arXiv subjects

Frédéric Chapoton

Publications and source records attributed to Frédéric Chapoton.

At least 19 recordsLinked to original sources

arXiv2026

Polytopes and posets associated to preorders

Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author. Preorder polytopes are shown to be lattice polytopes which satisfy a certain duality relating their Ehrhart polynomials with the zeta polynomials of their posets of lattice points. A combinatorial interpretation of the normalized volume of a preorder polytope is proven, together with formulas for the Ehrhart polynomial and the $h^\ast$-polynomial, and a combinatorial interpretation of the latter is conjectured. Several conjectures and results on the lattice point enumeration of arbor polytopes are generalized to preorder polytopes, new conjectures are proposed and new interesting examples of preorder polytopes are studied.

math.CO↗

arXiv2026

The Python Simulations of Chemistry Framework: 10 years of an open-source quantum chemistry project

Over the past decade, the Python-based Simulations of Chemistry Framework (PySCF) has developed into a widely used open-source platform for electronic structure theory and quantum chemical method development. This article reviews the major advances since the previous overview in 2020, covering new modules and methodology, infrastructure changes, and performance benchmarks.

physics.chem-ph↗

arXiv2026

Yamaguti algebras and noncrossing partitions

Recently, Das defined a new type of algebras, the Yamaguti algebras, which are supposed to serve as envelopes of Lie-Yamaguti algebras appearing naturally in differential geometry. We show that the nonsymmetric operad of Yamaguti algebras admit a simple combinatorial description via noncrossing partitions without singleton blocks.

math.AT↗

arXiv2025

On posets and polytopes attached to arbors

Starting from the data of an arbor, which is a rooted tree with vertices decorated by disjoint sets, we introduce a lattice polytope and a partial order on its lattice points. We give recursive algorithms for various classical invariants of these polytopes and posets, using the tree structure. For linear arbors, we propose a conjecture exchanging the Ehrhart polynomial of the polytope with the Zeta polynomial of the poset for the reverse arbor. The general motivation comes from the action of a transmutation operator acting on M -triangles, which should link the posets considered here with some kinds of generalized noncrossing partitions and generalized associahedra. We give some evidence for this relationship in several cases, including notably some polytopes, namely halohedra and Hochschild polytopes.

math.CO↗

arXiv2025

A note on Dirichlet-like series attached to polynomials

Some Dirichlet-like functions, attached to a pair (periodic function, polynomial) are introduced and studied. These functions generalize the standard Dirichlet L-functions of Dirichlet characters. They have similar properties, being holomorphic on thefull complex plane and having simple values on negative integers.

math.NT↗

arXiv2025

On a q-analogue of the Zeta polynomial of posets

We introduce a q-analogue of the classical Zeta polynomial of finite partially ordered sets, as a polynomial in one variable x with coefficients depending on the indeterminate q. We prove some properties of this polynomial invariant, including its behaviour with respect to duality, product and disjoint union. The leading term is a q-analogue of the number of maximal chains, but not always with non-negative coefficients. The value at q=0 turns out to be essentially the characteristic polynomial.

math.CO↗

arXiv2024

Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra

We introduce the shuffle of deformed permutahedra (a.k.a. generalized permutahedra), a simple associative operation obtained as the Cartesian product followed by the Minkowski sum with the graphical zonotope of a complete bipartite graph. Besides preserving the class of graphical zonotopes (the shuffle of two graphical zonotopes is the graphical zonotope of the join of the graphs), this operation is particularly relevant when applied to the classical permutahedra and associahedra. First, the shuffle of an $m$-permutahedron with an $n$-associahedron gives the $(m,n)$-multiplihedron, whose face structure is encoded by $m$-painted $n$-trees, generalizing the classical multiplihedron. We show in particular that the graph of the $(m,n)$-multiplihedron is the Hasse diagram of a lattice generalizing the weak order on permutations and the Tamari lattice on binary trees. Second, the shuffle of an $m$-associahedron with an $n$-associahedron gives the $(m,n)$-constrainahedron, whose face structure is encoded by $(m,n)$-cotrees, and reflects collisions of particles constrained on a grid. Third, the shuffle of an $m$-anti-associahedron with an $n$-associahedron gives the $(m,n)$-biassociahedron, whose face structure is encoded by $(m,n)$-bitrees, with relevant connections to bialgebras up to homotopy. We provide explicit vertex, facet, and Minkowski sum descriptions of these polytopes, as well as summation formulas for their $f$-polynomials based on generating functionology of decorated trees.

