Exhaustive Generation of Pattern-Avoiding s-Words

TL;DR

This paper extends Gray code generation techniques to s-words, introducing bumps as a new operation, and demonstrates their effectiveness in generating pattern-avoiding multiset permutations with applications to s-Stirling words and s-Catalan objects.

Contribution

It generalizes jump operations to bumps for s-words and proves their effectiveness in generating pattern-avoiding languages, leading to new Gray codes and Hamilton paths.

Findings

Efficient algorithms for generating s-Stirling words avoiding pattern 121.

New Gray codes for s-Catalan objects avoiding patterns 132 and 121.

Hamilton paths in s-permutahedra derived from pattern-avoiding s-words.

Abstract

The most well-known Gray code of permutations is plain changes. It was discovered in the 1600s by bell-ringers who wished to order the permutations of [n] by swaps (e.g., 123, 132, 312, 321, 231, 213 for n = 3). In other words, plain changes traces a Hamilton path in the permutohedron. In 2013 it was shown that plain changes can be generated by a greedy algorithm: swap the largest value. Algorithm J replaces the swap operation with the jump operation (which moves a larger digit past one or more smaller digits) and forms the basis of the wildly successful Combinatorial Generation via Permutation Languages series of papers. Here we further generalize this line of research to languages of s-words (i.e., multiset permutations). We generalize jumps to bumps, which moves a sequential run of the same larger digit past one or more smaller digits. Algorithm B greedily applies minimum-length bumps prioritized by largest value, then largest index, then rightward before leftward (e.g. 1122, 1221, 1212, 2112, 2121, 2211 for s = (2, 2)). We show that the algorithm works for s-word languages avoiding a wide variety of tame patterns. Specific applications include efficient algorithm for generating s-Stirling words (which avoid 121) and new Gray codes for various s-Catalan objects (which avoid 132 and 121). The former result leads to Hamilton paths in every s-permutahedron.