Summary: In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach 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 J. M. Lucas et al. [J. Algorithms 15, No. 3, 343–366 (1993; Zbl 0782.68090)]; the Gray code for generating all partitions of an \(n\)-element ground set by element exchanges due to R. Kaye [Inf. Process. Lett. 5, 171–173 (1976; Zbl 0357.94010)].
We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, boxed patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We also obtain new Gray codes for all the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into \(n\) rectangles subject to certain restrictions.
The second main application of our framework are lattice congruences of the weak order on the symmetric group \(S_n\). Recently, V. Pilaud and F. Santos [Bull. Lond. Math. Soc. 51, No. 3, 406–420 (2019; Zbl 1420.52015)] realized all those lattice congruences as \((n-1)\)-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of 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. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope.