### Talks in 2024

#### Combinatorics Seminar

**Title:**Kneser graphs are Hamiltonian

**Speaker:**Torsten Mütze (University of Warwick)

**Date:**Friday 26th April 12:30

**Room:**Online meeting (Webex)

**Abstract:**

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of an $n$-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph $K(5,2)$. This problem received considerable attention in the literature, including a recent solution for the sparsest case $n=2k+1$. The main contribution of our work is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph $J(n,k,s)$ has as vertices all k-element subsets of an n-element ground set, and an edge between any two sets whose intersection has size exactly $s$. Clearly, we have $K(n,k)=J(n,k,0)$, i.e., generalized Johnson graphs include Kneser graphs as a special case. Our results imply that all known families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovász’ conjecture on Hamilton cycles in vertex-transitive graphs from 1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway’s Game of Life, and to analyze this system combinatorially and via linear algebra.

This is joint work with Arturo Merino (TU Berlin) and Namrata (Warwick).

Meeting link:

\[

\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}

\]

#### Colloquium Discrete Geometry

**Title:**Subdivisions and invariants of matroids

**Speaker:**Benjamin Schröter (KTH Royal Institute of Technology)

**Date:**24.04.2024, 11:00 Uhr

**Room:**Seminarraum 1, Kopernikusgasse 24/IV.

**Abstract:**

Matroids are a combinatorial abstraction of both graphs and linear spaces who play a central role in tropical geometry. In my talk I will take a polyhedral view on matroids and demonstrate how this perspective can be used to study classical matroid and graph invariants, tropical moduli spaces as well as problems arising in real life applications, e.g., shortest path problems in optimization, scattering processes in physics or decompositions in phylogenetics.

#### Geometrisches Seminar

**Title:**Cup Product Persistence and Its Efficient Computation

**Speaker:**Abhishek Rathod (Ben Gurion University of the Negev)

**Date:**24.04.2024, 13:45

**Room:**Seminarraum 2, Kopernikusgasse 24/IV.

**Abstract:**

It is well-known that the cohomology ring has a richer structure than homology groups. However, until recently, the use of cohomology in the persistence setting has been limited to speeding up barcode computations. Some of the recently introduced invariants, namely, persistent cup-length, persistent cup modules and persistent Steenrod modules, to some extent, fill this gap. When added to the standard persistence barcode, they lead to invariants that are more discriminative than the standard persistence barcode. In this work, we devise an $O(dn^4)$ algorithm for computing the persistent $k$-cup modules for all $k\in\{2,\dots,d\}$, where $d$ denotes the dimension of the filtered complex, and $n$ denotes its size. Moreover, we note that since the persistent cup length can be obtained as a byproduct of our computations, this leads to a faster algorithm for computing it for $d \ge 3$. Finally, we introduce a new stable invariant called partition modules of cup product that is more discriminative than persistent $k$-cup modules and devise an $O(c(d)n^4)$ algorithm for computing it, where $c(d)$ is subexponential in $d$.

#### Colloquium Discrete Geometry

**Title:**Generalized Heawood graphs and triangulations of tori

**Speaker:**Cesar Ceballos (TU Graz)

**Date:**Tue 23.04.2024, 13:00

**Room:**Seminarraum 2, Kopernikusgasse 24/IV.

**Abstract:**

The Heawood graph is a remarkable graph that played a fundamental role in the development of the theory of graph colorings on surfaces in the 19th and 20th centuries. The purpose of this talk is to introduce a generalization of the classical Heawood graph indexed by a sequence of positive integers. The resulting generalized Heawood graphs are toroidal graphs with a rich combinatorial and geometric structure. In particular, they are dual to higher dimensional triangulated tori and we present explicit combinatorial formulas for their f-vectors.

This talk is based on joint work with Joseph Doolittle.

#### Colloquium Discrete Geometry

**Title:**From Tropical Geometry to Applications

**Speaker:**Georg Loho (FU Berlin)

**Date:**Tue 23.04.2024, 09:30

**Room:**Seminarraum 2, Kopernikusgasse 24/IV.

**Abstract:**

Tropical geometry by itself is a young and fast-growing area of mathematics.

