Related papers: Cayley's formula from middle school math

We exhibit an identity of abstract simplicial complexes between the well-studied complex of trees and the reduced minimal nested set complex of the partition lattice. We conclude that the order complex of the partition lattice can be…

These notes are a self-contained short proof of the stability of persistence diagrams.

We determine the Newton trees of the rational polynomials of simple type, thus filling a gap in the proof of the classification of these polynomials given by Neumann and Norbury.

A symbolic-computational algorithm, fully implemented in Maple, is described, that computes explicit expressions for generating functions that enable the efficient computations of the expectation, variance, and higher moments, of the random…

We consider the counting problem of the number of \textit{leaf-labeled increasing trees}, where internal nodes may have an arbitrary number of descendants. The set of all such trees is a discrete representation of the genealogies obtained…

We are interested in random uniform minimal factorizations of the $n$-cycle which are factorizations of $(1~2\dots n)$ into a product of $n-1$ transpositions. Our main result is an explicit formula for the joint probability that 1 and 2…

We present an elementary proof for Ljunggren equation

We reduce the calculation of the simplest Hodge integrals to some sums over decorated trees. Since Hodge integrals are already calculated, this gives a proof of a rather interesting combinatorial theorem and a new representation of…

The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.

In this short note, we briefly discuss the Borel-Cantelli lemma and propose a new generalization of the first part of it.

The purpose of this short note is to consider multi-variate Hasse-Schmidt derivations on exterior algebras and to show how they easily provide remarkable identities, holding in the algebra of square matrices, which generalise the classical…

Some simple facts are proved ruling the Collatz tree and the chains of vertices appearing in it, leading to the reduction of the number of significant elements appearing in the tree. Although the Collatz conjecture remains open, these fact…

Maxmin trees are labeled trees with the property that each vertex is either a local maximum or a local minimum. Such trees were originally introduced by Postnikov, who gave a formula to count them and different combinatorial interpretations…

A "simple trace formula" is used to derive an asymptotic result for class numbers of complex cubic orders.

In this note we prove a weighted version of the Khintchine inequalities.

An observation on Hall-Littlewood polynomials.

We introduce a class of group-like objects and prove that Cayley Theorem on groups has a counterpart in the class of group-like objects.

In the present note a generalization of Borel-Cantelli Lemma is proposed.

We present a new probabilistic proof of Otter's asymptotic formula for the number of unlabelled trees with a given number of vertices. We additionally prove a new approximation result, showing that the total variation distance between…

In this paper we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some…