Related papers: The combinatorics of frieze patterns and Markoff n…
We survey some of the mechanisms used to prove that naturally defined sequences in combinatorics are log-concave. Among these mechanisms are Alexandrov's inequality for mixed discriminants, the Alexandrov Fenchel inequality for mixed…
In a recent series of papers, an exact combinatorial solution was claimed for a variant of the so-called Marcus--Lushnikov model of aggregation. In this model, a finite number of aggregates, are initially assumed to be present in the form…
We find an explicit combinatorial gradient vector field on the well known complex S (Salvetti complex) which models the complement to an arrangement of complexified hyperplanes. The argument uses a total ordering on the facets of the…
We analyze a functor from cyclic operads to chain complexes first considered by Getzler and Kapranov and also Markl. This functor is a generalization of the graph homology considered by Kontsevich, which was defined for the three operads…
We introduce a new function on the set of pairs of cluster variables via $f$-vectors, which we call it the compatibility degree (of cluster complexes). The compatibility degree is a natural generalization of the classical compatibility…
Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of L-functions in families with the same…
Polynomial completeness results aim at characterizing those functions that are induced by polynomials. Each polynomial function is congruence preserving, but the opposite need not be true. A finite algebraic structure $\mathbf{A}$ is called…
We are interested in the enumeration of Fully Packed Loop configurations on a grid with a given noncrossing matching. By the recently proved Razumov--Stroganov conjecture, these quantities also appear as groundstate components in the…
It is well known how the linking number and framing can be extracted from the degree 1 part of the (framed) Kontsevich integral. This note gives a general formula expressing any product of powers of these two invariants as combination of…
Parametric models in vector spaces are shown to possess an associated linear map. This linear operator leads directly to reproducing kernel Hilbert spaces and affine- / linear- representations in terms of tensor products. From the…
Some enumerative aspects of the fans, called generalized associahedra, introduced by S. Fomin and A. Zelevinsky in their theory of cluster algebras are considered, in relation with a bicomplex and its two spectral sequences. A precise…
New conjectures are proposed on the numbers of FPL configurations pertaining to certain types of link patterns. Making use of the Razumov and Stroganov Ansatz, these conjectures are based on the analysis of the ground state of the…
Keevash and Mycroft [\emph{Mem.~Amer.~Math.~Soc., 2015}] developed a geometric theory for hypergraph matchings and characterized the dense simplicial complexes that contain a perfect matching. Their proof uses the hypergraph regularity…
Erman, Smith, and V\'arilly-Alvarado showed that the expected number of doubly monic Laurent polynomials $f(z) = z^{-m} + a_{-m+1}z^{-m+1} + \cdots + a_{n-1}z^{n-1} + z^n$ whose first $m+n-1$ powers have vanishing constant term is the…
We give the explicit algorithm computing the motivic generalization of the Poincare series of the plane curve singularity introduced by A. Campillo, F. Delgado and S. Gusein-Zade. It is done in terms of the embedded resolution of the curve.…
We review and present new studies on the relation between the partition functions of integrable lattice models and symmetric polynomials, and its combinatorial representation theory based on the correspondence, including our work. In…
Morier-Genoud, Ovsienko and Tabachnikov introduced supersymmetric frieze patterns (see arXiv:1501.07476). This note gives a solution to Problem 1 from that article: determine the formula for the entries of a superfrieze.
Di Francesco conjectured in 2021 that the number of domino tilings of a certain family of regions -- called Aztec triangles -- on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign…
We introduce an algorithm that exploits a combinatorial symmetry of an arrangement in order to produce a geometric reflection between two disconnected components of its moduli space. We apply this method to disqualify three real examples…
The classical congruences satisfied by the Fibonacci and Lucas sequences are reflected with the decomposition of primes in the ring generated by the gold number. This generalizes to establish a correspondence that we hope will be new…