RSK-Complete Cycle Decompositions

TL;DR

This paper characterizes cycle decompositions that can produce all Young tableau shapes via RSK, identifying specific classes of permutations that are RSK-complete depending on the parity of n.

Contribution

It provides a complete characterization of RSK-complete cycle decompositions, distinguishing between even and odd n cases and identifying unique classes of permutations.

Findings

Cyclic permutations are the only RSK-complete fixed cycle decompositions for even n.

For odd n, cyclic and almost cyclic permutations are RSK-complete.

The paper establishes a clear criterion for RSK-completeness based on permutation cycle structure.

Abstract

We characterize the class of cycle decompositions that can achieve all Young tableau shapes (except the trivial ones with a single row or a single column) under the Robinson--Schensted--Knuth (RSK) correspondence, a property that we call RSK-completeness. We prove that for even $n$, cyclic permutations comprise the only fixed cycle decomposition that is RSK-complete. For odd $n$, cyclic permutations and almost cyclic permutations which have a cycle of length $n-1$ are the only RSK-complete cycle decompositions.