An explicit universal cycle for the (n-1)-permutations of an n-set

We show how to construct an explicit Hamilton cycle in the directed Cayley graph Cay({\sigma_n, sigma_{n-1}} : \mathbb{S}_n), where \sigma_k = (1 2 >... k). The existence of such cycles was shown by Jackson (Discrete Mathematics, 149 (1996) 123-129) but the proof only shows that a certain directed graph is Eulerian, and Knuth (Volume 4 Fascicle 2, Generating All Tuples and Permutations (2005)) asks for an explicit construction. We show that a simple recursion describes our Hamilton cycle and that the cycle can be generated by an iterative algorithm that uses O(n) space. Moreover, the algorithm produces each successive edge of the cycle in constant time; such algorithms are said to be loopless

Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking

We propose a new camera-based method of robot identification, tracking and orientation estimation. The system utilises coloured lights mounted in a circle around each robot to create unique colour sequences that are observed by a camera. The number of robots that can be uniquely identified is limited by the number of colours available, $q$, the number of lights on each robot, $k$, and the number of consecutive lights the camera can see, $\ell$. For a given set of parameters, we would like to maximise the number of robots that we can use. We model this as a combinatorial problem and show that it is equivalent to finding the maximum number of disjoint $k$-cycles in the de Bruijn graph $\text{dB}(q,\ell)$. We provide several existence results that give the maximum number of cycles in $\text{dB}(q,\ell)$ in various cases. For example, we give an optimal solution when $k=q^{\ell-1}$. Another construction yields many cycles in larger de Bruijn graphs using cycles from smaller de Bruijn graphs: if $\text{dB}(q,\ell)$ can be partitioned into $k$-cycles, then $\text{dB}(q,\ell)$ can be partitioned into $tk$-cycles for any divisor $t$ of $k$. The methods used are based on finite field algebra and the combinatorics of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied Mathematic

Middle School Students’ Conceptual Understanding of Equations: Evidence from Writing Story Problems

This study investigated middle school students’ conceptual understanding of algebraic equations. 257 sixth- and seventh-grade students solved algebraic equations and generated story problems to correspond with given equations. Aspects of the equations’ structures, including number of operations and position of the unknown, influenced students’ performance on both tasks. On the story-writing task, students’ performance on two-operator equations was poorer than would be expected on the basis of their performance on one-operator equations. Students made a wide variety of errors on the story-writing task, including (1) generating story contexts that reflect operations different from the operations in the given equations, (2) failing to provide a story context for some element of the given equations, (3) failing to include mathematical content from the given equations in their stories, and (4) including mathematical content in their stories that was not present in the given equations. The nature of students’ story-writing errors suggests two main gaps in students’ conceptual understanding. First, students lacked a robust understanding of the connection between the operation of multiplication and its symbolic representation. Second, students demonstrated difficulty combining multiple mathematical operations into coherent stories. The findings highlight the importance of fostering connections between symbols and their referents