61 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
Combinatorially interpreting generalized Stirling numbers
Let be a word in alphabet with 's and 's.
Interpreting "" as multiplication by , and "" as differentiation with
respect to , the identity , valid
for any smooth function , defines a sequence , the terms of
which we refer to as the {\em Stirling numbers (of the second kind)} of .
The nomenclature comes from the fact that when , we have , the ordinary Stirling number of the second kind.
Explicit expressions for, and identities satisfied by, the have been
obtained by numerous authors, and combinatorial interpretations have been
presented. Here we provide a new combinatorial interpretation that retains the
spirit of the familiar interpretation of as a count of
partitions. Specifically, we associate to each a quasi-threshold graph
, and we show that enumerates partitions of the vertex set of
into classes that do not span an edge of . We also discuss some
relatives of, and consequences of, our interpretation, including -analogs
and bijections between families of labelled forests and sets of restricted
partitions.Comment: To appear in Eur. J. Combin., doi:10.1016/j.ejc.2014.07.00
Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas
We investigate Rees algebras and special fiber rings obtained by blowing up
specialized Ferrers ideals. This class of monomial ideals includes strongly
stable monomial ideals generated in degree two and edge ideals of prominent
classes of graphs. We identify the equations of these blow-up algebras. They
generate determinantal ideals associated to subregions of a generic symmetric
matrix, which may have holes. Exhibiting Gr\"obner bases for these ideals and
using methods from Gorenstein liaison theory, we show that these determinantal
rings are normal Cohen-Macaulay domains that are Koszul, that the initial
ideals correspond to vertex decomposable simplicial complexes, and we determine
their Hilbert functions and Castelnuovo-Mumford regularities. As a consequence,
we find explicit minimal reductions for all Ferrers and many specialized
Ferrers ideals, as well as their reduction numbers. These results can be viewed
as extensions of the classical Dedekind-Mertens formula for the content of the
product of two polynomials.Comment: 36 pages, 9 figures. In the updated version, section 7: "Final
remarks and open problems" is new; the introduction was updated accordingly.
References update
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