Identifying where similar techniques can be used leads to an even more exciting range of applications. I will focus on two directions. First, we will see how a tropical point of view on the geometric structure of linear programming and certain two-player games helps to derive new algorithms. Second, I will present how fundamental structures of tropical geometry can be generalized to discrete convex analysis, a framework of well-behaved polyhedral functions. The latter helps to address questions from economics on one hand and the study of flag varieties on the other hand.

#### Combinatorics Seminar

**Title:**The structure and density of (strongly) $k$-product-free sets in the free semigroup.

**Speaker:**Frederick Illingworth (University College London)

**Date:**Friday 19th April 12:30

**Room:**Online meeting (Webex)

**Abstract:**

The free semigroup} $\mathcal{F}$ over a finite alphabet $\mathcal{A}$ is the set of all finite words with letters in $\mathcal{A}$ equipped with the operation of concatenation. A subset $S$ of $\mathcal{F}$ is $k$-product-free} if no $k$ words in $S$ concatenate to another word in $S$. How dense can a $k$-product-free subset of $\mathcal{F}$ be? What is the structure of the densest $k$-product-free subsets?

Leader, Letzter, Narayanan, and Walters proved that $2$-product-fee subsets of the free semigroup have density at most $1/2$ and asked for the structure of the densest sets. In this talk I will discuss the answer to their question as well as the answer (both density and structure) for general $k$. This generalises results of {\L}uczak and Schoen for sum-free sets in the integers although the methods used are quite different.

This is joint work with Lukas Michel and Alex Scott.

Meeting link:

\[

\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}

\]

#### Vortrag im Rahmen des Seminars Operator Theorie

**Title:**Spectral cluster bounds for orthonormal functions

**Speaker:**Jean-Claude Cuenin (Lougborough, UK)

**Date:**Donnerstag, 18.4.2014, 12:15 Uhr

**Room:**TU Graz, Steyrergasse 30, EG, Seminarraum AE02 (STEG06)

**Abstract:**

Abstract: The topic of my talk are functional inequalities for systems of orthonormal functions. One wants to have an optimal dependence of the constant on the number of functions involved. In this talk we focus on so-called spectral cluster bounds, which are concerned with $L^p$ norms of (linear combinations of) eigenfunctions of the Laplace-Beltrami operator on a compact manifold without boundary. I will review what is known in the case of a single function and for systems of orthonormal functions, then I will present some new results for operators with non-smooth coefficients or on manifolds with boundary.

The talk is partly based on joint ongoing works with Ngoc Nhi Nguyen and Xiaoyan Su.

#### Colloquium Discrete Geometry

**Title:**The Wachspress Geometry of Polytopes --- a bridge between algebra, geometry and combinatorics

**Speaker:**Martin Winter (University of Warwick)

**Date:**Wed 17.04.2024, 11:00

**Room:**Seminarraum 1, Kopernikusgasse 24/IV.

**Abstract:**

Wachspress Geometry is a young field concerned with a family of closely related mathematical objects defined on polytopes -- the so-called ``Wachspress object''.

At its center, the ``Wachspress coordinates'' have their origin as generalized barycentric coordinates in geometric modeling and finite element analysis, but have since re-emerged in a wide range of seemingly unrelated contexts in algebraic geometry, statistics, spectral graph theory, rigidity theory and convex geometry. Explaining and exploiting this surprising ubiquity is a central motivation of Wachspress Geometry.

My first goal for this talk is to give an introduction to this fascinating subject and to elaborate on the potential of Wachspress Geometry to bridge between the algebra, geometry and combinatorics of polytopes. Subsequently I will focus on two particular applications that emerged in my own research: first, the rigidity and reconstruction of polytopes from partial combinatorial and metric data; second, a novel approach to algorithmically decide the polytopality of simplicial spheres.

