Many invertible actions $tau$ on a set ${mathcal{S}}$ of combinatorial objects, along with a natural statistic $f$ on ${mathcal{S}}$, exhibit the following property which we dub textbf{homomesy}: the average of $f$ over each $tau$-orbit in ${mathcal{
S}}$ is the same as the average of $f$ over the whole set ${mathcal{S}}$. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong, Stump, and Thomas proved Panyushevs conjecture in 2011. We describe a theoretical framework for results of this kind that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suters action on certain subposets of Youngs Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains.
We consider a large family of equivalence relations on permutations in Sn that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obta
ined from the other by a sequence of pattern-replacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, subject to the constraint that a third element of a suitable type be in a suitable position. For various instances of the problem, we compute the number of equivalence classes, determine how many n-permutations are equivalent to the identity permutation, or characterise this equivalence class. Although our results feature familiar integer sequences (e.g., Catalan, Fibonacci, and Tribonacci numbers) and special classes of permutations (layered, connected, and 123-avoiding), some of the sequences that arise appear to be new.
