Motzkin paths, Motzkin polynomials and recurrence relations

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial "counting" all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials) have been studied before, but here we deduce sonic properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix entries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof

Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

We consider the Fermi\u2013Pasta\u2013Ulam\u2013Tsingou (FPUT) chain composed by N 6b 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature \u3b2- 1. Given a fixed 1 64 m 6a N, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order \u3b2, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics

Valuative invariants for large classes of matroids

We study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a \emph{stressed subset}. This framework provides a new combinatorial characterization of the class of split matroids. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which in turn can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations on split matroids depend solely on the behavior of the invariant on tractable subclass of Schubert matroids that we call \emph{cuspidal matroids}. We address systematically the consequences of our approach for several invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan--Lusztig polynomials, the Whitney numbers of the first and second kind, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's $g$-polynomials, as well as Chow rings of matroids and their Hilbert--Poincar\'e series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.Comment: 69 pages, 10 figures. This is a substantially revised and condensed version. The structure and numbering of the results and definitions differs from v1 and v