Twenty-Five Comparators is Optimal when Sorting Nine Inputs (and Twenty-Nine for Ten)

This paper describes a computer-assisted non-existence proof of nine-input sorting networks consisting of 24 comparators, hence showing that the 25-comparator sorting network found by Floyd in 1964 is optimal. As a corollary, we obtain that the 29-comparator network found by Waksman in 1969 is optimal when sorting ten inputs. This closes the two smallest open instances of the optimal size sorting network problem, which have been open since the results of Floyd and Knuth from 1966 proving optimality for sorting networks of up to eight inputs. The proof involves a combination of two methodologies: one based on exploiting the abundance of symmetries in sorting networks, and the other, based on an encoding of the problem to that of satisfiability of propositional logic. We illustrate that, while each of these can single handed solve smaller instances of the problem, it is their combination which leads to an efficient solution for nine inputs.Comment: 18 page

Lattice paths of slope 2/5

We analyze some enumerative and asymptotic properties of Dyck paths under a line of slope 2/5.This answers to Knuth's problem \\#4 from his "Flajolet lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in June 2014.Our approach relies on the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths. A key ingredient in the proof is the generalization of an old trick of Knuth himself (for enumerating permutations sortable by a stack),promoted by Flajolet and others as the "kernel method". All the corresponding generating functions are algebraic,and they offer some new combinatorial identities, which can be also tackled in the A=B spirit of Wilf--Zeilberger--Petkov{\v s}ek.We show how to obtain similar results for other slopes than 2/5, an interesting case being e.g. Dyck paths below the slope 2/3, which corresponds to the so called Duchon's club model.Comment: Robert Sedgewick and Mark Daniel Ward. Analytic Algorithmics and Combinatorics (ANALCO)2015, Jan 2015, San Diego, United States. SIAM, 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), eISBN 978-1-61197-376-1, pp.105-113, 2015, 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO

A solution to one of Knuth's permutation problems

We answer a problem posed recently by Knuth: an n-dimensional box, with edges lying on the positive coordinate axes and generic edge lengths W_1 < W_2 < ... < W_n, is dissected into n! pieces along the planes x_i = x_j. We describe which pieces have the same volume, and show that there are C_n distinct volumes, where C_n denotes the nth Catalan number.Comment: 4 pages, 2 figures

Prospective K-8 Teachers’ Knowledge of Relational Thinking

The goal of this study was to examine two issues: First, pre-service teachers’ ability and inclination to think relationally prior to instruction about the role relational thinking plays in the K-8 mathematics curriculum. Second, to examine task specific variables possibly associated with pre-service teachers’ inclination to engage in relational thinking. The results revealed that preservice teachers engage in relational thinking about equality, however, their inclination to do so is rather limited. Furthermore, they tend to engage in relational thinking more frequently in the context of arithmetic than algebra-related tasks. Pre-service teachers’ inclination to engage in relational thinking appeared to also relate to the overall task complexity and the use of variables. Implications of these findings for pre-service teacher education are provided