Dilworth's theorem partitions posets into the so-called chains and states that a poset of width $k$ requires only $k$ disjoint chains to decompose. It is an existential statement but constructive approaches exist.
QUESTION: If we are restricted to nice posets (posets with a lot of structure), namely finite Young's lattices $Y_\mu$, is there something more known about the chain structure or properties or anything that would help me construct the chains not on a case-by-case basis? I'm especially interested in all $Y_\mu$, where $\mu=[n,n-1,\dots,1]$.
By playing with small $n$'s I noticed that it is not difficult to create a minimal chain decomposition.