7,435 research outputs found
The Brownian limit of separable permutations
We study random uniform permutations in an important class of
pattern-avoiding permutations: the separable permutations. We describe the
asymptotics of the number of occurrences of any fixed given pattern in such a
random permutation in terms of the Brownian excursion. In the recent
terminology of permutons, our work can be interpreted as the convergence of
uniform random separable permutations towards a "Brownian separable permuton".Comment: 45 pages, 14 figures, incorporating referee's suggestion
Counting real rational functions with all real critical values
We study the number of real rational degree n functions (considered up to
linear fractional transformations of the independent variable) with a given set
of 2n-2 distinct real critical values. We present a combinatorial reformulation
of this number and pose several related questions.Comment: 12 pages (AMSTEX), 3 picture
The Hopf algebra of diagonal rectangulations
We define and study a combinatorial Hopf algebra dRec with basis elements
indexed by diagonal rectangulations of a square. This Hopf algebra provides an
intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter
permutations, which previously had only been described extrinsically as a sub
Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We
describe the natural lattice structure on diagonal rectangulations, analogous
to the Tamari lattice on triangulations, and observe that diagonal
rectangulations index the vertices of a polytope analogous to the
associahedron. We give an explicit bijection between twisted Baxter
permutations and the better-known Baxter permutations, and describe the
resulting Hopf algebra structure on Baxter permutations.Comment: Very minor changes from version 1, in response to comments by
referees. This is the final version, to appear in JCTA. 43 pages, 17 figure
Permutads
We unravel the algebraic structure which controls the various ways of
computing the word ((xy)(zt)) and its siblings. We show that it gives rise to a
new type of operads, that we call permutads. It turns out that this notion is
equivalent to the notion of "shuffle algebra" introduced by the second author.
It is also very close to the notion of "shuffle operad" introduced by V.
Dotsenko and A. Khoroshkin. It can be seen as a noncommutative version of the
notion of nonsymmetric operads. We show that the role of the associahedron in
the theory of operads is played by the permutohedron in the theory of
permutads.Comment: Same results, re-arranged and more details. 38 page
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
Harmonic analysis on the infinite symmetric group
Let S be the group of finite permutations of the naturals 1,2,... The subject
of the paper is harmonic analysis for the Gelfand pair (G,K), where G stands
for the product of two copies of S while K is the diagonal subgroup in G. The
spherical dual to (G,K) (that is, the set of irreducible spherical unitary
representations) is an infinite-dimensional space. For such Gelfand pairs, the
conventional scheme of harmonic analysis is not applicable and it has to be
suitably modified.
We construct a compactification of S called the space of virtual
permutations. It is no longer a group but it is still a G-space. On this space,
there exists a unique G-invariant probability measure which should be viewed as
a true substitute of Haar measure. More generally, we define a 1-parameter
family of probability measures on virtual permutations, which are
quasi-invariant under the action of G.
Using these measures we construct a family {T_z} of unitary representations
of G depending on a complex parameter z. We prove that any T_z admits a unique
decomposition into a multiplicity free integral of irreducible spherical
representations of (G,K). Moreover, the spectral types of different
representations (which are defined by measures on the spherical dual) are
pairwise disjoint.
Our main result concerns the case of integral values of parameter z: then we
obtain an explicit decomposition of T_z into irreducibles. The case of
nonintegral z is quite different. It was studied by Borodin and Olshanski, see
e.g. the survey math.RT/0311369.Comment: AMS Tex, 80 pages, no figure
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