11 research outputs found
Combinatorial Invariants of Rational Polytopes
The first part of this dissertation deals with the equivariant Ehrhart theory of the permutahedron. As a starting point to determining the equivariant Ehrhart theory of the permutahedron, Ardila, Schindler, and I obtain a volume formula for the rational polytopes that are fixed by acting on the permutahedron by a permutation, which generalizes a result of Stanley’s for the volume for the standard permutahedron. Building from the aforementioned work, Ardila, Supina, and I determine the equivariant Ehrhart theory of the permutahedron, thereby resolving an open problem posed by Stapledon. We provide combinatorial descriptions of the Ehrhart quasipolynomial and Ehrhart series of the fixed polytopes of the permutahedron. Additionally, we answer questions regarding the polynomiality of the equivariant analogue of the h*-polynomial.
The second part of this dissertation deals with decompositions of the h*-polynomial for rational polytopes. An open problem in Ehrhart theory is to classify all Ehrhart quasipolynomials. Toward this classification problem, one may ask for necessary in- equalities among the coefficients of an h*-polynomial. Beck, Braun, and I contribute such inequalities when P is a rational polytope. Additionally, we provide two decompositions of the h*-polynomial for rational polytopes, thereby generalizing results of Betke and McMullen and Stapledon. We use our rational Betke–McMullen formula to provide a novel proof of Stanley’s Monotonicity Theorem for rational polytopes
A Brief Survey on Lattice Zonotopes
Zonotopes are a rich and fascinating family of polytopes, with connections to
many areas of mathematics. In this article we provide a brief survey of
classical and recent results related to lattice zonotopes. Our emphasis is on
connections to combinatorics, both in the sense of enumeration (e.g. Ehrhart
theory) and combinatorial structures (e.g. graphs and permutations)
Generalized parking function polytopes
A classical parking function of length is a list of positive integers
whose nondecreasing rearrangement satisfies . The convex hull of all parking
functions of length is an -dimensional polytope in , which
we refer to as the classical parking function polytope. Its geometric
properties have been explored in (Amanbayeva and Wang 2022) in response to a
question posed in (Stanley 2020). We generalize this family of polytopes by
studying the geometric properties of the convex hull of -parking
functions for , which we refer to as
-parking function polytopes. We explore connections between these
-parking function polytopes, the Pitman-Stanley polytope, and the
partial permutahedra of (Heuer and Striker 2022). In particular, we establish a
closed-form expression for the volume of -parking function
polytopes. This allows us to answer a conjecture of (Behrend et al. 2022) and
also obtain a new closed-form expression for the volume of the convex hull of
classical parking functions as a corollary.Comment: 29 pages, 3 figures, comments welcome
A Reflection on Growth Mindset and Meritocracy
As mathematicians working in higher education we reflect on meritocracy and growth mindset with a focus on the relationship between the two. We also note the subtle differences between growth mindset and grit. Our reflection ends with suggestions for how to move forward in the math classroom and throughout the collegiate level
Local -polynomials for one-row Hermite normal form simplices
The local -polynomial of a lattice polytope is an important invariant
arising in Ehrhart theory. Our focus in this work is on lattice simplices
presented in Hermite normal form with a single non-trivial row. We prove that
when the off-diagonal entries are fixed, the distribution of coefficients for
the local -polynomial of these simplices has a limit as the normalized
volume goes to infinity. Further, this limiting distribution is determined by
the coefficients for a relatively small normalized volume. We also provide a
thorough analysis of two specific families of such simplices, to illustrate and
motivate our main result
Triangulations, order polytopes, and generalized snake posets
This work regards the order polytopes arising from the class of generalized
snake posets and their posets of meet-irreducible elements. Among generalized
snake posets of the same rank, we characterize those whose order polytopes have
minimal and maximal volume. We give a combinatorial characterization of the
circuits in these order polytopes and then conclude that every regular
triangulation is unimodular. For a generalized snake word, we count the number
of flips for the canonical triangulation of these order polytopes. We determine
that the flip graph of the order polytope of the poset whose lattice of filters
comes from a ladder is the Cayley graph of a symmetric group. Lastly, we
introduce an operation on triangulations called twists and prove that twists
preserve regular triangulations.Comment: 39 pages, 26 figures, comments welcomed
Stack-sorting simplices: geometry and lattice-point enumeration
We study the polytopes that arise from the convex hulls of stack-sorting on
particular permutations. We show that they are simplices and proceed to study
their geometry and lattice-point enumeration. First, we prove some enumerative
results on permutations, i.e., permutations of length whose
penultimate and last entries are and , respectively. Additionally, we
then focus on a specific permutation, which we call , and show that the
convex hull of all its iterations through the stack-sorting algorithm share the
same lattice-point enumerator as that of the -dimensional unit cube and
lecture-hall simplex. Lastly, we detail some results on the real lattice-point
enumerator for variations of the simplices arising from stack-sorting
permutations. This then allows us to show that simplices are Gorenstein
of index .Comment: 25 pages, 5 figures, 1 table, comments welcomed
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Triangulations, Order Polytopes, and Generalized Snake Posets
This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in related order polytopes and then conclude that all of their triangulations are unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice of upper order ideals comes from a ladder is the Cayley graph of a symmetric group. Lastly, we introduce an operation on triangulations called twists and prove that twists preserve regular triangulations.Mathematics Subject Classifications: 52B20, 52B05, 52B12, 06A07Keywords: Order polytopes, triangulations, flow polytopes, circuit