Related papers: Introduction to the combinatorial atlas

The volume and the number of lattice points of the permutohedron P_n are given by certain multivariate polynomials that have remarkable combinatorial properties. We give several different formulas for these polynomials. We also study a more…

It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given; the…

We prove a conjecture of K. Marton, widely known as the polynomial Freiman--Ruzsa conjecture, in characteristic $2$. The argument extends to odd characteristic, with details to follow in a subsequent paper.

This paper contains the proof of difference counterparts of the conjectures due to Keven Kadell on symmetric and anti-symmetric Macdonald polynomials.

We present a new combinatorial formula for Hall-Littlewood functions associated with the affine root system of type $\tilde A_{n-1}$, i.e. corresponding to the affine Lie algebra $\hat{\mathfrak{sl}}_n$. Our formula has the form of a sum…

Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As a part of the proof, we also establish combinatorially the invariance of the linear-categorical…

In this article, we extend some results about algebra $A$ with the group of units $U(A)$ having a special polynomial identity, Laurent polynomial. And we present a new version of B. Hartley Conjecture with these identities.

In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials…

Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half…

In this paper, we pose many challenging conjectures on congruences involving binomial coefficients and Ap\'ery-like numbers.

Some natural inequalities related to rearrangement in matrix products can also be regarded as extensions of classical inequalities for sequences or integrals. In particular, we show matrix versions of Chebyshev and Kantorovich type…

We prove several combinatorial results on path algebras over discrete structures related to directed graphs. These results are motivated by Morse theory on a manifold with boundary and, more generally, by Floer theory on a configuration…

An important conjecture in additive combinatorics, number theory, and algebraic geometry posits that the partition rank and analytic rank of tensors are equal up to a constant, over any finite field. We prove the conjecture up to a…

We prove a metrical result on a family of conjectures related to the Littlewood conjecture, namely the original Littlewood conjecture, the mixed Littlewood conjecture of de Mathan and Teuli\'e and a hybrid between a conjecture of Cassels…

We give a combinatorial identity related to the Franel numbers involving the sum of fourth power of binomial coefficients. Furthermore, investigating in J. Mikic's proof of the first Strehl Identity, we provide a combinatorial proof of this…

For line arrangements in P^2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal…

We introduce a notion of intrinsically H\"older graphs in metric spaces. Following a recent paper of Le Donne and the author, we prove some relevant results as the Ascoli-Arzel\`a compactness Theorem, Ahlfors-David regularity and the…

We prove the "strong conjecture" expressed by Gazeau et al. in arXiv:1203.3936v1 [math-ph] about the coefficients of the Taylor expansion of the exponential of a polynomial. This implies the "weak conjecture" as a special case. The proof…

An introduction is given to the Littlewood-Richardson rule, and various combinatorial constructions related to it. We present a proof based on tableau switching, dual equivalence, and coplactic operations. We conclude with a section…

Using algebraic transformations and equivalent reformulations we derive a number of new results from some earlier ones (by the author) in more accepted terms closely related to well-known conjectures of Bondy and Jung including a number of…