20,811 research outputs found
Asymptotics of Selberg-like integrals by lattice path counting
We obtain explicit expressions for positive integer moments of the
probability density of eigenvalues of the Jacobi and Laguerre random matrix
ensembles, in the asymptotic regime of large dimension. These densities are
closely related to the Selberg and Selberg-like multidimensional integrals. Our
method of solution is combinatorial: it consists in the enumeration of certain
classes of lattice paths associated to the solution of recurrence relations
Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations
Defant, Engen, and Miller defined a permutation to be uniquely sorted if it
has exactly one preimage under West's stack-sorting map. We enumerate classes
of uniquely sorted permutations that avoid a pattern of length three and a
pattern of length four by establishing bijections between these classes and
various lattice paths. This allows us to prove nine conjectures of Defant.Comment: 18 pages, 16 figures, new version with updated abstract and
reference
Lattice path counting and the theory of queues
In this paper we will show how recent advances in the combinatorics of lattice paths can be applied to solve interesting and nontrivial problems in the theory of queues. The problems we discuss range from classical ones like M^a/M^b/1 systems to open tandem systems with and without global blocking and to queueing models that are related to random walks in a quarter plane like the Flatto-Hahn model or systems with preemptive priorities. (author´s abstract)Series: Research Report Series / Department of Statistics and Mathematic
On FPL configurations with four sets of nested arches
The problem of counting the number of Fully Packed Loop (FPL) configurations
with four sets of a,b,c,d nested arches is addressed. It is shown that it may
be expressed as the problem of enumeration of tilings of a domain of the
triangular lattice with a conic singularity. After reexpression in terms of
non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a
formula as a sum of determinants. This is made quite explicit when
min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates
the numbers of configurations with b=d.Comment: 22 pages, TeX, 16 figures; a new formula for a generating function
adde
Counting tropical elliptic plane curves with fixed j-invariant
In complex algebraic geometry, the problem of enumerating plane elliptic
curves of given degree with fixed complex structure has been solved by
R.Pandharipande using Gromov-Witten theory. In this article we treat the
tropical analogue of this problem, the determination of the number of tropical
elliptic plane curves of degree d and fixed ``tropical j-invariant''
interpolating an appropriate number of points in general position. We show that
this number is independent of the position of the points and the value of the
j-invariant and that it coincides with the number of complex elliptic curves.
The result can be used to simplify Mikhalkin's algorithm to count curves via
lattice paths in the case of rational plane curves.Comment: 34 pages; minor changes to match the published versio
Lattice path matroids: enumerative aspects and Tutte polynomials
Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North
steps with P never going above Q. We show that the lattice paths that go from
(0,0) to (m,r) and that remain in the region bounded by P and Q can be
identified with the bases of a particular type of transversal matroid, which we
call a lattice path matroid. We consider a variety of enumerative aspects of
these matroids and we study three important matroid invariants, namely the
Tutte polynomial and, for special types of lattice path matroids, the
characteristic polynomial and the beta invariant. In particular, we show that
the Tutte polynomial is the generating function for two basic lattice path
statistics and we show that certain sequences of lattice path matroids give
rise to sequences of Tutte polynomials for which there are relatively simple
generating functions. We show that Tutte polynomials of lattice path matroids
can be computed in polynomial time. Also, we obtain a new result about lattice
paths from an analysis of the beta invariant of certain lattice path matroids.Comment: 28 pages, 11 figure
Transmission and Spectral Aspects of Tight Binding Hamiltonians for the Counting Quantum Turing Machine
It was recently shown that a generalization of quantum Turing machines
(QTMs), in which potentials are associated with elementary steps or transitions
of the computation, generates potential distributions along computation paths
of states in some basis B. The distributions are computable and are thus
periodic or have deterministic disorder. These generalized machines (GQTMs) can
be used to investigate the effect of potentials in causing reflections and
reducing the completion probability of computations. This work is extended here
by determination of the spectral and transmission properties of an example GQTM
which enumerates the integers as binary strings. A potential is associated with
just one type of step. For many computation paths the potential distributions
are initial segments of a quasiperiodic distribution that corresponds to a
substitution sequence. The energy band spectra and Landauer Resistance (LR) are
calculated for energies below the barrier height by use of transfer matrices.
The LR fluctuates rapidly with momentum with minima close to or at band-gap
edges. For several values of the parameters, there is good transmission over
some momentum regions.Comment: 22 pages Latex, 13 postscript figures, Submitted to Phys. Rev.
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