COUNTING PATHS IN YOUNG LATTICE

Young's lattice is the lattice of partitions of integers, ordered by inclusion of diagrams. Standard young tableaux can be represented as paths in Young's lattice that go up by one square at each step, and more general paths in Young's lattice correspond to more general kinds of tableaux. Using the theory of symmetric functions, in particular Pieri's rule for multiplying a Schur function by a complete symmetric function, we derive formulas for counting paths in Young's lattice that go up or down by horizontal or vertical strips. Our results are related to Richard Stanley's theory of differential posets in the special case of Young's lattice.