#### First guest lecture within SS24 Elective subject mathematics (Linear Operators)

**Title:**Eigenfunctions of the Laplacian

**Speaker:**Jean-Claude Cuenin (Loughborough University, UK)

**Date:**Start: April 15, 12.00-12.30 (first organizatorial meeting)

**Room:**TU Graz, Steyrergasse 30, Seminarraum AE02 (STEG006), EG

**Abstract:**

Outline: The topic of the lecture is eigenfunctions of the Laplace-Beltrami operator on Riemannian manifolds. The first goal is to prove the sharp Weyl formula, which describes the asymptotic distribution of the eigenvalues. The second goal is to prove sharp bounds on Lp norms of eigenfunctions. Lp norms provide a convenient measure for the concentration of eigenfunctions and shed some light on the question `How do eigenfunctions look like’? To tackle this question we will develop tools from harmonic and microlocal analysis. If time permits, we will discuss improvements to these estimates under additional dynamical conditions on the geodesic flow.

#### Combinatorics Seminar (Irregular Time)

**Title:**Improved bounds for Szemerédi’s theorem

**Speaker:**Ashwin Sah (MIT)

**Date:**Friday 12th April 13:30

**Room:**Online meeting (Webex)

**Abstract:**

Let $r_k(N)$ denote the size of the largest subset of $[N] = \{1,\ldots,N\}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that

\[r_k(N)\ll N\exp(-(\log\log N)^{c_k}).\]

Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm as well as the density increment strategy of Heath-Brown and Szemerédi as reformulated by Green and Tao.

Meeting link:

\[

\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}

\]

#### Combinatorics Seminar

**Title:**$2$-neighbourhood bootstrap percolation on the Hamming graph

**Speaker:**Dominik Schmid (Graz University of Technology)

**Date:**Friday 22nd March 12:30

**Room:**AE06, Steyrergasse 30

**Abstract:**

We consider the following infection spreading process on graphs: At the beginning, each vertex is infected independently with a given probability $p$ and after that, new vertices get infected if at least $2$ of their neighbours have been infected. We say a graph percolates if eventually every vertex of the graph is infected. An interesting question in this context is above which threshold value of $p$ a graph percolates with high probability. In particular, this problem was studied on the $n$-dimensional hypercube by Balogh, Bollob\'{a}s and Morris, who established a sharp threshold when $n$ tends to infinity. We consider the process on a generalised version of the hypercube, the

Hamming graph}, which arises as the $n$-fold cartesian product of the complete graph $K_k$. We establish a threshold for percolation when $n$ tends to infinity and $k =k(n)$ is an arbitrary, positive function of $n$. The presented techniques are in parts applicable to more general classes of product graphs and might be of interest for studying the process in these cases.

This is joint work with Mihyun Kang and Michael Missethan.

#### Zahlentheoretisches Kolloquium

**Title:**UPDATE - Arithmetik an der A7

**Speaker:**()

**Date:**Thursday (March 21, 2024) und Friday (March 22, 2024)

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG

**Abstract:**

Homepage: https://www.math.tugraz.at/\string~mtechnau/2024-aaa7-graz.html

\noindentThursday}

\begin{description}

\item{14:10--14:40:} Lukas Spiegelhofer:

Thue--Morse along the sequence of cubes}

\item{14:40--15:10:} Pascal Jelinek:

Square-free values of polynomials on average}

\item{15:10--15:40:} Helmut Maier:

The problem of propinquity of divisors for Gaussian integers}

\item{16:00--16:30:} Martin Widmer:

A quantitative version of the primitive element theorem for number fields}

\item{16:30--17:00:} Sumaia Saad Eddin:

Asymptotic results for a class of arithmetic functions involving the greatest common divisor}

\item{17:00--17:30:} Christian Weiß:

Uniform distribution --- the $p$-adic viewpoint}

\end{description}

\noindentFriday}

\begin{description}

\item{09:20--09:50:} Ram\={u}nas Garunk\v{s}tis:

Discrete mean value theorems for the Riemann zeta-function over its shifted nontrivial zeros}

\item{09:50--10:20:} Rainer Dietmann:

