A constant-time algorithm for middle levels Gray codes

For any integer $n\geq 1$ a middle levels Gray code is a cyclic listing of all $n$-element and $(n+1)$-element subsets of $\{1,2,\ldots,2n+1\}$ such that any two consecutive subsets differ in adding or removing a single element. The question whether such a Gray code exists for any $n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T. M\"utze and J. Nummenpalo. An efficient algorithm for computing a middle levels Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence proof was turned into an algorithm that computes each new set in the Gray code in time $\mathcal{O}(n)$ on average. In this work we present an algorithm for computing a middle levels Gray code in optimal time and space: each new set is generated in time $\mathcal{O}(1)$ on average, and the required space is $\mathcal{O}(n)$

Hilbert C*-modules and related subjects - a guided reference overview I

The overview contains 450 references of books, chapters of monographs, papers, preprints and Ph.~D.~thesises which are concerned with the theory and/or various applications of Hilbert C*-modules. To show a way through this amount of literature a four pages guide is added clustering sources around major research problems and research fields, and giving information on the historical background. Two smaller separate parts list references treating Hilbert modules over Hilbert*-algebras and Hilbert modules over (non-self-adjoint) operator algebras. Any additions, corrections and forthcoming information are welcome.Comment: LaTeX 2.09, 23 page

Combinatorics of the Permutahedra, Associahedra, and Friends

I present an overview of the research I have conducted for the past ten years in algebraic, bijective, enumerative, and geometric combinatorics. The two main objects I have studied are the permutahedron and the associahedron as well as the two partial orders they are related to: the weak order on permutations and the Tamari lattice. This document contains a general introduction (Chapters 1 and 2) on those objects which requires very little previous knowledge and should be accessible to non-specialist such as master students. Chapters 3 to 8 present the research I have conducted and its general context. You will find: * a presentation of the current knowledge on Tamari interval and a precise description of the family of Tamari interval-posets which I have introduced along with the rise-contact involution to prove the symmetry of the rises and the contacts in Tamari intervals; * my most recent results concerning q, t-enumeration of Catalan objects and Tamari intervals in relation with triangular partitions; * the descriptions of the integer poset lattice and integer poset Hopf algebra and their relations to well known structures in algebraic combinatorics; * the construction of the permutree lattice, the permutree Hopf algebra and permutreehedron; * the construction of the s-weak order and s-permutahedron along with the s-Tamari lattice and s-associahedron. Chapter 9 is dedicated to the experimental method in combinatorics research especially related to the SageMath software. Chapter 10 describes the outreach efforts I have participated in and some of my approach towards mathematical knowledge and inclusion.Comment: 163 pages, m\'emoire d'Habilitation \`a diriger des Recherche

A constant-time algorithm for middle levels Gray codes

For any integer~$n\geq 1$, a \emph{middle levels Gray code} is a cyclic listing of all $n$-element and $(n+1)$-element subsets of $\{1,2,\ldots,2n+1\}$ such that any two consecutive sets differ in adding or removing a single element. The question whether such a Gray code exists for any~$n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T.~M\"utze. Proof of the middle levels conjecture. \textit{Proc. London Math. Soc.}, 112(4):677--713, 2016]. In a follow-up paper [T.~M\"utze and J.~Nummenpalo. An efficient algorithm for computing a middle levels Gray code. \textit{ACM Trans. Algorithms}, 14(2):29~pp., 2018] this existence proof was turned into an algorithm that computes each new set in the Gray code in time~\cO(n) on average. In this work we present an algorithm for computing a middle levels Gray code in optimal time and space: each new set is generated in time~\cO(1), and the required space is~\cO(n)

Quasi-Linear Cellular Automata

Simulating a cellular automaton (CA) for t time-steps into the future requires t^2 serial computation steps or t parallel ones. However, certain CAs based on an Abelian group, such as addition mod 2, are termed ``linear'' because they obey a principle of superposition. This allows them to be predicted efficiently, in serial time O(t) or O(log t) in parallel. In this paper, we generalize this by looking at CAs with a variety of algebraic structures, including quasigroups, non-Abelian groups, Steiner systems, and others. We show that in many cases, an efficient algorithm exists even though these CAs are not linear in the previous sense; we term them ``quasilinear.'' We find examples which can be predicted in serial time proportional to t, t log t, t log^2 t, and t^a for a < 2, and parallel time log t, log t log log t and log^2 t. We also discuss what algebraic properties are required or implied by the existence of scaling relations and principles of superposition, and exhibit several novel ``vector-valued'' CAs.Comment: 41 pages with figures, To appear in Physica