Regina is a software package for studying 3-manifold triangulations and normal surfaces. It includes a graphical user interface and Python bindings, and also supports angle structures, census enumeration, combinatorial recognition of triangulations, and high-level functions such as 3-sphere recognition, unknot recognition and connected sum decomposition. This paper brings 3-manifold topologists...

We present an implementation of Redelemeier’s algorithm for the enumeration of lattice animals in high-dimensional lattices. The implementation is lean and fast enough to allow us to extend the existing tables of animal counts, perimeter polynomials and series expansion coefficients in d-dimensional hypercubic lattices for 3 ≤ d ≤ 10. From the data we compute formulae for perimeter polynomials ...

The purpose of this paper is to count the number of isomorphism types of reduced Witt rings of nondegenerate symmetric bilinear forms over fields with a fixed number of orderings. All of the background material necessary for understanding the situation in which this counting problem arises will be described in Section 1. Using results from [2], this problem is restated in Section 2 in terms of ...

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences or generating functions of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane...

A seminal technique of theoretical physics calledWick’s theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogous to those obtained by combinatorial methods. In this paper we show that the enume...

We discuss the enumeration theory for flags in Eulerian partially ordered sets, emphasizing the two main geometric and algebraic examples, face posets of convex polytopes and regular CW -spheres, and Bruhat intervals in Coxeter groups. We review the two algebraic approaches to flag enumeration – one essentially as a quotient of the algebra of noncommutative symmetric functions and the other as ...

The enumeration of self-orthogonal Latin squares (SOLS) of a given order seems to be an open problem in the literature on combinatorial designs. The existence of at least one SOLS is guaranteed for any order except 2, 3 and 6, but it is not known how many of these squares of a given order exist. In this talk we present enumeration tables of unequal SOLS, idempotent SOLS, isomorphism classes of ...

We describe a system that integrates the SMT solver Z3 with the Scala programming language. The system supports the use of the SMT solver for checking satisfiability, unsatisfiability, as well as solution enumeration. The embedding of formula trees into Scala uses the host type system of Scala to prevent the construction of certain ill-typed constraints. The solution enumeration feature integra...

We consider the problem of minimizing a fixed-degree polynomial over the standard simplex. This problem is well known to be NP-hard, since it contains the maximum stable set problem in combinatorial optimization as a special case. In this paper, we revisit a known upper bound obtained by taking the minimum value on a regular grid, and a known lower bound based on Pólya’s representation theorem....

Symmetry forms the foundation of combinatorial theories and algorithms enumeration such as Möbius inversion, Euler totient functions, celebrated Pólya’s theory under symmetric group action. As machine learning artificial intelligence techniques play increasingly important roles in perception music to image processing that are central many disciplines, combinatorics, graph theory, symmetry act p...