Critical Groups of Graphs with Dihedral Actions II

In this paper we consider the critical group of finite connected graphs which admit harmonic actions by the dihedral group $D_n$, extending earlier work by the author and Criel Merino. In particular, we show that the critical group of such a graph can be decomposed in terms of the critical groups of the quotients of the graph by certain subgroups of the automorphism group. This is analogous to a theorem of Kani and Rosen which decomposes the Jacobians of algebraic curves with a $D_n$-action.Comment: Revised version includes new examples and increased detail in expositio

Composition of Integers with Bounded Parts

In this note, we consider ordered partitions of integers such that each entry is no more than a fixed portion of the sum. We give a method for constructing all such compositions as well as both an explicit formula and a generating function describing the number of k-tuples whose entries are bounded in this way and sum to a fixed value g

On Pi Day, A Serving of Why We Need Math

Today, our Facebook feeds will be peppered with references to Pi Day, a day of celebration that has long been acknowledged by math fans and that Congress recognized in 2009. Every high schooler learns that pi is the ratio of the circumference of a circle to its diameter and that its decimal expansion begins 3.14 and goes on infinitely without repeating. [excerpt

Klein Four Actions on Graphs and Sets

We consider how a standard theorem in algebraic geometry relating properties of a curve with a (ℤ/2ℤ)2-action to the properties of its quotients generalizes to results about sets and graphs that admit (ℤ/2ℤ)2-actions

Fair-Weather Fans: The Correlation Between Attendance and Winning Percentage

In Rob Neyer\u27s chapter on San Francisco in his Big Book of Baseball Lineups, he speculates that there aren\u27t really good baseball cities, and that attendance more closely correlates with winning percentage than with any other factor. He also suggests that a statistically minded person look at this. I took the challenge and have been playing with a lot of data

Solving the Debt Crisis on Graphs - Solutions

We begin by noting that solutions to these puzzles are not unique. In particular, doing the `lending\u27 action from each of the vertices once brings us back to where we started. Moreover, the act of doing the `borrowing\u27 action from one vertex is equivalent to doing the`lending\u27 action from each of the other vertices. In particular, without loss of generality one can assume that there is (at least) one vertex for which you do neither action and for all other vertices you do the `lending\u27 action a nonnegative number of times. Below we give possible solutions to four of the puzzles by showing the number of times one lends from each vertex in order to eliminate all debt

Communal Partitions of Integers

There is a well-known formula due to Andrews that counts the number of incongruent triangles with integer sides and a fixed perimeter. In this note, we consider the analagous question counting the number of k-tuples of nonnegative integers none of which is more than 1/(k−1) of the sum of all the integers. We give an explicit function for the generating function which counts these k-tuples in the case where they are ordered, unordered, or partially ordered. Finally, we discuss the application to algebraic geometry which motivated this question

Math Quiz on the Radio

What word, often spelled with an umlaut, is used to identify a point on a two-dimensional graph? Many of you probably already figured out the answer is coordinate. But that\u27s because you are sitting comfortably in your dorm room rather than being on a stage with bright lights in front of a few hundred people being recorded for national broadcast on public radio. [excerpt