James Propp
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View article: Whirling injections, surjections, and other functions between finite sets
Whirling injections, surjections, and other functions between finite sets Open
This paper analyzes a certain action called "whirling" that can be defined on any family of functions between two finite sets equipped with a linear (or cyclic) ordering. Many maps of interest in dynamical algebraic combinatorics…
View article: Tilings of Benzels via Generalized Compression
Tilings of Benzels via Generalized Compression Open
Defant, Li, Propp, and Young recently resolved two enumerative conjectures of Propp concerning the tilings of regions in the hexagonal grid called benzels using two types of prototiles called stones and bones (with varying constraints on a…
View article: Tilings of benzels via the abacus bijection
Tilings of benzels via the abacus bijection Open
Propp recently introduced regions in the hexagonal grid called benzels and stated several enumerative conjectures about the tilings of benzels using two types of prototiles called stones and bones. We resolve two of his conjectures and pro…
View article: Homomesy via toggleability statistics
Homomesy via toggleability statistics Open
The rowmotion operator acting on the set of order ideals of a finite poset has been the focus of a significant amount of recent research. One of the major goals has been to exhibit homomesies: statistics that have the same average along ev…
View article: A Pentagonal Number Theorem for Tribone Tilings
A Pentagonal Number Theorem for Tribone Tilings Open
Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we introduce a two-parameter family of roughly hexagonal regions in the hexagonal grid …
View article: Some 2-Adic Conjectures Concerning Polyomino Tilings of Aztec Diamonds
Some 2-Adic Conjectures Concerning Polyomino Tilings of Aztec Diamonds Open
For various sets of tiles, we count the ways to tile an Aztec diamond of order $n$ using tiles from that set. The resulting function $f(n)$ often has interesting behavior when one looks at $n$ and $f(n)$ modulo powers of 2.
View article: A greedy chip‐firing game
A greedy chip‐firing game Open
We introduce a deterministic analogue of Markov chains that we call the hunger game. Like rotor‐routing, the hunger game deterministically mimics the behavior of both recurrent Markov chains and absorbing Markov chains. In the case of recu…
View article: Tilings of Benzels via the Abacus Bijection
Tilings of Benzels via the Abacus Bijection Open
Propp recently introduced regions in the hexagonal grid called benzels and stated several enumerative conjectures about the tilings of benzels using two types of prototiles called stones and bones. We resolve two of his conjectures and pro…
View article: Conway’s Mathematics After Conway
Conway’s Mathematics After Conway Open
View article: Trimer covers in the triangular grid: twenty mostly open problems
Trimer covers in the triangular grid: twenty mostly open problems Open
In the past three decades, the study of rhombus tilings and domino tilings of various plane regions has been a thriving subfield of enumerative combinatorics. Physicists classify such work as the study of dimer covers of finite graphs. In …
View article: A pentagonal number theorem for tribone tilings
A pentagonal number theorem for tribone tilings Open
Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we study a two-parameter family of roughly hexagonal regions in the hexagonal grid and …
View article: Some 2-adic conjectures concerning polyomino tilings of Aztec diamonds
Some 2-adic conjectures concerning polyomino tilings of Aztec diamonds Open
For various sets of tiles, we count the ways to tile an Aztec diamond of order $n$ using tiles from that set. The resulting function $f(n)$ often has interesting behavior when one looks at $n$ and $f(n)$ modulo powers of 2.