Sieving with square conditions and applications to Hilbert cubes in arithmetic sets}

\item{10:20--10:50:} Roland Miyamoto:

Fixed-points of stribolic operators and their combinatorial aspects}

\item{10:50--11:20:} Robert F.\ Tichy:

Pseudorandom measures}

\item{11:40--12:10:} Christian Bernert:

Cubic forms over imaginary quadratic number fields (and why we should care)}

\item{12:10--12:40:} Shuntaro Yamagishi:

Rational curves on complete intersections}

\end{description}

#### Combinatorics Seminar

**Title:**Chromatic number is not tournament-local

**Speaker:**Michael Savery (University of Oxford)

**Date:**Friday 15th March 12:30

**Room:**Online meeting (Webex)

**Abstract:**

Scott and Seymour conjectured the existence of a function $f$ such that, for every graph $G$ and tournament $T$ on the same vertex set, $\chi(G)\geq f(k)$ implies that $\chi(G[N_T^+(v)])\geq k$ for some vertex $v$. We will disprove this conjecture even if $v$ is replaced by a vertex set of size $\mathcal{O}(\log{|V(G)|})$. As a consequence, we obtain a negative answer to a question of Harutyunyan, Le, Thomassé, and Wu concerning the analogous statement where the graph $G$ is replaced by another tournament. Time permitting, we will also discuss the setting in which chromatic number is replaced by degeneracy, where a quite different behaviour is exhibited.

This is joint work with António Girão, Kevin Hendrey, Freddie Illingworth, Florian Lehner, Lukas Michel, and Raphael Steiner.

Meeting link:

\[

\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}

\]

#### Zahlentheoretisches Kolloquium

**Title:**The number of prime factors of integers - classical results and recent variations of the classical theme

**Speaker:**Dr. Krishnaswami Alladi (University of Florida)

**Date:**15.03.2024, 15:00 Uhr

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG

**Abstract:**

Although prime numbers have been studied

since Greek antiquity, the first systematic study of the

number of prime factors is due to Hardy and Ramanujan

in 1917. Subsequently, with the work of Paul Turan, Paul

Erdos, and Marc Kac, the subject of Probabilistic Number

Theory was born around 1940 and has blossomed

considerably since then. We shall discuss classical

work on the number of prime factors of integers and more

recent results of ours with restrictions on the size of the

prime factors, that lead to very interesting variations of

the classical theme. My most recent results are jointly

with my former PhD student Todd Molnar. A variety of

techniques come into play - the classical Perron integral

approach, an analytic method of Atle Selberg, the behavior

of solutions to certain difference - differential equations,

and the Buchstab-de Bruijn iteration. The talk will be

accessible to non-experts.

#### Vortrag im Rahmen des Seminars Operator Theory

**Title:**An anti-maximum principle for the Dirichlet-to-Neumann operator

**Speaker:**Sahiba Arora (University of Twente, Netherlands)

**Date:**14.3.2024, 12:15 Uhr

**Room:**TU Graz, Steyrergasse 30, Seminarraum AE02 (STEG006), EG

**Abstract:**

Abstract:

Extensive literature has been devoted to study the operators for which the (anti-)maximum principle holds. Inspired by ideas from the recent theory of eventually positive $C_0$-semigroups, we characterise when the Dirichlet-to-Neumann operator satisfies an anti-maximum principle.

To be precise, let $\Omega \subset \mathbb{R}^d$ be a bounded domain with $C^\infty$-boundary and let $A$ be the Dirichlet-to-Neumann operator on $L^2(\partial \Omega)$. We consider the equation

\begin{equation*}

(\lambda − A)u = f

\end{equation*}

for real numbers $\lambda$ in the resolvent set of $A$. We find those $d$ for which $f \geq 0$ implies $u \leq 0$ for $\lambda$ in a ($f$-dependent) left neighbourhood of the spectral bound.

This is joint work with Jochen Gl\"uck.

