49,857 research outputs found
Finite Affine Groups: Cycle Indices, Hall-Littlewood Polynomials, and Probabilistic Algorithms
The asymptotic study of the conjugacy classes of a random element of the
finite affine group leads one to define a probability measure on the set of all
partitions of all positive integers. Four different probabilistic
understandings of this measure are given--three using symmetric function theory
and one using Markov chains. This leads to non-trivial enumerative results.
Cycle index generating functions are derived and are used to compute the large
dimension limiting probabilities that an element of the affine group is
separable, cyclic, or semisimple and to study the convergence to these limits.
This yields the first examples of such computations for a maximal parabolic
subgroup of a finite classical group.Comment: Revised version, to appear in J. Algebra. A few typos are fixed; no
substantive change
Polynomial tuning of multiparametric combinatorial samplers
Boltzmann samplers and the recursive method are prominent algorithmic
frameworks for the approximate-size and exact-size random generation of large
combinatorial structures, such as maps, tilings, RNA sequences or various
tree-like structures. In their multiparametric variants, these samplers allow
to control the profile of expected values corresponding to multiple
combinatorial parameters. One can control, for instance, the number of leaves,
profile of node degrees in trees or the number of certain subpatterns in
strings. However, such a flexible control requires an additional non-trivial
tuning procedure. In this paper, we propose an efficient polynomial-time, with
respect to the number of tuned parameters, tuning algorithm based on convex
optimisation techniques. Finally, we illustrate the efficiency of our approach
using several applications of rational, algebraic and P\'olya structures
including polyomino tilings with prescribed tile frequencies, planar trees with
a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures,
colours. Implementation and examples are available at [1]
https://github.com/maciej-bendkowski/boltzmann-brain [2]
https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler
Generating Random Elements of Finite Distributive Lattices
This survey article describes a method for choosing uniformly at random from
any finite set whose objects can be viewed as constituting a distributive
lattice. The method is based on ideas of the author and David Wilson for using
``coupling from the past'' to remove initialization bias from Monte Carlo
randomization. The article describes several applications to specific kinds of
combinatorial objects such as tilings, constrained lattice paths, and
alternating-sign matrices.Comment: 13 page
A generating algorithm for ribbon tableaux and spin polynomials
We describe a general algorithm for generating various families of ribbon
tableaux and computing their spin polynomials. This algorithm is derived from a
new matricial coding. An advantage of this new notation lies in the fact that
it permits one to generate ribbon tableaux with skew shapes. This algorithm
permits us to compute quickly big LLT polynomials in MuPAD-Combinat
Reformulating Space Syntax: The Automatic Definition and Generation of Axial Lines and Axial Maps
Space syntax is a technique for measuring the relative accessibility of different locations in a spatial system which has been loosely partitioned into convex spaces.These spaces are approximated by straight lines, called axial lines, and the topological graph associated with their intersection is used to generate indices of distance, called integration, which are then used as proxies for accessibility. The most controversial problem in applying the technique involves the definition of these lines. There is no unique method for their generation, hence different users generate different sets of lines for the same application. In this paper, we explore this problem, arguing that to make progress, there need to be unambiguous, agreed procedures for generating such maps. The methods we suggest for generating such lines depend on defining viewsheds, called isovists, which can be approximated by their maximum diameters,these lengths being used to form axial maps similar to those used in space syntax. We propose a generic algorithm for sorting isovists according to various measures,approximating them by their diameters and using the axial map as a summary of the extent to which isovists overlap (intersect) and are accessible to one another. We examine the fields created by these viewsheds and the statistical properties of the maps created. We demonstrate our techniques for the small French town of Gassin used originally by Hillier and Hanson (1984) to illustrate the theory, exploring different criteria for sorting isovists, and different axial maps generated by changing the scale of resolution. This paper throws up as many problems as it solves but we believe it points the way to firmer foundations for space syntax
Analyzing Boltzmann Samplers for Bose-Einstein Condensates with Dirichlet Generating Functions
Boltzmann sampling is commonly used to uniformly sample objects of a
particular size from large combinatorial sets. For this technique to be
effective, one needs to prove that (1) the sampling procedure is efficient and
(2) objects of the desired size are generated with sufficiently high
probability. We use this approach to give a provably efficient sampling
algorithm for a class of weighted integer partitions related to Bose-Einstein
condensation from statistical physics. Our sampling algorithm is a
probabilistic interpretation of the ordinary generating function for these
objects, derived from the symbolic method of analytic combinatorics. Using the
Khintchine-Meinardus probabilistic method to bound the rejection rate of our
Boltzmann sampler through singularity analysis of Dirichlet generating
functions, we offer an alternative approach to analyze Boltzmann samplers for
objects with multiplicative structure.Comment: 20 pages, 1 figur
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