arXiv subjects

Gregory S. Warrington

Publications and source records attributed to Gregory S. Warrington.

At least 19 recordsLinked to original sources

arXiv2026

Quantized rational chip-firing

This article introduces a quantized chip-firing model with close connections to the theory of rational lattice paths and rational parking functions. Given a graph with a sink and positive integers a,b,c with gcd(a,b)=1, a set S of vertices fires by the following rule. Each vertex in S provisionally sends c chips to the sink and a/b chips to each non-sink neighbor outside of S. The novel feature is that the total number of chips leaving from or arriving at any vertex gets rounded down to the nearest integer before being finalized. We define the notions of chip configurations being superstable, k-stable, or k-skeletal in this model. When c=1 and the graph is complete, superstable configurations correspond to rational parking functions. There is a bijection between superstable configurations and k-skeletal configurations for each k. We establish these results by building a combinatorial theory of k-skeletal rational lattice paths (both unlabeled and labeled) and translating that theory to chip configurations. There is a group structure on the set of chip configurations modulo firing and borrowing moves. We show that this group is isomorphic to the product of b-1 copies of the integers modulo a; and, for each k, each coset of chip configurations in this group contains a unique k-skeletal representative.

math.CO↗

arXiv2024

Skeletal generalizations of Dyck paths, parking functions, and chip-firing games

For $0\leq k\leq n-1$, we introduce a family of $k$-skeletal paths which are counted by the $n$-th Catalan number for each $k$, and specialize to Dyck paths when $k=n-1$. We similarly introduce $k$-skeletal parking functions which are equinumerous with the spanning trees on $n+1$ vertices for each $k$, and specialize to classical parking functions for $k=n-1$. The preceding constructions are generalized to paths lying in a trapezoid with base $c > 0$ and southeastern diagonal of slope $1/m$; $c$ and $m$ need not be integers. We give bijections among these families when $k$ varies with $m$ and $c$ fixed. Our constructions are motivated by chip firing and have connections to combinatorial representation theory and tropical geometry.

math.CO↗

arXiv2023

A rooted variant of Stanley's chromatic symmetric function

Richard Stanley defined the chromatic symmetric function $X_G$ of a graph $G$ and asked whether there are non-isomorphic trees $T$ and $U$ with $X_T=X_U$. We study variants of the chromatic symmetric function for rooted graphs, where we require the root vertex to either use or avoid a specified color. We present combinatorial identities and recursions satisfied by these rooted chromatic polynomials, explain their relation to pointed chromatic functions and rooted $U$-polynomials, and prove three main theorems. First, for all non-empty connected graphs $G$, Stanley's polynomial $X_G(x_1,\ldots,x_N)$ is irreducible in $\mathbb{Q}[x_1,\ldots,x_N]$ for all large enough $N$. The same result holds for our rooted variant where the root node must avoid a specified color. We prove irreducibility by a novel combinatorial application of Eisenstein's Criterion. Second, we prove the rooted version of Stanley's Conjecture: two rooted trees are isomorphic as rooted graphs if and only if their rooted chromatic polynomials are equal. In fact, we prove that a one-variable specialization of the rooted chromatic polynomial (obtained by setting $x_0=x_1=q$, $x_2=x_3=1$, and $x_n=0$ for $n>3$) already distinguishes rooted trees. Third, we answer a question of Pawlowski by providing a combinatorial interpretation of the monomial expansion of pointed chromatic functions.

math.CO↗

arXiv2021

Simulated packing and cracking

We introduce simulated packing and cracking as a technique for evaluating partisan-gerrymandering measures. We apply it to historical congressional and legislative elections to evaluate four measures: partisan bias, declination, efficiency gap, and mean-median difference. While the efficiency gap recognizes simulated packing and cracking in a completely predictable manner (a fact that follows immediately from the efficiency gap's definition) and the declination does a very good job of recording simulated packing and cracking, we conclude that both of the other two measures record it poorly. This deficiency is especially notable given the frequent use of such measures in outlier analyses.

