We give a combinatorial proof of the formula giving the number of representations of an even permutation σ in S n as a product of an n-cycle by an (n − 2)-cycle, such a number being (n − χ(σ ))(n − 3)!, where χ(σ ) is the number of fixed points of σ . This proof relies on the fact that any odd permutation in S n is the product of an n-cycle by an (n − 1)-cycle in exactly 2(n − 2)! different ways.Let σ be a permutation in S n . If σ is odd, then the number of different representations of σ as a product of an n-cycle by an (n − 1)-cycle is 2(n − 2)!. If σ is even, the number of the representations of σ as a product of an n-cycle by an (n −2)-cycle is (n −χ (σ ))(n −3)!, where χ(σ ) is the number of fixed points of σ . There exist several proofs of these two formulas, relying on the theory of characters [2] or not [1]. As regards the first formula, there is also a combinatorial proof by Machì [3], which contains a recursive algorithm to construct all the above representations. In this paper, we give a combinatorial proof of the second formula, assuming the validity of the first one. Since our proof constructively reduces the problem of the representation of an even σ in S n to the problem of the representation of an odd ρ in S n−1 , then if we put together our present result with the combinatorial proof of Machì, we obtain an algorithm to construct all the representations of any even σ as a product of an n-cycle by a (n − 2)-cycle.We adopt the following notations and definitions. We always assume n ≥ 3. Then we let [n] = {1, 2, . . . , n}, S n be the the group of all the permutations on [n], C m be the conjugacy class of all the m-cycles in S n ; for any subset X of [n], we let Sym (X ) be the the subgroup of all the permutations in S n which leave each element of [n] \ X fixed, and C m (X ) be the subset of all the m-cycles in C m which leave each element of [n] \ X fixed. We let σ be an even permutation in S n , and we putFor n ≥ 4, the number |E(σ )| is equal to the number of representations of σ as a product of an n-cycle by an (n − 2)-cycle (for n = 3 see Example 1).Note that if σ = γ δ, with (γ , δ) ∈ C n ×C n−2 , and if y is one of the fixed points of δ, then y is necessarily not fixed by σ . This shows that if σ is the identity, then |E(σ )| = 0. Now assume that σ is not the identity. In this case, we can take some x ∈ [n] \ Fix (σ ), we put y = x σ and we consider the transposition (x y). Then the permutation ρ x = σ (x y) is an odd permutation which leaves x fixed, hence it belongs to Sym ([n]\{x}), which is isomorphic to S n−1 . We shall show that we can recover all the representations σ = γ δ, with (γ , δ) ∈ C n × C n−2 , from all the representations ρ x = cd, with (c, d) ∈ C n−1 ([n] \ {x}) × C n−2 ([n] \ {x}), of all the ρ x s for x ∈ [n] \ Fix (σ ). We put, for any x ∈ [n] \ Fix (σ ), D x = {(c, d) ∈ C n−1 ([n] \ {x}) × C n−2 ([n] \ {x}) | ρ x = cd }.