math.CO↗

arXiv2023

Posets and Fractional Calabi-Yau Categories

This article deals with a relationship between derived categories of modules over some partially ordered sets and triangulated categories arising from quasi-homogeneous isolated singularities. It produces heuristics for the existence of derived equivalences between posets, using the geometric category as an auxiliary intermediate. The notion of Weight plays a central role as a simple footprint of the derived categories under consideration.

math.CO↗

arXiv2021

Zinbiel algebras and multiple zeta values

We build, using the notion of zinbiel algebra, some commutative subalgebras $C_{u,v}$ inside an algebra of formal iterated integrals. There is a quotient map from this algebra of formal iterated integrals to the algebra of motivic multiple zeta values. Restricting this quotient map to the subalgebras $C_{u,v}$ of A$_{1,0}$ gives a morphism of graded commutative algebras with the same graded dimension. This is conjectured to be generically an isomorphism. When u+v = 0, the image is instead a sub-algebra of the algebra of motivic multiple zeta values.

math.NT↗

arXiv2021

Multiple $T$-values with one parameter

This article introduces an algebra of functions in one variable $c$ defined by iterated integrals of two specific differential forms depending on $c$, where the product is the shuffle product. This algebra can be seen as a common deformation of multiple zeta values and of Kaneko-Tsumura's recent multiple $T$-values. The first few graded dimensions, assuming that a grading by the weight does hold, are computed.

math.NT↗

arXiv2021

Counting quadrant walks via Tutte's invariant method

In the 1970s, William Tutte developed a clever algebraic approach, based on certain "invariants", to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past 20 years to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps, taken in $\{-1, 0,1\}^2$. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic. This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity follows almost automatically. Then, we move to a complex analytic viewpoint that has already proved very powerful, leading in particular to integral expressions for the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions for the generating function, and a proof that this series is D-algebraic (that is, satisfies polynomial differential equations).

math.CO↗

arXiv2020

Moments of q-Jacobi Polynomials and q-Zeta Values

We explore some connections between moments of rescaled little q-Jacobi polynomials, q-analogues of values at negative integers for some Dirichlet series, and the q-Eulerian polynomials of wreath products of symmetric groups.

math.CO↗

arXiv2020

On the roots of the Poupard and Kreweras polynomials

The Poupard polynomials are polynomials in one variable with integer coefficients, with some close relationship to Bernoulli and tangent numbers. They also have a combinatorial interpretation. We prove that every Poupard polynomial has all its roots on the unit circle. We also obtain the same property for another sequence of polynomials introduced by Kreweras and related to Genocchi numbers. This is obtained through a general statement about some linear operators acting on palindromic polynomials.

math.CO↗

arXiv2019

Ramanujan-Bernoulli numbers as moments of Racah polynomials

The classical sequence of Bernoulli numbers is known to the the sequence of moments of a family of orthogonal polynomials. Some similar statements are obtained for another sequence of rational numbers, which is similar in many ways to the Bernoulli numbers.

math.NT↗

arXiv2018

Some properties of a new partial order on Dyck paths

We introduce and study a new partial order on Dyck paths. We prove that these posets are meet-semilattices. We show that their numbers of intervals are the same as the number of bicubic planar maps. We describe an unexpected connection with the Hochschild polytopes of Saneblidze.

math.CO↗

arXiv2018

A note on gamma triangles and local gamma vectors

This article introduces Gamma-triangles, which are closely related to and more fundamental than F-triangles and H-triangles that have been used in the combinatorics of cluster complexes. It is proved that Gamma-triangles can be expressed as sums of the local gamma-vectors introduced by Athanasiadis, which themselves refine the local h-vectors attached by Stanley to simplicial subdivisions. Gamma triangles of all cluster complexes are then explicitly computed.

math.CO↗

arXiv2017

A note on Tamari intervals

To every partial order P, one associates a polynomial $\mathbb{D}_P$ in 4 variables that enumerates the intervals of P according to 4 parameters. Some symmetry properties of this polynomial are obtained for a specific family of posets, the Tamari lattices. A ternary symmetry is proved for the polynomial in 3 variables obtained by setting one variable to 1. Another global symmetry is conjectured. The set of synchronized intervals is described using a facet of the Newton polytope. A relation to the statistics of the canopy of binary planar trees is described.

math.CO↗

arXiv2017

On the wildness of cambrian lattices

In this note, we investigate the representation type of the cambrian lattices and some other related lattices. The result is expressed as a very simple trichotomy. When the rank of the underlined Coxeter group is at most 2, the lattices are of finite representation type. When the Coxeter group is a reducible group of type A 3 1 , the lattices are of tame representation type. In all the other cases they are of wild representation type.

math.RT↗