physics.soc-ph↗

arXiv2020

Abacus-histories and the combinatorics of creation operators

Creation operators act on symmetric functions to build Schur functions, Hall--Littlewood polynomials, and related symmetric functions one row at a time. Haglund, Morse, Zabrocki, and others have studied more general symmetric functions $H_α$, $C_α$, and $B_α$ obtained by applying any sequence of creation operators to $1$. We develop new combinatorial models for the Schur expansions of these and related symmetric functions using objects called abacus-histories. These formulas arise by chaining together smaller abacus-histories that encode the effect of an individual creation operator on a given Schur function. We give a similar treatment for operators such as multiplication by $h_m$, $h_m^{\perp}$, $ω$, etc., which serve as building blocks to construct the creation operators. We use involutions on abacus-histories to give bijective proofs of properties of the Bernstein creation operator and Hall-Littlewood polynomials indexed by three-row partitions.

math.CO↗

arXiv2019

Accumulation charts for instant-runoff elections

We propose a new graphical format for instant-runoff voting election results. We call this proposal an "accumulation chart." This model, a modification of standard bar charts, is easy to understand, clearly indicates the winner, depicts the instant-runoff algorithm, and summarizes the votes cast. Moreover, it includes the pedigree of each accumulated vote and gives a clear depiction of candidates' coalitions.

physics.soc-ph↗

arXiv2019

A comparison of partisan-gerrymandering measures

We compare and contrast fourteen measures that have been proposed for the purpose of quantifying partisan gerrymandering. We consider measures that, rather than examining the shapes of districts, utilize only the partisan vote distribution among districts. The measures considered are two versions of partisan bias; the efficiency gap and several of its variants; the mean-median difference and the equal vote weight standard; the declination and one variant; and the lopsided-means test. Our primary means of evaluating these measures is a suite of hypothetical elections we classify from the start as fair or unfair. We conclude that the declination is the most successful measure in terms of avoiding false positives and false negatives on the elections considered. We include in an appendix the most extreme outliers for each measure among historical congressional and state legislative elections.

physics.soc-ph↗

arXiv2018

Packed voters and cracked voters

The actions of packing and cracking are central to the construction of gerrymandered district plans. The US Supreme Court opinion in Gill v. Whitford makes clear that vote dilution arguments require showing that individual voters have been packed or cracked. In this article we provide precise definitions of what it means for a voter to be packed or cracked. These definitions, which depend crucially on the existence of at least one comparator plan, are illustrated using a simple hypothetical example. We also explore who might be considered packed or cracked for congressional plans in Maryland and North Carolina, and for the current state assembly plan in Wisconsin.

physics.soc-ph↗

arXiv2018

Introduction to the declination function for gerrymanders

The declination is a quantitative method for identifying possible partisan gerrymanders by analyzing vote distributions. In this expository note we explain and motivate the definition of the declination. The minimal computer code required for computing the declination is included. We end by computing its value on several recent elections.

physics.soc-ph↗

arXiv2017

Gerrymandering and the net number of US House seats won due to vote-distribution asymmetries

Using the recently introduced declination function, we estimate the net number of seats won in the US House of Representatives due to asymmetries in vote distributions. Such asymmetries can arise from combinations of partisan gerrymandering and inherent geographic advantage. Our estimates show significant biases in favor of the Democrats prior to the mid 1990s and significant biases in favor of Republicans since then. We find net differences of 28, 20 and 25 seats in favor of the Republicans in the years 2012, 2014 and 2016, respectively. The validity of our results is supported by the technique of simulated packing and cracking. We also use this technique to show that the presidential-vote logistic regression model is insensitive to the packing and cracking by which partisan gerrymanders are achieved.

stat.AP↗

arXiv2017

Quantifying gerrymandering using the vote distribution

To assess the presence of gerrymandering, one can consider the shapes of districts or the distribution of votes. The "efficiency gap," which does the latter, plays a central role in a 2016 federal court case on the constitutionality of Wisconsin's state legislative district plan. Unfortunately, however, the efficiency gap reduces to proportional representation, an expectation that is not a constitutional right. We present a new measure of partisan asymmetry that does not rely on the shapes of districts, is simple to compute, is provably related to the "packing and cracking" integral to gerrymandering, and that avoids the constitutionality issue presented by the efficiency gap. In addition, we introduce a generalization of the efficiency gap that also avoids the equivalency to proportional representation. We apply the first function to US congressional and state legislative plans from recent decades to identify candidate gerrymanders.

stat.AP↗

arXiv2016

Orthogonal bases for transportation polytopes applied to Latin squares, magic squares and Sudoku boards

