We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern 2143. We use a generating tree approach to construct a recursive bijection between the set A 2n (2143) of alternating permutations of length 2n avoiding 2143 and standard Young tableaux of shape n, n, n and between the set A 2n+1 (2143) of alternating permutations of length 2n + 1 avoiding 2143 and shifted standard Young tableaux of shape n + 2, n + 1, n . We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizations thereof.

We define a class L n,k of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give bijections between the set A 2n (1234) of alternating permutations of length 2n with no four-term increasing subsequence and standard Young tableaux of shape 3 n , and between the set A 2n+1 (1234) and standard Young tableaux of shape 3 n−1 , 2, 1 . This represents the first enumeration of alternating permutations avoiding a pattern of length four. We also extend previous work on doubly-alternating permutations (alternating permutations whose inverses are alternating) to our more general context.The set L n,k may be viewed as the set of reading words of the standard Young tableaux of a certain skew shape. In the last section of the paper, we expand our study to consider pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape λ/μ whose reading words avoid 213 is a natural μ-analogue of the Catalan numbers (and in particular does not depend on λ, up to a simple technical condition), and that there are similar results for the patterns 132, 231 and 312.

International audience The number of shortest factorizations into reflections for a Singer cycle in $GL_n(\mathbb{F}_q)$ is shown to be $(q^n-1)^{n-1}$. Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given. Nous prouvons que le nombre de factorisations de longueur minimale d’un cycle de Singer dans $GL_n(\mathbb{F}_q)$ comme un produit de réflexions est $(q^n-1)^{n-1}$. Nous présentons aussi des formules donnant le nombre de factorisations de toutes les longueurs ainsi que des formules pour le nombre de factorisations comme produit de réflexions ayant des classes de conjugaison fixes.

Abstract. This paper studies a partial order on the general linear group GL(V ) called the absolute order, derived from viewing GL(V ) as a group generated by reflections, that is, elements whose fixed space has codimension one. The absolute order on GL(V ) is shown to have two equivalent descriptions, one via additivity of length for factorizations into reflections, the other via additivity of fixed space codimensions. Other general properties of the order are derived, including self-duality of its intervals.Working over a finite field F q , it is shown via a complex character computation that the poset interval from the identity to a Singer cycle (or any regular elliptic element) in GL n (F q ) has a strikingly simple formula for the number of chains passing through a prescribed set of ranks.

We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are q-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a q-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.