The weighted hook-length formula II: Complementary formulas

Let λ = (λ1, λ2, . . . , λl), λ1 ≥ λ2 ≥ . . . ≥ λl > 0, be a partition of n, λ ⊢ n, and let [λ] = {(i, j) ∈ Z : 1 ≤ i ≤ l, 1 ≤ j ≤ λi} be the corresponding Young diagram. The conjugate partition λ is defined by λj = max{i : λi ≥ j}. We will freely use implications such as i ≤ j ⇒ λi ≥ λj. The hook Hz ⊆ [λ] is the set of squares weakly to the right and below of z = (i, j) ∈ [λ], and the hook length hz = hij = |Hz| = λi + λ ′ j − i − j + 1 is the size of the hook. See Figure 1, left drawing.