Lemma 3 ([Young's rule: support and diagonal multiplicity]). Let θ=(θ_1,…,θ_ℓ) n . The row-tabloid permutation module has the equivalent induced realization M^θ= _ S_ θ_1 ×⋯×S_ θ_ℓ ^ S_n 1. Then S^τ occurs in M^θ if and only if τθ , and …

The case n=0 is immediate from the trivial representation of S_0 . Assume henceforth that n>0 . Transitivity of induction and Lemma 2 ([Pieri rule]) show that the multiplicity of S^τ counts chains ∅=ν^ (0) ⊆ν^ (1) ⊆⋯ ⊆ν^ (ℓ) =τ, |ν^ (k) /ν^ (k-1) |=θ_k, whose successive differences are horizontal strips. Each strip raises the first column's height by at most one, so ν^ (k) has at most k rows. Since it is contained in τ , ∑_ i=1 ^kθ_i=|ν^ (k) | ≤∑_ i=1 ^kτ_i. For k>ℓ , both initial sums equal n , since τ has at most ℓ rows. This proves necessity of dominance. If τ=θ , equality forces ν^ (k) to be exactly the first k rows of θ . These diagrams do form a chain of the required kind, proving multiplicity one. For sufficiency, suppose τθ and induct on ℓ . For ℓ=1 , necessarily τ=(n) . For ℓ>1 , put t=θ_ℓ . There are at least t columns of τ , since τ_1≥θ_1≥t . Delete a bottom box from each of the t tallest columns, choosing the rightmost columns first among equal heights, and call the remaining diagram ν . Its column heights are still nonincreasing: unequal adjacent heights differ by at least one, and within any group of equal heights precisely a suffix is lowered. Thus ν is a partition and τ/ν is a horizontal t -strip. For every k≥1 its loss in the first k rows is ∑_ i=1 ^kτ_i-∑_ i=1 ^kν_i =(0,t-τ_ k+1 ): there are τ_ k+1 columns taller than k , and the rule selects these before any column whose bottom box belongs to the first k rows. If the loss is zero, the desired dominance inequality for ν follows immediately. If it is positive and k<ℓ , then ∑_ i=1 ^kν_i =∑_ i=1 ^ k+1 τ_i-t ≥∑_ i=1 ^ k+1 θ_i-t ≥∑_ i=1 ^kθ_i. For completeness, dominance at ℓ gives (τ)≤ℓ , and dominance at ℓ-1 gives τ_ℓ≤t . All τ_ℓ columns of height ℓ are therefore selected, so the last row disappears and (ν)≤ℓ-1 . Hence all remaining initial sums equal the total n-t , and ν(θ_1,…,θ_ ℓ-1 ) . Induction gives a chain ending at ν ; appending the deleted strip gives ending at τ . Finally, the stabilizer of a word with prescribed letter multiplicities is the product of the symmetric groups on its equal-letter position sets. Reordering those multiplicities conjugates this subgroup, giving the asserted isomorphism of permutation modules.