View article: Homomesy via Toggleability Statistics
Homomesy via Toggleability Statistics Open
The rowmotion operator acting on the set of order ideals of a finite poset has been the focus of a significant amount of recent research. One of the major goals has been to exhibit homomesies: statistics that have the same average along ev…
View article: The Genius Box
The Genius Box Open
Who gets called a genius, and why? What effect does having a special category of people called "geniuses" have on an intellectual community and on individuals within it? Drawing on my own experience, and reflecting on writings by Moon Duch…
View article: A spectral theory for combinatorial dynamics
A spectral theory for combinatorial dynamics Open
This article proposes a framework for the study of periodic maps $T$ from a (typically finite) set $X$ to itself when the set $X$ is equipped with one or more real- or complex-valued functions. The main idea, inspired by the time-evolution…
View article: Combinatorial, piecewise-linear, and birational homomesy for products of two chains
Combinatorial, piecewise-linear, and birational homomesy for products of two chains Open
This article illustrates the dynamical concept of homomesy in three kinds of dynamical systems – combinatorial, piecewise-linear, and birational – and shows the relationship between these three settings. In particular, we show how the rowm…
View article: The Hunger Game
The Hunger Game Open
We introduce a deterministic analogue of Markov chains that we call the hunger game. Like rotor-routing, the hunger game gives a way to deterministically mimic the behavior of both recurrent Markov chains and absorbing Markov chains. In th…
View article: Brussels sprouts, noncrossing trees, and parking functions
Brussels sprouts, noncrossing trees, and parking functions Open
We consider a variant of the game of Brussels Sprouts that, like Conway's original version, ends in a predetermined number of moves.We show that the endstates of the game are in natural bijection with noncrossing trees and that the game hi…
View article: Brussels sprouts, noncrossing trees, and parking functions
Brussels sprouts, noncrossing trees, and parking functions Open
View article: Quantifying Noninvertibility in Discrete Dynamical Systems
Quantifying Noninvertibility in Discrete Dynamical Systems Open
Given a finite set $X$ and a function $f:X\to X$, we define the \emph{degree of noninvertibility} of $f$ to be $\displaystyle\deg(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. This is a natural measure of how far the function $f$ is from b…
View article: Noncrossing partitions, toggles, and homomesy
Noncrossing partitions, toggles, and homomesy Open
We introduce n(n − 1)/2 natural involutions (“toggles”) on the set S of noncrossing partitions π of size n, along with certain composite operations obtained by composing these involutions. We show that for many operations T of this kind, a…
View article: The Combinatorics of Frieze Patterns and Markoff Numbers
The Combinatorics of Frieze Patterns and Markoff Numbers Open
See the abstract in the attached pdf.
View article: One-Dimensional Packing: Maximality and Rationality
One-Dimensional Packing: Maximality and Rationality Open
Every set of natural numbers determines a generating function convergent for $q \in (-1,1)$ whose behavior as $q \rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sizes of sets of …
View article: Germ order for one-dimensional packings
Germ order for one-dimensional packings Open
Every set of natural numbers determines a generating function convergent for $q \in (-1,1)$ whose behavior as $q \rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sets of natural n…
View article: Brussels Sprouts, Noncrossing Trees, and Parking Functions
Brussels Sprouts, Noncrossing Trees, and Parking Functions Open
We consider a variant of the game of Brussels Sprouts that, like Conway's original version, ends in a predetermined number of moves. We show that the endstates of the game are in natural bijection with noncrossing trees and that the game h…
View article: Whirling injections, surjections, and other functions between finite sets
Whirling injections, surjections, and other functions between finite sets Open
This paper analyzes a certain action called "whirling" that can be defined on any family of functions between two finite sets equipped with a linear (or cyclic) ordering. Many maps of interest in dynamical algebraic combinatorics, such as …
View article: One-Dimensional Packing: Maximality Implies Rationality
One-Dimensional Packing: Maximality Implies Rationality Open
Every set of natural numbers determines a generating function convergent for $q \in (-1,1)$ whose behavior as $q \rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sizes of sets of …
View article: Noncrossing Partitions, Toggles, and Homomesies
Noncrossing Partitions, Toggles, and Homomesies Open
We introduce $n(n-1)/2$ natural involutions ("toggles") on the set $S$ of noncrossing partitions $\pi$ of size $n$, along with certain composite operations obtained by composing these involutions. We show that for many operations $T$ of th…
View article: Homomesy in Products of Two Chains
Homomesy in Products of Two Chains Open
Many invertible actions $\tau$ on a set $\mathcal{S}$ of combinatorial objects, along with a natural statistic $f$ on $\mathcal{S}$, exhibit the following property which we dub homomesy: the average of $f$ over each $\tau$-orbit in $\mathc…
View article: Formation of an interface by competitive erosion
Formation of an interface by competitive erosion Open
In 2006, the fourth author proposed a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either red or blue. New red and blue particles alternately get emi…