Cayley's formula is a fundamental result in combinatorics that counts the number of labeled trees on n vertices. While existing proofs use approaches such as Prufer sequences and the Matrix-Tree Theorem, we give a combinatorial proof that highlights the role of degree sequences and structural properties of labeled trees. Our goal is to provide an accessible perspective and suggest connections to related enumeration problems.
In a recent paper, Caracciolo, Sokal and Sportiello presented, inter alia, an algebraic/combinatorial proof for Cayley's identity. The purpose of the present paper is to give a "purely combinatorial" proof for this identity; i.e., a proof involving only combinatorial arguments together with a generalization of Laplace's Theorem, for which a "purely combinatorial" proof is already known.
The combinatorial heat and wave equations on all finite Cayley and coset graphs with discrete time variable was solved by Lal {\it et al}. In this paper, the results of the above paper are extended for infinite Cayley and coset graphs, whenever the associated groups are discrete, abelian and finitely generated. Furthermore, we study the solution of the combinatorial heat and wave equations on a $k$-regular tree whose associated group is a non-abelian free group on $k$ generators, each of order~$2$. It turns out that in case of Cayley graphs the solutions to combinatorial heat and wave equation…
We present two hypermatrix formulations of the Cayley Hamilton theorem. One of the proposed formulation naturally extends to hypermatrices the combinatorial interpretations of the classical Cayley Hamilton theorem. We conclude by discussing an application of the theorem to computing graph invariants which distinguish some non-isomorphic graphs with isospectral adjacency matrices.
We give combinatorial proofs of two multivariate Cayley--Hamilton type theorems. The first one is due to Phillips (Amer. J. Math., 1919) involving $2k$ matrices, of which $k$ commute pairwise. The second one regards the mixed discriminant, a matrix function which has generated a lot of interest in recent times. Recently, the Cayley--Hamilton theorem for mixed discriminants was proved by Bapat and Roy (Comb. Math. and Comb. Comp., 2017). We prove a Phillips-type generalization of the Bapat--Roy theorem involving $2nk$ matrices, where $n$ is the size of the matrices, among which $nk$ commute pai…
Recently Féray, Goulden and Lascoux gave a proof of a new hook summation formula for unordered increasing trees by means of a generalization of the Prüfer code for labelled trees and posed the problem of finding a bijection between weighted increasing trees and Cayley trees. We give such a bijection, providing an answer to the problem posed by Féray, Goulden and Lascoux as well as showing a combinatorial connection to the theory of tree volumes defined by Kelmans. In addition we give two simple proofs of the hook summation formula. As an application we describe how the hook summation formula g…
Let f be a function mapping an n dimensional vector space over GF(p) to GF(p). When p is 2, Bernasconi et al. have shown that there is a correspondence between certain properties of f (e.g., if it is bent) and properties of its associated Cayley graph. Analogously, but much earlier, Dillon showed that f is bent if and only if the "level curves" of f had certain combinatorial properties (again, only when p is 2). The attempt is to investigate an analogous theory when p is greater than 2 using the (apparently new) combinatorial concept of a weighted partial difference set. More precisely, we try…
The classic Cayley identity states that \det(\partial) (\det X)^s = s(s+1)...(s+n-1) (\det X)^{s-1} where X=(x_{ij}) is an n-by-n matrix of indeterminates and \partial=(\partial/\partial x_{ij}) is the corresponding matrix of partial derivatives. In this paper we present straightforward combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann-Berezin integration. Among the new identities proven here are a pair of "diagonal-parametrized" Cayley identities, a pair of "Laplacian-par…
Problem 10 of the Landelijke Interuniversitaire Mathematische Olympiade 2010 asks for a proof that all matrices in a certain family are nilpotent. Both model solutions prove this using the Cayley-Hamilton theorem. I give a purely combinatorial proof.
We give a short proof of Cayley's tree formula for counting the number of different labeled trees on $n$ vertices. The following nonlinear recursive relation for the number of labeled trees on $n$ vertices is deduced from a combinatorial argument, $$ T_n = \frac{n}{2} \sum_{k=0}^{n-2} \left ( \begin {array} {c} n-2 \\ k \end {array} \right ) T_{k+1} T_{n-k-1}; \ \ for \ n > 1 \ and \ T_1 = 1, $$ and then it is proved that $T_n = n^{n-2}$, which gives yet another proof of the celebrated Cayley's tree formula.