#### Zahlentheoretisches Kolloquium

**Title:**Graphs and limits of Riemann surfaces

**Speaker:**Dr. Noema Nicolussi (TU Graz)

**Date:**08.03.2024, 14:00 Uhr

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG

**Abstract:**

In the last decades, there has been an immense interest in the behavior of

analytic objects, (e.g., invariants, measures, Green functions or Laplacian operators) on degenerating Riemann surfaces, that is, Riemann surfaces undergoing a degeneration to a singular Riemann surface. Due to their connection to geometry, the canonical measure and Arakelov Green function have been intensively studied in this context. Recently, it has become clear that their limits under degeneration are closely related to analogous analytic objects on graphs.

In this talk, we discuss recent results on degenerations of the canonical measure and related objects. A new geometric object - called hybrid curve - which mixes graphs and Riemann surfaces, plays an important role in our approach.

Based on joint work with O. Amini (Ecole Polytechnique).

#### Combinatorics Seminar

**Title:**Ramsey problems for monotone (powers of) paths in graphs and hypergraphs

**Speaker:**Lior Gishboliner (ETH Zurich)

**Date:**Friday 8th March 12:30

**Room:**Online meeting (Webex)

**Abstract:**

The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdos and Szekeres in the early days of Ramsey theory. We obtain several results in this area, establishing two conjectures of Mubayi and Suk. This talk is based on a joint work with Zhihan Jin and Benny Sudakov.

Meeting link:

\[

\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}

\]

#### Vortrag im Rahmen des Seminars Operator Theory

**Title:**Local energy decay and low frequency asymptotics for the Schrödinger equation

**Speaker:**Julien Royer (Universite Toulouse, France)

**Date:**7.3.2024, 13:00 Uhr

**Room:**TU Graz, Steyrergasse 30, Seminarraum AE02 (STEG006), EG

**Abstract:**

Abstract:

We are interested in the local energy decay for the Schrödinger equation in an asymptotically Euclidean setting. For this, we study in particular the behavior of the corresponding resolvent for low frequencies. We will see how to use some ideas coming from the analysis of the damped wave equation to get the asymptotic profile for the resolvent, and then for the large time behavior of the solution.

#### Vortrag im Rahmen des Seminars Operator Theory

**Title:**The Maslov index in spectral theory: an overview

**Speaker:**Selim Sukhtaiev (Auburn University, USA)

**Date:**7.3.2024, 12:15 Uhr

**Room:**TU Graz, Steyrergasse 30, Seminarraum AE02 (STEG006), EG

**Abstract:**

Abstract:

This talk is centered around a symplectic approach to eigenvalue problems for systems of ordinary differential operators (e.g., Sturm-Liouville operators,

canonical systems, and quantum graphs), multidimensional elliptic operators on bounded domains, and abstract self-adjoint extensions of symmetric operators in Hilbert spaces. The symplectic view naturally relates spectral counts for self-adjoint problems to the topological invariant called the Maslov index. In this talk, the notion of the Malsov index will be introduced in analytic terms and an overview of recent results on its role in spectral theory will be given.

#### Combinatorics Seminar

**Title:**Semirestricted Rock, Paper, Scissors

**Speaker:**Svante Janson (Uppsala University)

**Date:**Friday 1st March 12:30

**Room:**AE06, Steyrergasse 30

**Abstract:**

A semirestricted variant of the well-known game Rock, Paper, Scissors was recently studied by Spiro et al (Electronic J. Comb. 30 (2023), #P4.32). They assume that two players $R$ (restricted) and $N$ (normal) agree to play $3n$ rounds, where $R$ is restricted to use each of the three choices exactly $n$ times each, while $N$ can choose freely. Obviously, this gives an advantage to $N$. How large is the advantage?

The main result of Spiro et al is that the optimal strategy for $R$ is the greedy strategy, playing each round as if it were the last. (I will not give the proof, and I cannot improve on this.) They also show that with optimal play, the expected net score of $N$ is $\Theta(\sqrt{n})$. In the talk, I will show that with optimal play, the game can be regarded as a twice stopped random walk, and I will show that the expected score is asymptotic to $c \sqrt{n}$, where $c = 3\sqrt{3}/(2\sqrt{\pi}})$.

