Related papers: On intervals in some posets of forests

In this short note we discuss recent results on hook length formulas of trees unifying some earlier results, and explain hook length formulas naturally associated to families of increasingly labelled trees.

We introduce an infinitesimal Hopf algebra of planar trees, generalising the construction of the non-commutative Connes-Kreimer Hopf algebra. A non-degenerate pairing and a dual basis are defined, and a combinatorial interpretation of the…

The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees…

This paper proves that two differently defined rooted binary trees are isomorphic. The first tree is one associated to a version of Farey sequences where the vertices correspond to the open intervals formed by two successive terms in the…

We define and prove isomorphisms between three combinatorial classes involving labeled trees. We also give an alternative proof by means of generating functions.

We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T0, T1] where T0 and T1 are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and…

Answering a question of Frank Calegari, we extend some of our earlier results on dimension of fixed point spaces of elements in irreducible linear groups. We consider characteristic polynomials rather than just fixed spaces.

We characterize those countable rooted trees whose full automorphism group has uncountable strong cofinality or contains an open subgroup with ample generics.

We introduce a monoid structure on a certain set of labelled binary trees, by a process similar to the construction of the plactic monoid. This leads to a new interpretation of the algebra of planar binary trees of Loday-Ronco.

Rotation distance between rooted binary trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomial-time algorithms for computing rotation distance. We give an efficient,…

When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…

We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial…

We give combinatorial criteria for predicting the transcendental weight of Feynman integrals of certain graphs in $\phi^4$ theory. By studying spanning forest polynomials, we obtain operations on graphs which are weight-preserving, and a…

We show that the category of pastures has arbitrary limits and colimits of diagrams indexed by a small category.

The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex…

Meadows are a sort of commutative rings with a multiplicative identity element and a total multiplicative inverse operation. In this paper we study algebraic properties of common meadows, which are meadows that introduce, as the inverse of…

We determine the tree which maximizes the distance between characteristic set and subtree core over all trees on $n$ vertices. The asymptotic nature of this distance is also discussed. The problem of extremizing the distance between…

We present a decomposition of the generalized binomial coefficients associated with Jack polynomials into two factors: a stem, which is described explicitly in terms of hooks of the indexing partitions, and a leaf, which inherits various…

Phylogenetic invariants are certain polynomials in the joint probability distribution of a Markov model on a phylogenetic tree. Such polynomials are of theoretical interest in the field of algebraic statistics and they are also of practical…

We revisit the model of the ballistic deposition studied in \cite{bdeposition} and prove several combinatorial properties of the random tree structure formed by the underlying stochastic process. Our results include limit theorems for the…