occurrences d'une large classe de schémas de permutation consécutive aux évènements du même modèle dans les cycles de permutations. Nous développons une méthode générale qui résout ces deux problèmes et beaucoup plus. Nous avons également utiliser le principe d'involution Garsia-Milne pour obtenir des preuves purement bijectives de ces résultats.

On a Conjecture about Pattern Avoidance of Cyclic Permutations

In this paper, we establish a natural bijection between the almost-increasing cyclic permutations of length $n$ and unimodal permutations of length $n-1$. This map is used to give a new characterization, in terms of pattern avoidance, of almost-increasing cycles. Additionally, we use this bijection to enumerate several statistics on almost-increasing cycles. Such statistics include descents, inversions, peaks and excedances, as well as the newly defined statistic called low non-inversions. Furthermore, we refine the enumeration of unimodal permutations by descents, inversions and inverse valleys. We conclude this paper with a theorem that characterizes the standard cycle notation of almost-increasing permutations.