Related papers: A Family of Invariants of Rooted Forests

We consider three bivariate polynomial invariants $P$, $A$, and $S$ for rooted trees, as well as a trivariate polynomial invariant $M$. These invariants are motivated by random destruction processes such as the random cutting model or site…

We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to…

We define a bivariate polynomial for unlabeled rooted trees and show that the polynomial of an unlabeled rooted tree $T$ is the generating function of a class of subtrees of $T$. We prove that the polynomial is a complete isomorphism…

Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial…

Given a permutation $\pi\in \Sn\_n$, construct a graph $G\_\pi$ on the vertex set $\{1,2, ..., n\}$ by joining $i$ to $j$ if (i) $i<j$ and $\pi(i)<\pi(j)$ and (ii) there is no $k$ such that $i < k < j$ and $\pi(i)<\pi(k)<\pi(j)$. We say…

In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted $U$-polynomial for every $k$; by this we mean that the polynomials agree on terms with degree at most $k+1$. The main tool for this…

We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…

Let $K$ be an arbitrary field of characteristic zero, $P_n:= K[ x_1, ..., x_n]$ be a polynomial algebra, and $P_{n, x_1}:= K[x_1^{-1}, x_1, ..., x_n]$, for $n\geq 2$. Let $\s' \in {\rm Aut}_K(P_n)$ be given by $$ x_1\mapsto x_1-1, \quad…

We propose a new arithmetic for non-empty rooted unordered trees simply called trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication and stretch, prove their properties, and…

We consider a family of skew-products of the form $(Tx, g_x(t)) : X \times \mathbb{R} \to X \times \mathbb{R}$ where $T$ is a continuous expanding Markov map and $g_x : \mathbb{R} \to \mathbb{R}$ is a family of homeomorphisms of…

Invariants for complicated objects such as those arising in phylogenetics, whether they are invariants as matrices, polynomials, or other mathematical structures, are important tools for distinguishing and working with such objects. In this…

The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of…

We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We…

We consider fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds and the conditions on a manifold for admissibility of such involutions. We construct a class $\Psi$ of fiber-preserving,…

We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such algebras are described in terms of labeled reduced planar rooted trees. This allows…

We investigate a trivariate polynomial associated with rooted trees. It generalises a bivariate polynomial for rooted trees that was recently introduced by Liu. We show that this polynomial satisfies a deletion-contraction recursion and can…

We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex…

The main result of this paper is a recursive description of all decompositions \[ \Delta^+ = \Phi_1 \sqcup \Phi_2 \sqcup \dots \sqcup \Phi_k \] of the positive roots $\Delta^+$ of an arbitrary root system $\Delta$ into a disjoint union of…

Let FI denote the category whose objects are the sets $[n] = \{1,\ldots, n\}$, and whose morphisms are injections. We study functors from the category FI into the category of sets. We write $\mathfrak{S}_n$ for the symmetric group on $[n]$.…

We study $I(T)$, the number of inversions in a tree $T$ with its vertices labeled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of $I(T)$ have explicit formulas involving the…