Odd and Even Major Indices and One-Dimensional Characters for Classical Weyl Groups

Abstract

We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz’s identity, of the Gessel–Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs.

Odd diagrams, Bruhat order, and pattern avoidance

“…Odd analogues of well-known combinatorial objects and statistics associated with permutations (and, more generally, with Weyl and Coxeter group elements) have been recently considered and studied (see, for instance, [3,4,5,6,7,10,13,14,15]). In particular, odd analogues of permutation diagrams, called odd diagrams, were introduced and studied in [6].…”

Odd diagrams, Bruhat order, and pattern avoidance

“…Odd analogues of well-known combinatorial objects and statistics associated with permutations (and, more generally, with Weyl and Coxeter group elements) have been recently considered and studied (see, for instance, [3,4,5,6,7,10,13,14,15]). In particular, odd analogues of permutation diagrams, called odd diagrams, were introduced and studied in [6].…”