281 research outputs found
Bruhat order, smooth Schubert varieties, and hyperplane arrangements
The aim of this article is to link Schubert varieties in the flag manifold
with hyperplane arrangements. For a permutation, we construct a certain
graphical hyperplane arrangement. We show that the generating function for
regions of this arrangement coincides with the Poincare polynomial of the
corresponding Schubert variety if and only if the Schubert variety is smooth.
We give an explicit combinatorial formula for the Poincare polynomial. Our main
technical tools are chordal graphs and perfect elimination orderings.Comment: 14 pages, 2 figure
Elliptic rook and file numbers
Utilizing elliptic weights, we construct an elliptic analogue of rook numbers
for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's
q-rook numbers by two additional independent parameters a and b, and a nome p.
These are shown to satisfy an elliptic extension of a factorization theorem
which in the classical case was established by Goldman, Joichi and White and
later was extended to the q-case by Garsia and Remmel. We obtain similar
results for our elliptic analogues of Garsia and Remmel's q-file numbers for
skyline boards. We also provide an elliptic extension of the j-attacking model
introduced by Remmel and Wachs. Various applications of our results include
elliptic analogues of (generalized) Stirling numbers of the first and second
kind, Lah numbers, Abel numbers, and r-restricted versions thereof.Comment: 45 pages; 3rd version shortened (elliptic rook theory for matchings
has been taken out to keep the length of this paper reasonable
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
Jensen polynomials for the Riemann zeta function and other sequences
In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to the
hyperbolicity of Jensen polynomials for the Riemann zeta function at
its point of symmetry. This hyperbolicity has been proved for degrees . We obtain an asymptotic formula for the central derivatives
that is accurate to all orders, which allows us to prove
the hyperbolicity of a density subset of the Jensen polynomials of each
degree. Moreover, we establish hyperbolicity for all . These results
follow from a general theorem which models such polynomials by Hermite
polynomials. In the case of the Riemann zeta function, this proves the GUE
random matrix model prediction in derivative aspect. The general theorem also
allows us to prove a conjecture of Chen, Jia, and Wang on the partition
function.Comment: 11 page
- …