23,206 research outputs found

### 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

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