310 research outputs found
Odd length for even hyperoctahedral groups and signed generating functions
We define a new statistic on the even hyperoctahedral groups which is a
natural analogue of the odd length statistic recently defined and studied on
Coxeter groups of types and . We compute the signed (by length)
generating function of this statistic over the whole group and over its maximal
and some other quotients and show that it always factors nicely. We also
present some conjectures
Alternating subgroups of Coxeter groups
We study combinatorial properties of the alternating subgroup of a Coxeter
group, using a presentation of it due to Bourbaki.Comment: 39 pages, 3 figure
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres
Recurrence Relations for Strongly q-Log-Convex Polynomials
We consider a class of strongly q-log-convex polynomials based on a
triangular recurrence relation with linear coefficients, and we show that the
Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the
Dowling polynomials are strongly q-log-convex. We also prove that the Bessel
transformation preserves log-convexity.Comment: 15 page
Mixed Bruhat operators and Yang-Baxter equations for Weyl groups
We introduce and study a family of operators which act in the span of a Weyl
group and provide a multi-parameter solution to the quantum Yang-Baxter
equations of the corresponding type. Our operators generalize the "quantum
Bruhat operators" that appear in the explicit description of the multiplicative
structure of the (small) quantum cohomology ring of .
The main combinatorial applications concern the "tilted Bruhat order," a
graded poset whose unique minimal element is an arbitrarily chosen element
. (The ordinary Bruhat order corresponds to the case .) Using the
mixed Bruhat operators, we prove that these posets are lexicographically
shellable, and every interval in a tilted Bruhat order is Eulerian. This
generalizes well known results of Verma, Bjorner, Wachs, and Dyer.Comment: 19 page
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