Restricted Grassmannian permutations

Parity Permutation Pattern Matching

“…odd) elements of π. For one thing, pattern avoidance with additional constraints [3,9], including parity restrictions [20,14], has emerged as a promising research area. For another, Parity Permutation Pattern Matching aims at providing concrete use cases of the 2-colored extension of PPM introduced in [16].…”

Pattern-avoiding even and odd Grassmannian permutations

“…This set is in bijection with the set of Dyck paths of semilength n having at most one long ascent. 1 We will call them Grassmannian Dyck paths. Further, let G n (σ) denote the set of elements in G n that avoid the pattern σ, and let G * n (σ) = G n (σ)\{id n }.…”

“…Further, let G n (σ) denote the set of elements in G n that avoid the pattern σ, and let G * n (σ) = G n (σ)\{id n }. The enumeration of G n (σ) for a pattern σ of arbitrary size was studied in [1]. Clearly, G n (σ) = G n whenever des(σ) > 1.…”

“…Recall that a permutation is said to be even if it has an even number of inversions (occurrences of the pattern 21); otherwise the permutation is said to be odd. As discussed in [1,Remark 5.2],…”

“…As it turns out, the enumeration of G odd n (σ) (or G even n (σ)) leads to multiple Wilf equivalence classes. Because of Theorem 1.1, it suffices to only enumerate the odd or the even 1 An explicit bijection can be found in [1, Section 4].…”