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A decomposition of \(m\)-Dyck paths,
preprint (arXiv, 2025).
An \(m\)-Dyck path is a lattice path from (0,0) to \((mn,n)\) taking steps (1,0) or (0,1) and never go below the \(my=x\) diagonal.
We construct a decomposition of an \(m\)-Dyck path into an \(m\)-tuple of Dyck paths such that the area sequence and bounce sequence of the \(m\)-Dyck path are entrywise the sum of the area sequences and bounce sequences of the Dyck paths in the tuple.
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Rowmotion and Echelonmotion,
with Colin Defant,
Rene Marczinzik,
Adrien Segovia,
David E Speyer,
Hugh Thomas,
Nathan Williams,
preprint (arXiv, 2025).
Given a linear extension \(\sigma\) of a finite poset \(R\), we consider the
permutation matrix indexing the Bruhat cell containing the Cartan matrix of
\(R\) with respect to \(\sigma\). This yields a bijection
\(\mathrm{Ech}_\sigma: R\to R\) that we call echelonmotion; it is the
inverse of the Coxeter permutation studied by Klász, Marczinzik, and Thomas.
Those authors proved that echelonmotion agrees with rowmotion when \(R\) is a
distributive lattice. We generalize this result to semidistributive lattices.
In addition, we prove that every trim lattice has a linear extension with
respect to which echelonmotion agrees with rowmotion. We also show that
echelonmotion on an Eulerian poset (with respect to any linear extension) is an
involution. Finally, we initiate the study of echelon-independent posets, which
are posets for which echelonmotion is independent of the chosen linear
extension. We prove that a lattice is echelon-independent if and only if it is
semidistributive. Moreover, we show that echelon-independent connected posets
are bounded and have semidistributive MacNeille completions.
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The Ehrhart series of alcoved polytopes,
with Elisabeth Bullock, preprint (arXiv, 2024).
Alcoved polytopes are convex polytopes, which are the closure of a union of
alcoves in an affine Coxeter arrangement. They are rational polytopes and,
therefore, have Ehrhart quasipolynomials. Here we describe a method for
computing the generating function of the Ehrhart quasipolynomial, or Ehrhart
series, of any alcoved polytope via a particular shelling order of its alcoves.
We also show a connection between Early’s decorated ordered set partitions
and this shelling order for the hypersimplex \(\Delta_{2,n}\).
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The Ehrhart \(h^*\)-polynomial of positroid polytopes,
extended abstract (FPSAC 2025), accepted by
Combinatorial Theory.
A positroid is a matroid realized by a matrix such that all maximal minors are
non-negative. Positroid polytopes are matroid polytopes of positroids. In
particular, they are lattice polytopes. The Ehrhart polynomial of a lattice
polytope counts the number of integer points in the dilation of that polytope.
The Ehrhart series is the generating function of the Ehrhart polynomial, which
is a rational function with the numerator called the \(h^*\)-polynomial. We
compute the \(h^*\)-polynomial of an arbitrary positroid polytope and an
arbitrary half-open positroid polytope. Our result generalizes that of Katzman,
Early, Kim, and Li for hypersimplices.
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The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model,
extended abstract (FPSAC 2024).
The multispecies asymmetric simple exclusion process (mASEP) is a Markov chain
in which particles of different species hop along a one-dimensional lattice.
This paper studies the doubly asymmetric simple exclusion process DASEP(n,p,q)
in which q particles with species 1,…,p hop along a circular lattice with
n sites, but also the particles are allowed to spontaneously change from one
species to another. In this paper, we introduce two related Markov chains called
the colored Boolean process and the restricted random growth model, and we show
that the DASEP lumps to the colored Boolean process, and the colored Boolean
process lumps to the restricted random growth model. This allows us to
generalize a theorem of David Ash on the relations between sums of steady state
probabilities. We also give explicit formulas for the stationary distribution of
DASEP(n,2,2).
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Linear Spaces of Symmetric Matrices with Non-Maximal Maximum Likelihood Degree,
with Kathlén Kohn and
Rosa Winter, Le Matematiche, 76 no. 2 (2021): 461–481.
We study the maximum likelihood degree of linear concentration models in
algebraic statistics. We relate the geometry of the reciprocal variety to that
of semidefinite programming. We show that the Zariski closure in the Grassmanian
of the set of linear spaces that do not attain their maximal possible maximum
likelihood degree coincides with the Zariski closure of the set of linear spaces
defining a projection with non-closed image of the positive semidefinite cone.
In particular, this shows that this closure is a union of coisotropic
hypersurfaces.
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Bad Projections of the PSD cone,
with Bernd Sturmfels, Collectanea Mathematica, 72 (2021): 261–280.
The image of the cone of positive semidefinite matrices under a linear map is a
convex cone. Pataki characterized the set of linear maps for which that image is
not closed. The Zariski closure of this set is a hypersurface in the
Grassmannian. Its components are the coisotropic hypersurfaces of symmetric
determinantal varieties. We develop the convex algebraic geometry of such bad
projections, with focus on explicit computations.
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Singularities and genus of the k-ellipse,
with Weiqiao Han, Journal of Symbolic Computation, 104 (2021): 343–355.
A k-ellipse is a plane curve consisting of all points whose distances from k
fixed foci sum to a constant. We determine the singularities and genus of its
Zariski closure in the complex projective plane. The paper resolves an open
problem stated by Nie, Parrilo and Sturmfels in 2008.