103 research outputs found
The Eulerian Distribution on Involutions is Indeed Unimodal
Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point
free involutions) of {1,...,n} with k descents. Motivated by Brenti's
conjecture which states that the sequence I_{n,0}, I_{n,1},..., I_{n,n-1} is
log-concave, we prove that the two sequences I_{n,k} and J_{2n,k} are unimodal
in k, for all n. Furthermore, we conjecture that there are nonnegative integers
a_{n,k} such that This statement is stronger than
the unimodality of I_{n,k} but is also interesting in its own right.Comment: 12 pages, minor changes, to appear in J. Combin. Theory Ser.
The derived moduli space of stable sheaves
We construct the derived scheme of stable sheaves on a smooth projective
variety via derived moduli of finite graded modules over a graded ring. We do
this by dividing the derived scheme of actions of Ciocan-Fontanine and Kapranov
by a suitable algebraic gauge group. We show that the natural notion of
GIT-stability for graded modules reproduces stability for sheaves
A good leaf order on simplicial trees
Using the existence of a good leaf in every simplicial tree, we order the
facets of a simplicial tree in order to find combinatorial information about
the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire
splitting of the ideal, as well as a refinement of a recursive formula of H\`a
and Van Tuyl for computing the graded Betti numbers of simplicial trees.Comment: 17 pages, to appear; Connections Between Algebra and Geometry,
Birkhauser volume (2013
Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
We consider the problem of finding the number of matrices over a finite field
with a certain rank and with support that avoids a subset of the entries. These
matrices are a q-analogue of permutations with restricted positions (i.e., rook
placements). For general sets of entries these numbers of matrices are not
polynomials in q (Stembridge 98); however, when the set of entries is a Young
diagram, the numbers, up to a power of q-1, are polynomials with nonnegative
coefficients (Haglund 98).
In this paper, we give a number of conditions under which these numbers are
polynomials in q, or even polynomials with nonnegative integer coefficients. We
extend Haglund's result to complements of skew Young diagrams, and we apply
this result to the case when the set of entries is the Rothe diagram of a
permutation. In particular, we give a necessary and sufficient condition on the
permutation for its Rothe diagram to be the complement of a skew Young diagram
up to rearrangement of rows and columns. We end by giving conjectures
connecting invertible matrices whose support avoids a Rothe diagram and
Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl
Governing Singularities of Schubert Varieties
We present a combinatorial and computational commutative algebra methodology
for studying singularities of Schubert varieties of flag manifolds.
We define the combinatorial notion of *interval pattern avoidance*. For
"reasonable" invariants P of singularities, we geometrically prove that this
governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties
are globally not P. The prototypical case is P="singular"; classical pattern
avoidance applies admirably for this choice [Lakshmibai-Sandhya'90], but is
insufficient in general.
Our approach is analyzed for some common invariants, including
Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness,
extending [Woo-Yong'05]; the description of the singular locus (which was
independently proved by [Billey-Warrington '03], [Cortez '03],
[Kassel-Lascoux-Reutenauer'03], [Manivel'01]) is also thus reinterpreted.
Our methods are amenable to computer experimentation, based on computing with
*Kazhdan-Lusztig ideals* (a class of generalized determinantal ideals) using
Macaulay 2. This feature is supplemented by a collection of open problems and
conjectures.Comment: 23 pages. Software available at the authors' webpages. Version 2 is
the submitted version. It has a nomenclature change: "Bruhat-restricted
pattern avoidance" is renamed "interval pattern avoidance"; the introduction
has been reorganize
Actions on permutations and unimodality of descent polynomials
We study a group action on permutations due to Foata and Strehl and use it to
prove that the descent generating polynomial of certain sets of permutations
has a nonnegative expansion in the basis ,
. This property implies symmetry and unimodality. We
prove that the action is invariant under stack-sorting which strengthens recent
unimodality results of B\'ona. We prove that the generalized permutation
patterns and are invariant under the action and use this to
prove unimodality properties for a -analog of the Eulerian numbers recently
studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams.
We also extend the action to linear extensions of sign-graded posets to give
a new proof of the unimodality of the -Eulerian polynomials of
sign-graded posets and a combinatorial interpretations (in terms of
Stembridge's peak polynomials) of the corresponding coefficients when expanded
in the above basis.
Finally, we prove that the statistic defined as the number of vertices of
even height in the unordered decreasing tree of a permutation has the same
distribution as the number of descents on any set of permutations invariant
under the action. When restricted to the set of stack-sortable permutations we
recover a result of Kreweras.Comment: 19 pages, revised version to appear in Europ. J. Combi
Generalizing Tanisaki's ideal via ideals of truncated symmetric functions
We define a family of ideals in the polynomial ring
that are parametrized by Hessenberg functions
(equivalently Dyck paths or ample partitions). The ideals generalize
algebraically a family of ideals called the Tanisaki ideal, which is used in a
geometric construction of permutation representations called Springer theory.
To define , we use polynomials in a proper subset of the variables
that are symmetric under the corresponding permutation
subgroup. We call these polynomials {\em truncated symmetric functions} and
show combinatorial identities relating different kinds of truncated symmetric
polynomials. We then prove several key properties of , including that if
in the natural partial order on Dyck paths then ,
and explicitly construct a Gr\"{o}bner basis for . We use a second family
of ideals for which some of the claims are easier to see, and prove that
. The ideals arise in work of Ding, Develin-Martin-Reiner, and
Gasharov-Reiner on a family of Schubert varieties called partition varieties.
Using earlier work of the first author, the current manuscript proves that the
ideals generalize the Tanisaki ideals both algebraically and
geometrically, from Springer varieties to a family of nilpotent Hessenberg
varieties.Comment: v1 had 27 pages. v2 is 29 pages and adds Appendix B, where we include
a recent proof by Federico Galetto of a conjecture given in the previous
version. We also add some connections between our work and earlier results of
Ding, Gasharov-Reiner, and Develin-Martin-Reiner. v3 corrects a typo in
Valibouze's citation in the bibliography. To appear in Journal of Algebraic
Combinatoric
Matrix Schubert varieties and Gaussian conditional independence models
Seth Sullivant was partially supported by the David and Lucille Packard Foundation and the US National Science Foundation (DMS 0954865)
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