Odd diagrams, Bruhat order, and pattern avoidance

Abstract

The odd diagram of a permutation is a subset of the classical diagram with additional parity conditions. In this paper, we study classes of permutations with the same odd diagram, which we call odd diagram classes. First, we prove a conjecture relating odd diagram classes and 213-and 312-avoiding permutations. Secondly, we show that each odd diagram class is a Bruhat interval. Instrumental to our proofs is an explicit description of the Bruhat edges that link permutations in a class.

Poincaré Polynomials of Odd Diagram Classes

“…Let Γ w (resp., Γ w ) denote the bipartite graph on P w 1 ∪ P w 2 (resp., P w ℓ(w)−1 ∪ P w ℓ(w)−2 ) with edges given by the covering relations in the Bruhat order. As noticed by Brenti, Carnevale and Tenner [6], odd diagram classes are not self-dual in general. For example, the following odd diagram class [654172839,958172634] is not self-dual.…”

“…In this section, we give an overview of the Bruhat order for the symmetric group. We also describe the legal move operation introduced by Brenti, Carnevale and Tenner [6], which plays a fundamental role in the study of odd diagram classes.…”

“…Assume that u = v ∈ S n have the same odd diagram. It was observed in [6] that one can apply a legal move to u to obtain a permutation which is "closer" to v. Define…”

“…The odd diagram of w is a diagram representation of its odd inversions, which can be viewed as an odd analogue of the classical Rothe diagram of w. Specifically, the odd diagram D o (w) of w is a subset of boxes in an n × n square grid defined by D o (w) = {(i, j) : w i > j, i < w −1 (j), i ≡ w −1 (j) (mod 2)}, where w −1 is the inverse of w. Here, we use the matrix coordinates, and use (i, j) to denote the box in row i and column j. Brenti, Carnevale and Tenner [6] proved that odd diagram classes partition the symmetric group in an extremely pleasant way. They conjectured that Perm(D) satisfies a stronger symmetry property.…”

“…They conjectured that Perm(D) satisfies a stronger symmetry property. Conjecture 1.2 (Brenti-Carnevale-Tenner [6,Conjecture 6.12]). Each odd diagram class is rank-symmetric in the Bruhat order.…”