In this talk I present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations, which provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an n-element set by adjacent transpositions; the binary reflected Gray code to generate all n-bit strings by flipping a single bit in each step; the Gray code for generating all n-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an n-element ground set by element exchanges due to Kaye. The first main application of our framework are permutation patterns, yielding new Gray codes for different pattern-avoiding permutations, such as vexillary, skew-merged, X-shaped, separable, Baxter and twisted Baxter permutations etc. We also obtain new Gray codes for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into n rectangles subject to different restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group S_n. Recently, Pilaud and Santos realized all those lattice congruences as (n-1)-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian.

This is joint work with Liz Hartung, Hung P. Hoang, and Aaron Williams.