Abstract :

: Young's lattice is the graph whose vertices are Integer Partitions. Two partitions are connected by an edge if they differ in one part, and the difference in that part is equal to one. An important statistic of an Integer Partition is the so-called f-statistic, which is given by the celebrated hook-length formula of Frame, Robinson and Thrall. This statistic can be interpreted as the dimension of a representation of a symmetric group. Macdonald has characterized integer partitions with odd f-statistic. In this talk, we will explain Macdonald's characterization of partitions with f-statistic. We will also show that the vertex induced subgraph of Young's lattice consisting of partitions with odd f-statistic is a binary tree. The results and their proofs are purely combinatorial, and no background in representation theory is expected.