Mathematics

We prove that a tournament and its complement contain the same number of oriented Hamiltonian paths (resp. cycles) of any given type, as a generalization of Rosenfeld's result proved for antidirected paths.

Poset games are a class of combinatorial game that remain unsolved. Soltys and Wilson proved that computing wining strategies is in \textbf{PSPACE} and aside from special cases such as Nim and N-Free games, \textbf{P} time algorithms for finding ideal play are unknown. This paper presents methods calculate the nimber of posets games allowing for the classification of winning or losing positions. The results present an equivalence of ideal strategies on posets that are seemingly unrelated.

Aldous-Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph G, but it is more general: it states that given a reversible M Markov chain on G started at r, the tree rooted at r formed by the steps of successive first entrance in each node (different from the root) has a probability proportional to ??e=(e1,e2)?�Edges(t,r) M e1,e2 , where the edges are directed toward r. As stated, it allows to sample many distributions on the set of spanning trees. In this paper we extend Aldous-Broder theorem by dropping the reversibility condition on M. We highlight that the general statement we prove is not the same as the original one (but it coincides in the reversible case with that of Aldous and Broder). We prove this extension in two ways: an adaptation of the classical argument, which is purely probabilistic, and a new proof based on combinatorial arguments. On the way we introduce a new combinatorial object that we call the golf sequences.

Let G be any group and k?? be an integer number. The ordered configuration set of k points in G is given by the subset F(G,k)={( g 1 ,?? g k )?�G??�×G: g i ??g j for i?�j}??G k . In this work, we will study the configuration set F(G,k) in algebraic terms as a subset of the product G k =G??�×G . As we will see, we develop practical tools for dealing with the configuration set of k points in G , which, to our knowledge, can not be found in literature.

We prove various results on the size and structure of subsets of vector spaces over finite fields which, in some sense, have too many mutually orthogonal pairs of vectors. In particular, we obtain sharp finite field variants of a theorem of Rosenfeld and an almost version of a theorem of Berlekamp.

Motivated by the definition of super Teichmüller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super Teichmüller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super λ -lengths associated to arcs in a bordered surface. In the special case of a disk, we are able to give combinatorial expansion formulas for the super λ -lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's T -path formulas for type A cluster algebras. We further connect our formulas to the super-friezes of Morier-Genoud, Ovsienko, and Tabachnikov, and obtain partial progress towards defining super cluster algebras of type A n . In particular, following Penner-Zeitlin, we are able to get formulas (up to signs) for the μ -invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra.

We prove that the partition rank and the analytic rank of tensors are equivalent up to a constant, over any large enough finite field (independently of the number of variables). The proof constructs rational maps computing a partition rank decomposition for successive derivatives of the tensor, on a carefully chosen subset of the kernel variety associated with the tensor. Proving the equivalence between these two quantities is the main question in the "bias implies low rank" line of work in higher-order Fourier analysis, and was reiterated by multiple authors.

Genetic programming is the practice of evolving formulas using crossover and mutation of genes representing functional operations. Motivated by genetic evolution we develop and solve two combinatorial games, and we demonstrate some advantages and pitfalls of using genetic programming to investigate Grundy values. We conclude by investigating a combinatorial game whose ruleset and starting positions are inspired by genetic structures.

A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. The problem of finding edge-disjoint Hamiltonian cycles in a given regular graph has many applications in combinatorial optimization and operations research. Our motivation for this problem comes from the field of polyhedral combinatorics, as a sufficient condition for vertex nonadjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph with one forbidden decomposition. In our approach, the algorithm starts by solving the relaxed 2-matching problem, then iteratively generates subtour elimination constraints for all subtours in the solution and solves the corresponding ILP-model to optimality. The procedure is enhanced by the local search heuristic based on chain edge fixing and cycle merging operations. In the computational experiments, the iterative ILP algorithm showed comparable results with the previously known heuristics on undirected multigraphs and significantly better performance on directed multigraphs.

In this paper, we discuss a few recent conjectures made by George Beck related to the ranks and cranks of partitions. The conjectures for the rank of a partition were proved by Andrews by using results due to Atkin and Swinnerton-Dyer on a suitable generating function, while the conjectures related to cranks were studied by Shane Chern using weighted partition moments. We revisit the conjectures on the crank of a partition by decomposing the relevant generating function and further explore connections with Apple-Lerch series and tenth order mock theta functions.