Related papers: Cayley's formula from middle school math
We prove an infinitary disjoint union theorem for level products of trees. To implement the proof we develop a Hales-Jewett type result for words indexed by a level product of trees.
It is shown that the formula for the variance of combined series yields surprisingly simple proofs of some well known variance bounds.
We consider the rooted trees which not have isomorphic representation and introduce a conception of complexity a natural number also. The connection between quantity such trees with $n$ edges and a complexity of natural number $n$ is…
We show that Pinney's equation [2] with a constant coefficient can be reduced to its linear part by a simple change of variables. Also, Pinney's original solution is simplified slightly.
A Cayley permutation is a word of positive integers such that if a letter appears in this word, then all positive integers smaller than that letter also appear. We initiate a systematic study of pattern avoidance on Cayley permutations…
In this article, we prove various properties of Calkin-Wilf tree. We also see how the Minkowski question mark function will act on Calkin-Wilf tree and its diagonals.
We present an algorithmic mapping from permutations of length dn to labeled n-node d-ary trees and back again. Given such a bijection, one can interpret each of the factorials in the formula for the Catalan numbers as a count of…
In this work, we present a series of bijections that reveal the deep connections between the concepts of tree records, the girth of a connected endofunction, and the genesis sequence, the first sequence in the OEIS. We use these results to…
The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the…
We give a presentation of abelian class field theory.
We give a precise description of combed trees in terms of Kelly-Mac Lane graphs. We show that any combed tree is uniquely expressed as an allowable Kelly-Mac Lane graph of a certain shape. Conversely, we show that any such Kelly-Mac Lane…
In this paper, we prove a general simple relative trace formula. As an application, we prove a relative analogue of the Weyl law.
In the paper, the authors provide four alternative proofs of an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.
In this note, the first-order Dickson polynomials are introduced through a particular case of the expression of the trace of the $n^{th}$ power of a matrix in terms of powers of the trace and determinant of the matrix itself. The technique…
In this article we derive a simple twisted relative trace formula.
This note gives a simple approach to q-analogues of some results associated with Abel polynomials.
It is proved that the restriction of a $k$ and $(k-1)$-component directed spanning forest of minimal weight to an atom of the subset algebra generated by the sets of vertices of trees of $k$-component minimal spanning forests is a tree. For…
Recently, Han obtained two hook length formulas for binary trees and asked for combinatorial proofs. One of Han's formulas has been generalized to k-ary trees by Yang. Sagan has found a probabilistic proof of Yang's extension. We give…
An increasing 1,2-tree is a labeled graph formed by starting with a vertex and then repeatedly attaching a leaf to a vertex or a triangle to an edge, the labeling of the vertices corresponding to the order in which the vertices are added.…
We provide direct elementary proofs of several explicit expressions for Bernoulli numbers and Bernoulli polynomials. As a byproduct of our method of proof, we provide natural definitions for generalized Bernoulli numbers and polynomials of…