Hooks and Powers of Parts in Partitions

Many textbooks contain material on partitions. Two standard references are [A] and [S]. A partition of a natural integer n with parts λ1, . . . , λk is a finite decreasing sequence λ = (λ1 ≥ λ2 ≥ · · · ≥ λk > 0) of natural integers λ1, . . . , λk > 0 such that n = ∑k i=1 λi. We denote by |λ| the content n of λ. Partitions are also written as sums: n = λ1 + · · ·+ λk and one uses also the (abusive) multiplicative notation λ = 11 · 22 · · ·nn where νi denotes the number of parts equal to i in the partition λ. A partition is graphically represented by its Young diagram obtained by drawing λ1 adjacent boxes of identical size on a first row, followed by λ2 adjacent boxes of identical size on a second row and so on with all first boxes (of different rows) aligned along a common first column. In the sequel we identify a partition with its Young diagram. A hook in a partition is a choice of a box H in the corresponding Young diagram together with all boxes at the right of the same row and all boxes below of the same column. The total number of boxes in a hook is its hooklength, the number of boxes in a hook to the right of H is its armlength and the number of boxes of a hook below H is the leglength. The Figure below displays the Young diagram of the partition (5, 4, 3, 1) of 13 together with a hook of length 4 having armlength 2 and leglength 1. We call the couple (armlength,leglength) of a hook its hooktype and denote it by τ = τ(α, k − 1− α) if its armlength is α and its leglength k− 1−α. Such a hook has hence total length k and there are exactly k different hooktypes for hooks of length k.