Related papers: A combinatorial conjecture from PAC-Bayesian machi…
Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to…
Dumont has conjectured a marvellous identity, which generalizes, in particular, the classical results of Lagrange, Gauss, Jacobi and Kronecker on the sums of two, three and four squares. We give a combinatorial proof of Dumont's conjecture.
A new combinatorial-probabilistic diagnostic entropy has been introduced. It describes the pair-wise sum of probabilities of system conditions that have to be distinguished during the diagnosing process. The proposed measure describes the…
Combinatorial mixed valuations associated to translation-invariant valuations on polytopes are introduced. In contrast to the construction of mixed valuations via polarization, combinatorial mixed valuations reflect and often inherit…
This paper presents an offering of some of the myriad connections between Combinatorics and Probability, directed in particular toward combinatorialists. The choice of material was dictated by the author's own interests, tastes and…
In this paper we improve drastically the estimate for the multiplicity of a binary recurrence. The main contribution comes from an effective version of the Faltings' Product Theorem.
Some generalized multi-sum Chu-Vandermonde identities are presented and proved, generalizing some known multi-sum Chu-Vandermonde identities from literature and adding some quadratic and cubic examples of these identities. Some other…
Combinatorial dimensions play an important role in the theory of machine learning. For example, VC dimension characterizes PAC learning, SQ dimension characterizes weak learning with statistical queries, and Littlestone dimension…
Based on an interesting identity of Bat{\i}r we derive new identities for double sums involving famous number sequences. We also prove some double sum identities for binomial transform pairs.
In this article, we verify the additivity for rank of a sum of coprime monomials and bivariate polynomials generalizing the result in (\cite{CCG}). We also show similar results hold for cactus rank.
The combinatorial invariance conjecture (due independently to G. Lusztig and M. Dyer) predicts that if $[x,y]$ and $[x',y']$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan-Lusztig…
In this work we prove a new combinatorial identity and applying it we establish many finite harmonic sum identities. Among many others, we prove that \begin{equation*}…
We present a PAC-Bayes-style generalization bound which enables the replacement of the KL-divergence with a variety of Integral Probability Metrics (IPM). We provide instances of this bound with the IPM being the total variation metric and…
The study of pinnacle sets has been a recent area of interest in combinatorics. Given a permutation, its pinnacle set is the set of all values larger than the values on either side of it. Largely inspired by conjectures posed by Davis,…
We establish some new bilateral double-sum Rogers-Ramanujan identities involving parameters. As applications, these identities yield several new multi-sum Rogers-Ramanujan type identities. Our proofs utilize the theory of basic…
Recent advances in statistical learning theory have revealed profound connections between mutual information (MI) bounds, PAC-Bayesian theory, and Bayesian nonparametrics. This work introduces a novel mutual information bound for…
Probabilistic applicative bisimulation is a recently introduced coinductive methodology for program equivalence in a probabilistic, higher-order, setting. In this paper, the technique is applied to a typed, call-by-value, lambda-calculus.…
We upgrade [1] to a complete proof of the conjecture NP = PSPACE. [1]: L. Gordeev, E. H. Haeusler, Proof Compression and NP Versus PSPACE, Studia Logica (107) (1): 55-83 (2019)
In this paper we give a computer proof of a new polynomial identity, which extends a recent result of Alladi and the first author. In addition, we provide computer proofs for new finite analogs of Jacobi and Euler formulas. All computer…
In this paper, we show combinatorial identities that represent powers of positive integers using multinomial coefficients, which do not come from the multinomial theorem and the multinomial Vandermonde's convolution.