3,650 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

### 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

### The Rank and Minimal Border Strip Decompositions of a Skew Partition

Nazarov and Tarasov recently generalized the notion of the rank of a
partition to skew partitions. We give several characterizations of the rank of
a skew partition and one possible characterization that remains open. One of
the characterizations involves the decomposition of a skew shape into a minimal
number of border strips, and we develop a theory of these MBSD's as well as of
the closely related minimal border strip tableaux. An application is given to
the value of a character of the symmetric group S_n indexed by a skew shape z
at a permutation whose number of cycles is the rank of z.Comment: 31 pages, 10 figure

### An equivalence relation on the symmetric group and multiplicity-free flag h-vectors

We consider the equivalence relation ~ on the symmetric group S_n generated
by the interchange of two adjacent elements a_i and a_{i+1} of w=a_1 ... a_n in
S_n such that |a_i - a_{i+1}|=1. We count the number of equivalence classes and
the sizes of equivalence classes. The results are generalized to permutations
of multisets using umbral techniques. In the original problem, the equivalence
class containing the identity permutation is the set of linear extensions of a
certain poset. Further investigation yields a characterization of all finite
graded posets whose flag h-vector takes on only the values -1, 0, 1.Comment: 19 pages, 7 figure

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