A unification of permutation patterns related to Schubert varieties

Abstract

International audience We prove new connections between permutation patterns and singularities of Schubert varieties, by giving a new characterization of factorial and Gorenstein varieties in terms of so called bivincular patterns. These are generalizations of classical patterns where conditions are placed on the location of an occurrence in a permutation, as well as on the values in the occurrence. This clarifies what happens when the requirement of smoothness is weakened to factoriality and further to Gorensteinness, extending work of Bousquet-Mélou and Butler (2007), and Woo and Yong (2006). We also prove results that translate some known patterns in the literature into bivincular patterns. Nous démontrons de nouveaux liens entre les motifs de permutation et les singularités des variétés de Schubert, par la méthode de donner une nouvelle caractérisation des variétés factorielles et de Gorenstein par rapport à les motifs bivinculaires. Ces motifs sont généralisations des motifs classiques où des conditions se posent sur la position d'une occurrence dans une permutation, aussi bien que sur les valeurs qui se présentent dans l'occurrence. Ceci éclaircit les phénomènes où la condition de nonsingularité s'affaiblit á factorialité et même à Gorensteinité, et augmente les travaux de Bousquet-Mélou et Butler (2007), et de Woo et Yong (2006). Nous démontrons également des résultats qui traduisent quelques motifs connus en la littérature en motifs bivinculaires.

Coincidental pattern avoidance

“…Vincular patterns were also surveyed in [9]. (For the sake of completeness, we point out that the other types of patterns mentioned above can be found in [3,4,13,15], respectively. More generally, patterns in both permutations and words have been explored in the text [7].)…”

Permutation classes and polyomino classes with excluded submatrices

“…A first generalization of pattern avoidance in permutations introduces adjacency constraints among the elements of a permutation that should form an occurrence of a (otherwise classical) pattern. Such patterns with adjacency constraints are known as vincular and bivincular patterns [8], and a generalization with additional border constraints has recently been introduced by [31].…”

“…Mesh patterns are another generalization of permutation patterns that has been introduced more recently by Brändén and Claesson [9]. It has itself been generalized in several way, in particular by Úlfarsson who introduced in [31] the notion of marked mesh patterns. It is very easy to see that the avoidance of a quasi-permutation matrix with no uncovered 0 entries 4 can be expressed as the avoidance of a special form of marked-mesh pattern.…”

Pattern Avoidance in Task-Precedence Posets

“…If π does not contain ρ, then π is said to avoid ρ. The classical definition of pattern avoidance in permutations has shown itself to be worthwhile in many fields including algebraic geometry [17] and theoretical computer science [9]. Analogues of pattern avoidance have been developed for a variety of combinatorial objects including Dyck paths [1], tableaux [11], set partitions [15], trees [14], posets [8], and many more.…”