I am not sure I understand your question. What is true is that: starting from the root of the Young lattice, a forward path of n steps corresponds to a non-inc

I am not sure I understand your question. What is true is that: starting from the root of the Young lattice, a forward path of n steps corresponds to a non-increasing partition of n. For instance, in the wiki picture of the lattice, the path up-left-left-right from the root corresponds to the partition 4=3+1. Now, if you are asking whether these paths/partitions are related to actual paths in the vector spaces that are representations of the symmetric groups, no; the connection between these Young diagrams to representations is much more indirect. The wikipedia section https://en.wikipedia.org/wiki/Young_tableau#Applications_in_representation_theory https://en.wikipedia.org/wiki/Young_tableau#Applications_in_... has a decent (maybe) exposition of the connection, but to get the full picture of the connection, you need a dedicated study of this topic.