20,015 research outputs found
Complexity spectrum of some discrete dynamical systems
We first study birational mappings generated by the composition of the matrix
inversion and of a permutation of the entries of matrices. We
introduce a semi-numerical analysis which enables to compute the Arnold
complexities for all the possible birational transformations. These
complexities correspond to a spectrum of eighteen algebraic values. We then
drastically generalize these results, replacing permutations of the entries by
homogeneous polynomial transformations of the entries possibly depending on
many parameters. Again it is shown that the associated birational, or even
rational, transformations yield algebraic values for their complexities.Comment: 1 LaTex fil
Generating Permutations with Restricted Containers
We investigate a generalization of stacks that we call
-machines. We show how this viewpoint rapidly leads to functional
equations for the classes of permutations that -machines generate,
and how these systems of functional equations can frequently be solved by
either the kernel method or, much more easily, by guessing and checking.
General results about the rationality, algebraicity, and the existence of
Wilfian formulas for some classes generated by -machines are
given. We also draw attention to some relatively small permutation classes
which, although we can generate thousands of terms of their enumerations, seem
to not have D-finite generating functions
Inflations of geometric grid classes of permutations
All three authors were partially supported by EPSRC via the grant EP/J006440/1.Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than Îș â 2.20557 (a specific algebraic integer at which infinite antichains first appear). Using language- and order-theoretic methods, we prove that the substitution closures of geometric grid classes are well partially ordered, finitely based, and that all their subclasses have algebraic generating functions. We go on to show that the inflation of a geometric grid class by a strongly rational class is well partially ordered, and that all its subclasses have rational generating functions. This latter fact allows us to conclude that every permutation class with growth rate less than Îș has a rational generating function. This bound is tight as there are permutation classes with growth rate Îș which have nonrational generating functions.PostprintPeer reviewe
The algebraic combinatorics of snakes
Snakes are analogues of alternating permutations defined for any Coxeter
group. We study these objects from the point of view of combinatorial Hopf
algebras, such as noncommutative symmetric functions and their generalizations.
The main purpose is to show that several properties of the generating functions
of snakes, such as differential equations or closed form as trigonometric
functions, can be lifted at the level of noncommutative symmetric functions or
free quasi-symmetric functions. The results take the form of algebraic
identities for type B noncommutative symmetric functions, noncommutative
supersymmetric functions and colored free quasi-symmetric functions.Comment: 29 pages, Late
Transitive factorizations of permutations and geometry
We give an account of our work on transitive factorizations of permutations.
The work has had impact upon other areas of mathematics such as the enumeration
of graph embeddings, random matrices, branched covers, and the moduli spaces of
curves. Aspects of these seemingly unrelated areas are seen to be related in a
unifying view from the perspective of algebraic combinatorics. At several
points this work has intertwined with Richard Stanley's in significant ways.Comment: 12 pages, dedicated to Richard Stanley on the occasion of his 70th
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