Characterization of the Group Association Scheme of the Symmetric Group

Letnbe a non-zero positive integer and Λ(n) the set of all partitions ofn. There is a one-to-one correspondence between Λ(n) and the set of the conjugacy classes ofSn, the symmetric group of degreen. Let X(Sn) = (Sn, {R*λ}λ∈Λ(n)) be the group association scheme ofSnand X = (X, {Rλ}λ∈Λ(n)) be an association scheme having intersection numbers identical to those of X(Sn). Suppose there exists no set of four vertices {x1,x2,x3,x4} withx1,x2,x3,x4 ∈ Xsatisfying (x1,x2), (x2,x3), (x3,x4), (x4,x1) ∈ R(2), (x1,x3) ∈ R(3)and (x2,x4) ∈ R(2,2). Then X is shown to be isomorphic to X (Sn). (In [17], the authors show that ifn ≥ 5, X does not possess four vertices of this type.)