The Weighted Hook Length Formula III: Shifted Tableaux

The Weighted Hook Length Formula III: Shifted Tableaux

Recently, a simple proof of the hook length formula was given via the branching rule. In this paper, we extend the results to shifted tableaux. We give a bijective proof of the branching rule for the hook lengths for shifted tableaux; present variants of this rule, including weighted versions; and make the first tentative steps toward a bijective proof of the hook length formula for d-complete ...

The weighted hook length formula

Abstract. Based on the ideas in [CKP], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk proof of Green-Nijenhuis-Wi...

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 len...

Swiching Rule on the Shifted Rim Hook Tableaux

When the Schur function sλ corresponding to a partition λ is defined as the generating function of the column strict tableaux of shape λ it is not at all obvious that sλ is symmetric. In [BK] Bender and Knuth showed that sλ is symmetric by describing a switching rule for column strict tableaux, which is essentially equivalent to the jeu de taquin of Schützenberger (see [Sü]). Bender and Knuth’s...

Hook Length Formula and Geometric Combinatorics