Functions of the Binomial Coefficient

The well known binomial coefficient is the building block of Pascal’s triangle. We explore the relationship between functions of the binomial coefficient and Pascal’s triangle, providing proofs of connections between Catalan numbers, determinants, non-intersecting paths, and Baxter permutations

Extreme Walrasian Dynamics: The Gale Example in the Lab

We study the classic Gale (1963) economy using laboratory markets. Tatonnement theory predicts prices will diverge from an equitable interior equilibrium towards infinity or zero depending only on initial prices. The inequitable equilibria determined by these dynamics give all gains from exchange to one side of the market. Our results show surprisingly strong support for these predictions. In most sessions one side of the market eventually outgains the other by more than twenty times, leaving the disadvantaged side to trade for mere pennies. We also find preliminary evidence that these dynamics are sticky, resisting exogenous interventions designed to reverse their trajectories

A Combinatorial Approach to Fibonomial Coefficients

A combinatorial argument is used to explain the integrality of Fibonomial coefficients and their generalizations. The numerator of the Fibonomial coeffcient counts tilings of staggered lengths, which can be decomposed into a sum of integers, such that each integer is a multiple of the denominator of the Fibonomial coeffcient. By colorizing this argument, we can extend this result from Fibonacci numbers to arbitrary Lucas sequences

Tiling Proofs of Recent Sum Identities Involving Pell Numbers

In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of a number of the identities

A New Framework for Network Disruption

Traditional network disruption approaches focus on disconnecting or lengthening paths in the network. We present a new framework for network disruption that attempts to reroute flow through critical vertices via vertex deletion, under the assumption that this will render those vertices vulnerable to future attacks. We define the load on a critical vertex to be the number of paths in the network that must flow through the vertex. We present graph-theoretic and computational techniques to maximize this load, firstly by removing either a single vertex from the network, secondly by removing a subset of vertices.Comment: Submitted for peer review on September 13, 201

The 99th Fibonacci Identity

We provide elementary combinatorial proofs of several Fibonacci and Lucas number identities left open in the book Proofs That Really Count [1], and generalize these to Gibonacci sequences Gn that satisfy the Fibonacci recurrence, but with arbitrary real initial conditions. We offer several new identities as well. [1] A. T. Benjamin and J. J. Quinn, Proofs That Really Count: The Art of Combinatorial Proof, The Dolciani Mathematical Expositions, 27, Mathematical Association of America, Washington, DC, 200

Gradual polyploid genome evolution revealed by pan-genomic analysis of Brachypodium hybridum and its diploid progenitors

Our understanding of polyploid genome evolution is constrained because we cannot know the exact founders of a particular polyploid. To differentiate between founder effects and post polyploidization evolution, we use a pan-genomic approach to study the allotetraploid Brachypodium hybridum and its diploid progenitors. Comparative analysis suggests that most B. hybridum whole gene presence/absence variation is part of the standing variation in its diploid progenitors. Analysis of nuclear single nucleotide variants, plastomes and k-mers associated with retrotransposons reveals two independent origins for B. hybridum, ~1.4 and ~0.14 million years ago. Examination of gene expression in the younger B. hybridum lineage reveals no bias in overall subgenome expression. Our results are consistent with a gradual accumulation of genomic changes after polyploidization and a lack of subgenome expression dominance. Significantly, if we did not use a pan-genomic approach, we would grossly overestimate the number of genomic changes attributable to post polyploidization evolution

Gradual polyploid genome evolution revealed by pan-genomic analysis of Brachypodium hybridum and its diploid progenitors

Our understanding of polyploid genome evolution is constrained because we cannot know the exact founders of a particular polyploid. To differentiate between founder effects and post polyploidization evolution, we use a pan-genomic approach to study the allotetraploid Brachypodium hybridum and its diploid progenitors. Comparative analysis suggests that most B. hybridum whole gene presence/absence variation is part of the standing variation in its diploid progenitors. Analysis of nuclear single nucleotide variants, plastomes and k-mers associated with retrotransposons reveals two independent origins for B. hybridum, ~1.4 and ~0.14 million years ago. Examination of gene expression in the younger B. hybridum lineage reveals no bias in overall subgenome expression. Our results are consistent with a gradual accumulation of genomic changes after polyploidization and a lack of subgenome expression dominance. Significantly, if we did not use a pan-genomic approach, we would grossly overestimate the number of genomic changes attributable to post polyploidization evolution

Tiling Proofs of Recent Sum Identities Involving Pell Numbers

In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of a number of the identities