Daniel Ullman

On the structure of the lattice of noncrossing partitions

Abstract We show that the lattice of noncrossing (set) partitions is self-dual and that it admits a symmetric chain decomposition. The self-duality is proved via an order-reversing involution. Two proofs are given of the existence of the symmetric chain decomposition, one recursive and one constructive. Several identities involving Catalan numbers emerge from the construction of the symmetric chain decomposition.

The fractional chromatic number of Mycielski's graphs

James Propp The most familiar construction of graphs whose clique number is much smaller than their chromatic number is due to Mycielski, who constructed a sequence Gn of triangle-free graphs with X(Gn) = n. In this article, we calculate the fractional chromatic number of Gn and show that this sequence of numbers satisfies the unexpected recurrence an+1 = an + (1/an).

Undirected edge geography

Abstract The game of edge geography is played by two players who alternately move a token on a graph from one vertex to an adjacent vertex, erasing the edge in between. The player who first has no legal move is the loser. We show that the decision problem of determining whether a position in this game is a win for the first player is PSPACE-complete. Further, the problem remains PSPACE-complete when restricted to planar graphs with maximum degree 3. However, if the underlying graph is bipartite we provide (1) a linear algebraic characterization of the P- and N-positions, yielding (2) a polynomial time algorithm for deciding whether any given position is P or N, and also (3) a polynomial time algorithm to find winning moves.

Fractional isomorphism of graphs

Abstract Let the adjacency matrices of graphs G and H be A and B . These graphs are isomorphic provided there is a permutation matrix P with AP = PB , or equivalently, A = PBP T . If we relax the requirement that P be a permutation matrix, and, instead, require P only to be doubly stochastic, we arrive at two new equivalence relations on graphs: linear fractional isomorphism (when we relax AP = PB ) and quadratic fractional isomorphism (when we relax A = PBP T ). Further, if we allow the two instances of P in A = PBP T to be different doubly stochastic matrices, we arrive at the concept of semi-isomorphism . We present necessary and sufficient conditions for graphs to be linearly fractionally isomorphic, we prove that quadratic fractional isomorphism is the same as isomorphism and we relate semi-isomorphism to isomorphism of bipartite graphs.

A mathematical look at politics

Preface, for the Student Preface, for the Instructor Voting Two Candidates Scenario Two-candidate methods Supermajority and status quo Weighted voting and other methods Criteria Mays Theorem Exercises and problems Social Choice Functions Scenario Ballots Social choice functions Alternatives to plurality Some methods on the edge Exercises and problems Criteria for Social Choice Scenario Weakness and strength Some familiar criteria Some new criteria Exercises and problems Which Methods are Good? Scenario Methods and criteria Proofs and counterexamples Summarizing the results Exercises and problems Arrows Theorem Scenario The Condorcet paradox Statement of the result Decisiveness Proving the theorem Exercises and problems Variations on the Theme Scenario Inputs and outputs Vote-for-one ballots Approval ballots Mixed approval/preference ballots Cumulative voting . Condorcet methods Social ranking functions Preference ballots with ties Exercises and problems Notes on Part I Apportionment Hamiltons Method Scenario The apportionment problem Some basic notions A sensible approach The paradoxes Exercises and problems Divisor Methods Scenario Jeffersons method Critical divisors Assessing Jeffersons method Other divisor methods Rounding functions Exercises and problems Criteria and Impossibility Scenario Basic criteria Quota rules and the Alabama paradox Population monotonicity Relative population monotonicity The new states paradox Impossibility Exercises and problems The Method of Balinski and Young Scenario Tracking critical divisors Satisfying the quota rule Computing the Balinski-Young apportionment Exercises and problems Deciding Among Divisor Methods Scenario Why Webster is best Why Dean is best Why Hill is best Exercises and problems History of Apportionment in the United States Scenario The fight for representation Summary Exercises and problems Notes on Part II Conflict Strategies and Outcomes Scenario Zero-sum games The naive and prudent strategies Best response and saddle points Dominance Exercises and problems Chance and Expectation Scenario Probability theory All outcomes are not created equal Random variables and expected value Mixed strategies and their payouts Independent processes Expected payouts for mixed strategies Exercises and Problems Solving Zero-Sum Games Scenario The best response Prudent mixed strategies An application to counterterrorism The -by- case Exercises and problems Conflict and Cooperation Scenario Bimatrix games Guarantees, saddle points, and all that jazz Common interests Some famous games Exercises and Problems Nash Equilibria Scenario Mixed strategies The -by- case The proof of Nashs Theorem Exercises and Problems The Prisoners Dilemma Scenario Criteria and Impossibility Omnipresence of the Prisoners Dilemma Repeated play Irresolvability Exercises and problems Notes on Part III The Electoral College Weighted Voting Scenario Weighted voting methods Non-weighted voting methods Voting power Power of the states Exercises and problems Whose Advantage? Scenario Violations of criteria People power Interpretation Exercises and problems Notes on Part IV Solutions to Odd-Numbered Exercises and Problems Bibliography Index

Sequential compounds of combinatorial games

Given combinatorial games G and H, define a new game G→H to be the game played by two players who alternately make moves of game G until G is exhausted and then proceed to game H. As usual, the player who has no move (in H) loses. Misere games are a special case of this construction. We explore the theory of these sequential compounds and determine the outcomes and Grundy values of certain games of this form.

On tensor powers of integer programs

A natural product on integer programming problems with nonnegative coefficients is defined. Hypergraph covering problems are a special case of such integer programs, and the product defined is a generalization of the usual hypergraph product. The main theorem of this paper gives a sufficient condition under which the solution to the nth power of an integer program is asymptotically as good as the solution to the same nth power when the variables are not necessarily integral but may be arbitrary nonnegative real numbers.

On the cookie game

On alternate days, each of two players eats a cookie from a cookie jar. Every cookie has a spoilage date after which it can not be eaten. The object of the game is to eat the last edible cookie. In this paper, we produce a strategy for winning this game when the cookies have distinct spoilage dates.

On point-halfspace graphs

The following definition is motivated by the study of circle orders and their connections to graphs. A graphs G is called a point-halfspace graph (in Rk) provided one can assign to each vertex v ϵ (G) a point pvRk and to each edge e ϵ E(G) a closed halfspace He ϵ Rk so that v is incident with e if and only if pv ϵ He. Let Hk denote the set of point-halfspace graphs (in Rk). We give complete forbidden subgraph and structural characterizations of the classes Hk for every k. Surprisingly, these classes are closed under taking minors and we give forbidden minor characterizations as well.

The Seventy-Eighth William Lowell Putnam Mathematical Competition

The 78th William Lowell Putnam Mathematical Competition took place on December 2, 2017. There were 4,638 undergraduates who participated in the competition at 575 institutions across the United Sta...