Related papers: An identity conjectured by Lacasse via the tree fu…
In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.
We examine an identity originally stated in Ramanujan's ``lost notebook'' and first proven algebraically by Andrews and combinatorially by Kim. We give two independent combinatorial proofs and interpretations of this identity, which also…
We give a proof of a conjecture of A. Lacasse in his doctoral thesis which has applications in machine learning algorithms. The proof relies on some interesting binomial sums identities introduced by Abel (1839), and on their generalization…
We derive a combinatorial identity which is useful in studying the distribution of Fourier coefficients of L-functions by allowing us to pass from knowledge of moments of the coefficients to the distribution of the coefficients.
In this paper, we give a decomposition of triply rooted trees into three doubly rooted trees. This leads to a combinatorial interpretation of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory, and…
In the process of studying a conjecture of Holly M. Green and Martin W. Liebeck, we obtain two interesting identities by elementary methods, one is a combinatorial identity, and the other is a number theoretic identity.
In this note we present a method for obtaining a wide class of combinatorial identities. We give several examples, in particular, based on the Gamma and Beta functions. Some of them have already been considered by previously, and other are…
Recently the second named author discovered a combinatorial identity in the context of vertex representations of quantum Kac-Moody algebras. We give a direct and elementary proof of this identity. Our method is to show a related identity of…
Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special…
In this note we give some identities which involve the Mertens function M(n). Our proofs are combinatorial with relatively prime subsets as a main tool.
In this paper, we explore some interesting applications of the matrix tree theorem. In particular, we present a combinatorial interpretation of a distribution of $(n-1)^{n-1}$, in the context of uprooted spanning trees of the complete graph…
We give an elementary proof of an interesting combinatorial identity which is of particular interest in graph theory and its applications. Two applications to enumeration of forests with closed-form expressions are given.
By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph…
We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are…
We provide bijective proofs of two classic identities that are very simple to prove using generating functions, but surprisingly difficult to prove combinatorially. The problem of finding a bijective proof for the first identity was first…
Using a probabilistic approach, we derive several interesting identities involving beta functions. Our results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.
In his lost notebook, Ramanujan listed 5 identities related to the false theta function $$f(q)=\sum_{n=0}^\infty (-1)^nq^{n(n+1)/2}.$$ A new combinatorial interpretation and proof of one of these identities is given. The methods of the…
We give a simple statistical proof of a binomial identity, by evaluating the Laplace transform of the maximum of n independent exponential random variables in two different ways. As a by product, we obtain a simple proof of an interesting…
We present a proof of a combinatorial conjecture from the second author's Ph.D. thesis. The proof relies on binomial and multinomial sums identities. We also discuss the relevance of the conjecture in the context of PAC-Bayesian machine…
We define and prove isomorphisms between three combinatorial classes involving labeled trees. We also give an alternative proof by means of generating functions.