4 research outputs found
Matriochka symmetric Boolean functions
International audienceWe present the properties of a new class of Boolean functions defined as the sum of m symmetric functions with decreasing number of variables and degrees. The choice of this construction is justified by the possibility to study these functions by using tools existing for symmetric functions. On the one hand we show that the synthesis is well understood and give an upper bound on the gate complexity. On the other hand, we investigate the Walsh spectrum of the sum of two functions and get explicit formulae for the case of degree at most three
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
Cambrian Hopf Algebras
Cambrian trees are oriented and labeled trees which fulfill local conditions
around each node generalizing the conditions for classical binary search trees.
Based on the bijective correspondence between signed permutations and leveled
Cambrian trees, we define the Cambrian Hopf algebra generalizing J.-L. Loday
and M. Ronco's algebra on binary trees. We describe combinatorially the
products and coproducts of both the Cambrian algebra and its dual in terms of
operations on Cambrian trees. We also define multiplicative bases of the
Cambrian algebra and study structural and combinatorial properties of their
indecomposable elements. Finally, we extend to the Cambrian setting different
algebras connected to binary trees, in particular S. Law and N. Reading's
Baxter Hopf algebra on quadrangulations and S. Giraudo's equivalent Hopf
algebra on twin binary trees, and F. Chapoton's Hopf algebra on all faces of
the associahedron.Comment: 60 pages, 43 figures. Version 2: New Part 3 on Schr\"oder Cambrian
Algebra. The title change reflects this modificatio
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