44 research outputs found
Combinatorial generation via permutation languages
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations.
This approach provides a unified view on many known results and allows us to prove many new ones.
In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an -element set by adjacent transpositions; the binary reflected Gray code to generate all -bit strings by flipping a single bit in each step; the Gray code for generating all -vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an -element ground set by element exchanges due to Kaye.
We present two distinct applications for our new framework:
The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others.
We thus also obtain new Gray code algorithms for the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into rectangles subject to certain restrictions.
The second main application of our framework are lattice congruences of the weak order on the symmetric group~.
Recently, Pilaud and Santos realized all those lattice congruences as -dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc.
Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian.
We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope
Realizing the -permutahedron via flow polytopes
Ceballos and Pons introduced the -weak order on -decreasing trees, for
any weak composition . They proved that it has a lattice structure and
further conjectured that it can be realized as the -skeleton of a polyhedral
subdivision of a polytope. We answer their conjecture in the case where is
a strict composition by providing three geometric realizations of the
-permutahedron. The first one is the dual graph of a triangulation of a flow
polytope of high dimension. The second one, obtained using the Cayley trick, is
the dual graph of a fine mixed subdivision of a sum of hypercubes that has the
conjectured dimension. The third one, obtained using tropical geometry, is the
-skeleton of a polyhedral complex for which we can provide explicit
coordinates of the vertices and whose support is a permutahedron as
conjectured.Comment: 39 pages, 14 figure
Trees, functional equations, and combinatorial Hopf algebras
One of the main virtues of trees is to represent formal solutions of various
functional equations which can be cast in the form of fixed point problems.
Basic examples include differential equations and functional (Lagrange)
inversion in power series rings. When analyzed in terms of combinatorial Hopf
algebras, the simplest examples yield interesting algebraic identities or
enumerative results.Comment: 14 pages, LaTE
On pattern avoiding indecomposable permutations
Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various contexts. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of length 3. Further, we provide a recursive formula for enumerating 12 ••• k-avoiding indecomposable permutations for k ≥ 3. Several of our results involve the descent statistic. We also provide a bijective proof of a fact relevant to our studies
Permutrees
We introduce permutrees, a unified model for permutations, binary trees,
Cambrian trees and binary sequences. On the combinatorial side, we study the
rotation lattices on permutrees and their lattice homomorphisms, unifying the
weak order, Tamari, Cambrian and boolean lattices and the classical maps
between them. On the geometric side, we provide both the vertex and facet
descriptions of a polytope realizing the rotation lattice, specializing to the
permutahedron, the associahedra, and certain graphical zonotopes. On the
algebraic side, we construct a Hopf algebra on permutrees containing the known
Hopf algebraic structures on permutations, binary trees, Cambrian trees, and
binary sequences.Comment: 43 pages, 25 figures; Version 2: minor correction
Random induced subgraphs of Cayley graphs induced by transpositions
In this paper we study random induced subgraphs of Cayley graphs of the
symmetric group induced by an arbitrary minimal generating set of
transpositions. A random induced subgraph of this Cayley graph is obtained by
selecting permutations with independent probability, . Our main
result is that for any minimal generating set of transpositions, for
probabilities where , a random induced subgraph has a.s. a unique
largest component of size , where
is the survival probability of a specific branching process.Comment: 18 pages, 1 figur
Lineup polytopes of product of simplices
Consider a real point configuration of size and an integer
. The vertices of the -lineup polytope of correspond
to the possible orderings of the top points of the configuration obtained
by maximizing a linear functional. The motivation behind the study of lineup
polytopes comes from the representability problem in quantum chemistry. In that
context, the relevant point configurations are the vertices of hypersimplices
and the integer points contained in an inflated regular simplex. The central
problem consists in providing an inequality representation of lineup polytopes
as efficiently as possible. In this article, we adapt the developed techniques
to the quantum information theory setup. The appropriate point configurations
become the vertices of products of simplices. A particular case is that of
lineup polytopes of cubes, which form a type analog of hypersimplices,
where the symmetric group of type~ naturally acts. To obtain the
inequalities, we center our attention on the combinatorics and the symmetry of
products of simplices to obtain an algorithmic solution. Along the way, we
establish relationships between lineup polytopes of products of simplices with
the Gale order, standard Young tableaux, and the Resonance arrangement.Comment: 19 pages, 8 figure
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure