A new proof from the Australian science fiction writer Greg Egan and a 2011 proof anonymously posted online are now being hailed as significant advances on a puzzle mathematicians have been studying ...

Combinatorial structures encompass an array of discrete objects—permutations, set partitions, matchings, trees and lattice paths—whose enumeration and intrinsic properties are often captured by ...

Group theory provides the abstract language for describing symmetry through the concept of a set equipped with an associative binary operation, identity and inverses. Permutation structures arise when ...

After previously discussing permutations, Dr. James McCaffrey of Microsoft Research uses step-by-step examples and full code presentations to explore combinations. A zero-based mathematical (n, k) ...