We give a simple construction of an orthogonal basis for the space of m by n matrices with row and column sums equal to zero. This vector space corresponds to the affine space naturally associated with the Birkhoff polytope, contingency tables and Latin squares. We also provide orthogonal bases for the spaces underlying magic squares and Sudoku boards. Our construction combines the outer (i.e., tensor or dyadic) product on vectors with certain rooted, vector-labeled, binary trees. Our bases naturally respect the decomposition of a vector space into centrosymmetric and skew-centrosymmetric pieces; the bases can be easily modified to respect the usual matrix symmetry and skew-symmetry as well.

math.CO↗

arXiv2014

Sweep maps: A continuous family of sorting algorithms

We define a family of maps on lattice paths, called sweep maps, that assign levels to each step in the path and sort steps according to their level. Surprisingly, although sweep maps act by sorting, they appear to be bijective in general. The sweep maps give concise combinatorial formulas for the q,t-Catalan numbers, the higher q,t-Catalan numbers, the q,t-square numbers, and many more general polynomials connected to the nabla operator and rational Catalan combinatorics. We prove that many algorithms that have appeared in the literature (including maps studied by Andrews, Egge, Gorsky, Haglund, Hanusa, Jones, Killpatrick, Krattenthaler, Kremer, Orsina, Mazin, Papi, Vaille, and the present authors) are all special cases of the sweep maps or their inverses. The sweep maps provide a very simple unifying framework for understanding all of these algorithms. We explain how inversion of the sweep map (which is an open problem in general) can be solved in known special cases by finding a "bounce path" for the lattice paths under consideration. We also define a generalized sweep map acting on words over arbitrary alphabets with arbitrary weights, which is also conjectured to be bijective.

math.CO↗

arXiv2014

Rational parking functions and Catalan numbers

The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the n'th Catalan number. In this paper, we will generalize this setup to rational parking functions indexed by a pair (a,b) of coprime positive integers. We show that these parking functions, which are counted by b^(a-1), carry a permutation representation of S_a in which the number of orbits is a rational Catalan number. We compute the Frobenius characteristic of the S_a-module of (a,b)-parking functions. Next we propose a combinatorial formula for a q-analogue of the rational Catalan numbers and relate this formula to a new combinatorial model for q-binomial coefficients. Finally, we discuss q,t-analogues of rational Catalan numbers and parking functions (generalizing the shuffle conjecture for the classical case) and present several conjectures.

math.CO↗

arXiv2014

Martin Gardner's minimum no-3-in-a-line problem

In Martin Gardner's October, 1976 Mathematical Games column in Scientific American, he posed the following problem: "What is the smallest number of [queens] you can put on a board of side n such that no [queen] can be added without creating three in a row, a column, or a diagonal?" We use the Combinatorial Nullstellensatz to prove that this number is at least n, except in the case when n is congruent to 3 modulo 4, in which case one less may suffice. A second, more elementary proof is also offered in the case that n is even.

math.CO↗

arXiv2013

Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials

We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials P_lambda/mu(x;t) and Hivert's quasisymmetric Hall-Littlewood polynomials G_gamma(x;t). More specifically, we provide: 1) the G-expansions of the Hall-Littlewood polynomials P_lambda, the monomial quasisymmetric polynomials M_alpha, the quasisymmetric Schur polynomials S_alpha, and the peak quasisymmetric functions K_alpha; 2) an expansion of P_lambda/mu in terms of the F_alpha's. The F-expansion of P_lambda/mu is facilitated by introducing starred tableaux.

math.CO↗

arXiv2013

Optimized random chemistry

The random chemistry algorithm of Kauffman can be used to determine an unknown subset S of a fixed set V. The algorithm proceeds by zeroing in on S through a succession of nested subsets V=V_0,V_1,...,V_m=S. In Kauffman's original algorithm, the size of each V_i is chosen to be half the size of V_{i-1}. In this paper we determine the optimal sequence of sizes so as to minimize the expected run time of the algorithm.

math.PR↗

arXiv2012

Matching expectations

The game of memory is played with a deck of n pairs of cards. The cards in each pair are identical. The deck is shuffled and the cards laid face down. A move consists of flipping over first one card then another. The cards are removed from play if they match. Otherwise, they are flipped back over and the next move commences. A game ends when all pairs have been matched. We determine that, when the game is played optimally, as n tends to infinity: 1) The expected number of moves is (3 - 2 ln 2)n + 7/8 - 2 ln 2 (approximately 1.61 n), 2) The expected number of times two matching cards are unwittingly flipped over is ln 2, and 3) The expected number of flips until two matching cards have been seen is asymptotically sqrt{pi n}.

math.PR↗