Smith Normal Form of a Multivariate Matrix Associated with Partitions

Consideration of a question of E. R. Berlekamp led Carlitz, Roselle, and Scoville to give a combinatorial interpretation of the entries of certain matrices of determinant~1 in terms of lattice paths. Here we generalize this result by refining the matrix entries to be multivariate polynomials, and by determining not only the determinant but also the Smith normal form of these matrices. A priori the Smith form need not exist but its existence follows from the explicit computation. It will be more convenient for us to state our results in terms of partitions rather than lattice paths.Comment: 12 pages; revised version (minor changes on first version); to appear in J. Algebraic Combinatoric

The Descent Set and Connectivity Set of a Permutation

The descent set D(w) of a permutation w of 1,2,...,n is a standard and well-studied statistic. We introduce a new statistic, the connectivity set C(w), and show that it is a kind of dual object to D(w). The duality is stated in terms of the inverse of a matrix that records the joint distribution of D(w) and C(w). We also give a variation involving permutations of a multiset and a q-analogue that keeps track of the number of inversions of w.Comment: 12 page

Effect of the Diurnal Atmospheric Bulge on Satellite Accelerations

Formulas are developed to express the secular acceleration of a satellite on passing through an atmosphere which bulges in the sunward direction and in which the scale height increases with height, these two properties of the high atmosphere having previously been established from satellite observations. Comparison of the new formulas with those for a spherically symmetric atmosphere of constant scale height indicates that deduced atmospheric densities may be systematically incorrect by up to 50 or 60 percent at heights of 500 to 600 km when the earlier and simpler equations are used

Portrait: Anatole Krattiger—Intellectual Property Management in The Global Public Interest

[Excerpt] What do cows in green Alpine landscapes have in common with IP? Not much unless you ask Dr. Krattiger. As a young farmer in his native Switzerland, and later in the South of France where he cultivated vineyards, he developed a practical approach to solving problems. During these formative years as a farmer, Dr. Krattiger particularly enjoyed tending dairy herds in the green pastures of the Swiss Alps. There he learned and practiced the art of fine cheese making: an age-old and fundamental application of traditional biotechnology. Working in sight of the sublime peaks of the Alps must have spurred his mind to lofty goals, for Dr. Krattiger has since gone on to pursue a career focused on providing developing countries with access to new agricultural and health technologies. This idealism, however, remains rooted in a farmer’s sensibility: his professional life has been grounded in a results driven pragmatism.

The Smith Normal Form of a Specialized Jacobi-Trudi Matrix

Let $\mathrm{JT}_\lambda$ be the Jacobi-Trudi matrix corresponding to the partition $\lambda$, so $\det\mathrm{JT}_\lambda$ is the Schur function $s_\lambda$ in the variables $x_1,x_2,\dots$. Set $x_1=\cdots=x_n=1$ and all other $x_i=0$. Then the entries of $\mathrm{JT}_\lambda$ become polynomials in $n$ of the form ${n+j-1\choose j}$. We determine the Smith normal form over the ring $\mathbb{Q}[n]$ of this specialization of $\mathrm{JT}_\lambda$. The proof carries over to the specialization $x_i=q^{i-1}$ for $1\leq i\leq n$ and $x_i=0$ for $i>n$, where we set $q^n=y$ and work over the ring $\mathbb{Q}(q)[y]$.Comment: 5 pages, 2 figure

Ordering Events in Minkowski Space

We are given k points (events) in (n+1)-dimensional Minkowski space. Using the theory of hyperplane arrangments and chromatic polynomials, we obtain information the number of different orders in which the events can occur in different reference frames if the events are sufficiently generic. We consider the question of what sets of orderings of the points are possible and show a connection with sphere orders and the allowable sequences of Goodman and Pollack.Comment: 17 page

Spanning trees and a conjecture of Kontsevich

Kontsevich conjectured that the number f(G,q) of zeros over the finite field with q elements of a certain polynomial connected with the spanning trees of a graph G is polynomial function of q. We have been unable to settle Kontsevich's conjecture. However, we can evaluate f(G,q) explicitly for certain graphs G, such as the complete graph. We also point out the connection between Kontsevich's conjecture and such topics as the Matrix-Tree Theorem and orthogonal geometry.Comment: 18 pages. This version corrects some minor inaccuracies and adds some computational information provided by John Stembridg

Valid Orderings of Real Hyperplane Arrangements

Given a real finite hyperplane arrangement A and a point p not on any of the hyperplanes, we define an arrangement vo(A,p), called the *valid order arrangement*, whose regions correspond to the different orders in which a line through p can cross the hyperplanes in A. If A is the set of affine spans of the facets of a convex polytope P and p lies in the interior of P, then the valid orderings with respect to p are just the line shellings of p where the shelling line contains p. When p is sufficiently generic, the intersection lattice of vo(A,p) is the *Dilworth truncation* of the semicone of A. Various applications and examples are given. For instance, we determine the maximum number of line shellings of a d-polytope with m facets when the shelling line contains a fixed point p. If P is the order polytope of a poset, then the sets of facets visible from a point involve a generalization of chromatic polynomials related to list colorings.Comment: 15 pages, 2 figure