**Title:**Graz-ISTA Number Theory Days

**Speaker:**()

**Date:**1.2.2024, 13:30

**Room:**SR AE02

**Abstract:**

13:30 - 14:30: Ingrid Vukusic (Universität Salzburg)}

Consecutive triples of multiplicatively dependent integers

15:00 - 16:00: Victor Wang (ISTA)}

Sums of three cubes over a function field

16:15 - 17:15: Pierre-Yves Bienvenu (TU Wien)}

Intersectivity with respect to sparse sets of integers

Seminar webpage for further information:

https://sites.google.com/view/gntd/start

#### Combinatorics Seminar

**Title:**Almost partitioning every $2$-edge-coloured complete $k$-graph into~$k$ monochromatic tight cycles

**Speaker:**Vincent Pfenninger (Graz University of Technology)

**Date:**Friday 26th January 12:30

**Room:**AE06, Steyrergasse 30

**Abstract:**

A $k$-uniform tight cycle} is a $k$-graph ($k$-uniform hypergraph) with a cyclic order of its vertices such that every~$k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on~$n$ vertices contains~$k$ vertex-disjoint monochromatic tight cycles that together cover $n - o(n)$ vertices.

This talk is based on joint work with Allan Lo.

#### Vortrag im Rahmen des Seminars Operator Theory

**Title:**Pointwise eigenvector estimates by landscape functions

**Speaker:**Delio Mugnolo (Fernuniversität Hagen)

**Date:**Donnerstag, 25.1.2024, 12:15 Uhr

**Room:**TU Graz, Steyrergasse 30, Seminarraum AE02 (STEG006), EG

**Abstract:**

Abstract:

Landscape functions are a popular tool used to provide upper bounds for eigenvectors of Schrödinger operators on domains. After reviewing some results obtained in the last 10 years, I will show how several approaches used to achieve such bounds can be unified and extended to a large class of linear and even nonlinear operators. We also use landscape functions to derive lower estimates on the principal eigenvalue - much in the spirit of earlier results by Donsker–Varadhan and Banuelos–Carrol. Our methods solely rely on order properties of operators, which I will briefly remind. As an application, I will show how to derive hitherto unknown lower bounds on the principal eigvenalue of nonlinear Laplacian-type operators on graphs.

#### Combinatorics Seminar

**Title:**Counting graphic sequences via integrated random walks

**Speaker:**Paul Balister (University of Oxford)

**Date:**Friday 12th January 12:30

**Room:**Online meeting (Webex)

**Abstract:**

Given an integer $n$, let $G(n)$ be the number of integer sequences $n-1\ge d_1\ge d_2\ge \dots \ge d_n \ge 0$ that are the degree sequence of some graph. We show that $G(n)=(c+o(1))4^n/n^{3/4}$ for some constant $c>0$, improving both the previously best upper and lower bounds by a factor of $n^{1/4+o(1)}$. The proof relies on a translation of the problem into one concerning integrated random walks.

Joint work with Serte Donderwinkel, Carla Groenland, Tom Johnston and Alex Scott.

Meeting link:

\[

\text{https://tugraz.webex.com/tugraz/j.php?MTID=mab523a645de428d5301998280dc510ed}

\]

#### Geometrie-Seminar

**Title:**The s-weak order and s-Permutahedron

**Speaker:**Viviane Pons (Université Paris-Saclay)

**Date:**Thu Jan 11, 2024, 8:30-9:30

**Room:**Seminarraum 2 Geometrie, Kopernikusgasse 24/IV, TU Graz

**Abstract:**

The weak order on permutations and permutahedron are classical

objects at the intersection of combinatorics and discrete geometry. Their

deep and interesting connections with the associahedron and the Tamari

lattice on binary trees have been an active subject of research. We present

a generalization of the weak order, the s-weak order on so-called

s-decreasing trees and their geometrical counter part the s-Permutahedron

with a possible realization as polytopal complex.