We present a nice result on the probability of a cycle occurring in a randomly generated graph. We then provide some extensions and applications, including the proof of the famous Cayley formula, which states that the number of labeled trees on $n$ vertices is $n^{n-2}.$
Cayley's formula states that the number of labelled trees on $n$ vertices is $n^{n-2}$, and many of the current proofs involve complex structures or rigorous computation. We present a bijective proof of the formula by providing an elementary calculation of the probability that a cycle occurs in a random map from an $n$-element set to an $n+1$-element set.
Renteln proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. We prove the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provide a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups.
Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map a mapical regular representation or MRR) and with what probability. The first question was answered by Jajcay. In this paper we tackle the probabilistic version, and prove that as groups get larger the proportion of MRRs among all Cayley Maps approaches 1.
We give a short proof for a formula for the number of divisions of a convex (sn+2)-gon along non-crossing diagonals into (sj+2)-gons, where 1<=j<=n-1. In other words, we consider dissections of an (sn+2)-gon into pieces which can be further subdivided into (s+2)-gons. This formula generalizes the formulas for classical numbers of polygon dissections: Euler-Catalan number, Fuss number and Kirkman-Cayley number. Our proof is elementary and does not use the method of generating functions.
We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon--Roichman about expanding properties of random Cayley graphs. Applying the general setting we show e.g., that with high probability (in a strong exp…
We give a combinatorial formula for LLT cumulants of unicellular unilevelled shapes in terms of parking functions and Cayley trees. Our formula implies previously conjectured Schur positivity.
We consider the relationship between a matroidal analogue of the degree $a$ Cayley-Bacharach property (finite sets of points failing to impose independent conditions on degree $a$ hypersurfaces) and geometric properties of matroids. If the matroid polytopes in question are nestohedra, we show that the minimal degree matroidal Cayley-Bacharach property denoted $MCB(a)$ is determined by the structure of the building sets used to construct them. This analysis also applies for other degrees $a$. Also, it does not seem to affect the combinatorial equivalence class of the matroid polytope. However, …
Aarhus University（奥胡斯大学）；University of Minnesota（明尼苏达大学）；University of Illinois Urbana Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Texas at Dallas（德克萨斯大学达拉斯分校）
We introduce a deformation of Cayley's second hyperdeterminant for even-dimensional hypermatrices. As an application, we formulate a generalization of the Jacobi-Trudi formula for Macdonald functions of rectangular shapes generalizing Matsumoto's formula for Jack functions.
In this paper we show a a proof by explicit bijections of the famous Kirkman-Cayley formula for the number of dissections of a convex polygon. Our starting point is the bijective correspondence between the set of nested sets made by \(k\) subsets of \(\{1,2,...,n\}\) with cardinality \(\geq 2\) and the set of partitions of \(\{1,2,...,n+k-1\}\) into \(k\) parts with cardinality \(\geq 2\).
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more standard approach to Cayley's formula. Moreover, this algebraic structure sheds light on the problem of counting the unlabeled trees. In particular, it indicates how counting the number of unlabeled trees on $n$ vertices is connected to finding the number of partitions of $n-2$
A proper vertex of a rooted tree with totally ordered vertices is a vertex that is less than all its proper descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials P_n(a,b,c)= c(a+(n-1)b+c)(2a+(n-2)b+c)...((n-1)a+b+c) which reduce to (n+1)^{n-1} for a=b=c=1. Our study of proper vertices was motivated by A. Postnikov's hook length formula for binary trees (arXiv:math.CO/0507163), which was also proved by W. Y. C. Chen and L. L. M. Yang (arXiv:math.CO/0507163) and generalized by R. R. X. D…
We present a very simple bijective proof of Cayley's formula due to Foata and Fuchs (1970). This bijection turns out to be very useful when seen through a probabilistic lens; we explain some of the ways in which it can be used to derive probabilistic identities, bounds, and growth procedures for random trees with given degrees, including random d-ary trees. We also introduce a partial order on the degree sequences of rooted trees, and conjecture that it induces a stochastic partial order on heights of random rooted trees with given degrees.
We provide a new derivation of the well-known generating function counting the number of walks on a regular tree that start and end at the same vertex, and more generally, a generating function for the number of walks that end at a vertex a distance i from the start vertex. These formulas seem to be very old, and go back, in an equivalent form, at least to Harry Kesten's work on symmetric random walks on groups from 1959, and in the present form to Brendan McKay (1983).
Using a certain Pick-type formula for the mixed volume of Cayley polyhedra, we compute the multiplicity of the isolated common zero of the maximal minors for a matrix of generic homogeneous polynomials of given degrees.
