Osculating Paths and Oscillating Tableaux

The combinatorics of certain osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. More specifically, the paths being considered have fixed start and end points on respectively the lower and right boundaries of a rectangle in the square lattice, each path can take only unit steps rightwards or upwards, and two different paths are permitted to share lattice points, but not to cross or share lattice edges. Such paths correspond to configurations of the six-vertex model of statistical mechanics with appropriate boundary conditions, and they include cases which correspond to alternating sign matrices and various subclasses thereof. Referring to points of the rectangle through which no or two paths pass as vacancies or osculations respectively, the case of primary interest is tuples of paths with a fixed number $l$ of vacancies and osculations. It is then shown that there exist natural bijections which map each such path tuple $P$ to a pair $(t,\eta)$, where $\eta$ is an oscillating tableau of length $l$ (i.e., a sequence of $l+1$ partitions, starting with the empty partition, in which the Young diagrams of successive partitions differ by a single square), and $t$ is a certain, compatible sequence of $l$ weakly increasing positive integers. Furthermore, each vacancy or osculation of $P$ corresponds to a partition in $\eta$ whose Young diagram is obtained from that of its predecessor by respectively the addition or deletion of a square. These bijections lead to enumeration formulae for osculating paths involving sums over oscillating tableaux.Comment: 65 pages; expanded versio

Deflationary Language

In the following recessionary story, numbers and letters deflate instead of inflate, and any word that has a comparative element is changed to its lower or lesser counterpart

Fractional Perfect b-Matching Polytopes. I: General Theory

The fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v is a prescribed nonnegative number b_v. General theorems which provide conditions for nonemptiness, give a formula for the dimension, and characterize the vertices, edges and face lattices of such polytopes are obtained. Many of these results are expressed in terms of certain spanning subgraphs of G which are associated with subsets or elements of the polytope. For example, it is shown that an element u of the fractional perfect b-matching polytope of G is a vertex of the polytope if and only if each component of the graph of u either is acyclic or else contains exactly one cycle with that cycle having odd length, where the graph of u is defined to be the spanning subgraph of G whose edges are those at which u is positive.Comment: 37 page

Disturbing Verbing and Pre-verbing

Humorous commentary on the English language trend of creating verbs out of nouns which he calls verbing.

Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions

We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related factorizations involving sums of two Schur polynomials, and certain odd-sized sets of variables. Our results generalize the factorization identities proved by Ciucu and Krattenthaler (Advances in combinatorial mathematics, 39-59, 2009) for partitions of rectangular shape. We observe that if, in some of the results, the partitions are taken to have rectangular or double-staircase shapes and all of the variables are set to 1, then factorization identities for numbers of certain plane partitions, alternating sign matrices and related combinatorial objects are obtained.Comment: 22 pages; v2: minor changes, published versio

Mountain Goat Removal in Olympic National Park: A Case Study of the Role of Organizational Culture in Individual Risk Decisions and Behavior

Using a case study, the authors explore the mediating role of organizational culture in individual Risk-taking decisions and behaviors. They argue that organizational culture can establish unique conditions that lead to highly reliable performance of high-Risk, undesired tasks. The authors also discuss the need for further research and its implications for Risk management

Graptemys barbouri

Number of Pages: 2Integrative BiologyGeological Science

Aggregate Demand and Supply

This paper is part of a broader project that provides a microfoundation to the General Theory of J.M. Keynes. I call this project 'old Keynesian economics' to distinguish it from new-Keynesian economics, a theory that is based on the idea that to make sense of Keynes we must assume that prices are sticky. I describe a multi-good model in which I interpret the definitions of aggregate demand and supply found in the General Theory through the lens of a search theory of the labor market. I argue that Keynes' aggregate supply curve can be interpreted as the aggregate of a set of first order conditions for the optimal choice of labor and, using this interpretation, I reintroduce a diagram that was central to the textbook teaching of Keynesian economics in the immediate post-war period.

Higher Spin Alternating Sign Matrices

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions. The case r=1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are the integer points of the r-th dilate of an integral convex polytope of dimension (n-1)^2 whose vertices are the standard alternating sign matrices of size n. It then follows that, for fixed n, these matrices are enumerated by an Ehrhart polynomial in r.Comment: 41 pages; v2: minor change