Combinatorial generation via permutation languages

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an $n$-element set by adjacent transpositions; the binary reflected Gray code to generate all $n$-bit strings by flipping a single bit in each step; the Gray code for generating all $n$-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an $n$-element ground set by element exchanges due to Kaye. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We thus also obtain new Gray code algorithms for the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into $n$ rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group~$S_n$. Recently, Pilaud and Santos realized all those lattice congruences as $(n-1)$-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope

Efficient generation of elimination trees and graph associahedra

An elimination tree for a connected graph~$G$ is a rooted tree on the vertices of~$G$ obtained by choosing a root~$x$ and recursing on the connected components of~$G-x$ to produce the subtrees of~$x$. Elimination trees appear in many guises in computer science and discrete mathematics, and they encode many interesting combinatorial objects, such as bitstrings, permutations and binary trees. We apply the recent Hartung-Hoang-M\"utze-Williams combinatorial generation framework to elimination trees, and prove that all elimination trees for a chordal graph~$G$ can be generated by tree rotations using a simple greedy algorithm. This yields a short proof for the existence of Hamilton paths on graph associahedra of chordal graphs. Graph associahedra are a general class of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov, whose vertices correspond to elimination trees and whose edges correspond to tree rotations. As special cases of our results, we recover several classical Gray codes for bitstrings, permutations and binary trees, and we obtain a new Gray code for partial permutations. Our algorithm for generating all elimination trees for a chordal graph~$G$ can be implemented in time~\cO(m+n) per generated elimination tree, where $m$ and~$n$ are the number of edges and vertices of~$G$, respectively. If $G$ is a tree, we improve this to a loopless algorithm running in time~\cO(1) per generated elimination tree. We also prove that our algorithm produces a Hamilton cycle on the graph associahedron of~$G$, rather than just Hamilton path, if the graph~$G$ is chordal and 2-connected. Moreover, our algorithm characterizes chordality, i.e., it computes a Hamilton path on the graph associahedron of~$G$ if and only if $G$ is chordal

Combinatorial generation via permutation languages. I. Fundamentals

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an $n$-element set by adjacent transpositions; the binary reflected Gray code to generate all $n$-bit strings by flipping a single bit in each step; the Gray code for generating all $n$-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an $n$-element ground set by element exchanges due to Kaye. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, boxed patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We also obtain new Gray codes for all the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into $n$ rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group~$S_n$. Recently, Pilaud and Santos realized all those lattice congruences as $(n-1)$-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope

All your bases are belong to us : listing all bases of a matroid by greedy exchanges

You provide us with a matroid and an initial base. We say that a subset of the bases "belongs to us" if we can visit each one via a sequence of base exchanges starting from the initial base. It is well-known that "All your base are belong to us". We refine this classic result by showing that it can be done by a simple greedy algorithm. For example, the spanning trees of a graph can be generated by edge exchanges using the following greedy rule: Minimize the larger label of an edge that enters or exits the current spanning tree and which creates a spanning tree that is new (i.e., hasn't been visited already). Amazingly, this works for any graph, for any labeling of its edges, for any initial spanning tree, and regardless of how you choose the edge with the smaller label in each exchange. Furthermore, by maintaining a small amount of information, we can generate each successive spanning tree without storing the previous trees. In general, for any matroid, we can greedily compute a listing of all its bases matroid such that consecutive bases differ by a base exchange. Our base exchange Gray codes apply a prefix-exchange on a prefix-minor of the matroid, and we can generate these orders using "history-free" iterative algorithms. More specifically, we store O(m) bits of data, and use O(m) time per base assuming O(1) time independence and coindependence oracles. Our work generalizes and extends a number of previous results. For example, the bases of the uniform matroid are combinations, and they belong to us using homogeneous transpositions via an Eades-McKay style order. Similarly, the spanning trees of fan graphs belong to us via face pivot Gray codes, which extends recent results of Cameron, Grubb, and Sawada [Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan, COCOON 2021]

Weathering:Ecologies of Exposure

Weathering is atmospheric, geological, temporal, transformative. It implies exposure to the elements and processes of wearing down, disintegration, or accrued patina. Weathering can also denote the ways in which subjects and objects resist and pass through storms and adversity. This volume contemplates weathering across many fields and disciplines; its contributions examine various surfaces, environments, scales, temporalities, and vulnerabilities. What does it mean to weather or withstand? Who or what is able to pass through safely? What is lost or gained in the process

Research on Teaching and Learning In Biology, Chemistry and Physics In ESERA 2013 Conference

This paper provides an overview of the topics in educational research that were published in the ESERA 2013 conference proceedings. The aim of the research was to identify what aspects of the teacher-student-content interaction were investigated frequently and what have been studied rarely. We used the categorization system developed by Kinnunen, Lampiselkä, Malmi and Meisalo (2016) and altogether 184 articles were analyzed. The analysis focused on secondary and tertiary level biology, chemistry, physics, and science education. The results showed that most of the studies focus on either the teacher’s pedagogical actions or on the student - content relationship. All other aspects were studied considerably less. For example, the teachers’ thoughts about the students’ perceptions and attitudes towards the goals and the content, and the teachers’ conceptions of the students’ actions towards achieving the goals were studied only rarely. Discussion about the scope and the coverage of the research in science education in Europe is needed.Peer reviewe

Proceedings of the 10th International Conference on NDE in Relation to Structural Integrity for Nuclear and Pressurized Components

This conference, the tenth in a series on NDE in relation to structural integrity for nuclear and pressurized components, was held from 1st October to 3 October 2013, in Cannes, France. The scientific programme was co-produced by the European Commission’s Joint Research Centre, Institute for Energy and Transport (EC-JRC/IET). The Conference has been coordinated by the Confédération Française pour les Essais Non Destructifs (COFREND). The first conference, under the sole responsibility of EC-JRC was held in Amsterdam, 20-22 October 1998. The second conference was locally organized by the EPRI NDE Center in New Orleans, 24-26 May 2000, the third one by Tecnatom in Seville, 14-16 November 2001, the fourth one by the British Institute of Non-Destructive Testing in London, 6-8 December 2004, the fifth by EPRI in San Diego, 10-12 May 2006, the sixth by Marovisz in Budapest, 8-10 October 2007, the seventh by the University of Tokyo and JAPEIC in Yokohama, the eight by DGZfP, 29 September to 1st October 2010, the ninth by Epri NDE Center, 22-24 May 2012 in Seattle. The theme of this conference series is to provide the link between the information originated by NDE and the use made of this information in assessing structural integrity. In this context, there is often a need to determine NDE performance against structural integrity requirements through a process of qualification or performance demonstration. There is also a need to develop NDE to address shortcomings revealed by such performance demonstration or otherwise. Finally, the links between NDE and structural integrity require strengthening in many areas so that NDE is focussed on the components at greatest risk and provides the precise information required for assessment of integrity. These were the issues addressed by the papers selected for the conference.JRC.F.5-Nuclear Reactor Safety Assessmen