We denote a polygon as a connected component in Cayley tree of order 2 containing certain number of fix vertices. We found an exact formula for a polygon counting problem for two cases, in which, for the first case the polygon contain a full connected component of a Cayley tree and for the second case the polygon contain two fixed vertices. From these formulas, which is in the form of finite linear combination of Catalan numbers, one can find the asymptotic estimation for a counting problem.
In this paper, we determine an explicit formula for the number of walks in $X_n = \textsf{Cay}(\mathbb{Z}_n,\mathbb{U}_n)$, the unitary Cayley Graphs of order $n$, between any pair of its vertices. With this result, we give the number of representations of a fixed residue class $\bmod{}n$ as the sum of $k$ units of $\mathbb{Z}_n$.
We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group $\C{A}$. Using these characterizations, we find degree distribution polynomials for weak equivalence of some graphs including 1) circulant graphs of prime power order, 2) circulant graphs of order $4p$, 3) circulant graphs of square free order and 4) Cayley graphs of order $p$ or $2p$. As an application, we find an enumeration formula for the number of weak equivalence classes of circulant graphs of prime power order, order $4p$ and square free order and Cayley graphs of order $p$ or $2p$.
Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $η$-closed if the average degree of the subgraph induced by $A$ is at least $ηd$. This says that if we choose a random vertex $x\in A$ and a random neighbour $y$ of $x$, then the probability that $y\in A$ is at least $η$. The work of this paper was motivated by an attempt to obtain a qualitative description of closed subsets of the Cayley graph $Γ$ whose vertex set is $\mathbb F_2^{n_1}\otimes \dots \otimes \mathbb F_2^{n_d}$ with two vertices joined by an edge if their difference is of the form $u_1\otimes \cdot…
We prove a combinatorial formula for the alternating descent polynomials of type A and B. Combining with Josuat-Vergès' combinatorial interpretation for Hoffman's derivative polynomials for tangent and secant functions, we obtain a unified combinatorial proof of Ma-Yeh's two formulae linking these two families of polynomials.
We consider polynomial maps, which we call degree $d$-linear maps, that satisfy the Jacobian condition. We prove that certain infinite families of elements, which appear in the coefficients of the formal inverse of such maps, are in the ideal determined by the Jacobian condition. Using the Cayley-Hamilton theorem, we provide expressions for these elements in terms of the generators of that ideal. We also give a combinatorial proof of the Cayley-Hamilton theorem similar to that of Straubing \cite{Straubing}. We also include results of Gröbner basis computations regarding other elements.
This thesis presents new applications of Gale duality to the study of polytopes, point configurations and oriented matroids with extremal combinatorial properties. The first part of the thesis explores construction techniques for neighborly polytopes and oriented matroids. First, we provide a new interpretation of Shemer's classical Sewing Construction for neighborly polytopes in terms of lexicographic extensions of oriented matroids. This allows us to provide a simplified proof and to generalize it to oriented matroids in two ways: the Extended Sewing Construction and the Gale Sewing Construc…
In this article, a short combinatorial proof of the Capelli's identity is given. It also leads to an easy proof of the Capelli--Cauchy--Binet identity, a more general form of Capelli's identity. With the technique introduced, the Turnbull's identity can be proven by sum with signs. At the end of the article, the general Cayley identity is discussed as a dual version of Capelli's identity.
Cayley's formula states that there are $n^{n-2}$ spanning trees in the complete graph on $n$ vertices; it has been proved in more than a dozen different ways over its 150 year history. The complete graphs are a special case of threshold graphs, and using Merris' Theorem and the Matrix Tree Theorem, there is a strikingly simple formula for counting the number of spanning trees in a threshold graph on $n$ vertices; it is simply the product, over $i=2,3, ...,n-1$, of the number of vertices of degree at least $i$. In this manuscript, we provide a direct combinatorial proof for this formula which d…
Choose a random linear operator on a vector space of finite cardinality N: then the probability that it is nilpotent is 1/N. This is a linear analogue of the fact that for a random self-map of a set of cardinality N, the probability that some iterate is constant is 1/N. The first result is due to Fine, Herstein and Hall, and the second is essentially Cayley's tree formula. We give a new proof of the result on nilpotents, analogous to Joyal's beautiful proof of Cayley's formula. It uses only general linear algebra and avoids calculation entirely.
Based on an exact trace formula for a one-dimensional ray-splitting system, we derive novel combinatorial identities for cyclic binary sequences (Pólya necklaces).
In 1992 Gelfand and MacPherson gave a local and combinatorial formula for the Pontrjagin classes of a differential manifold. We give an expanded version of their discussion and highlight the origins of combinatorial differential manifolds in their work.
