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On association schemes of the symmetric group \(S_{2n}\) acting on partitions of type \(2^ n\). (English) Zbl 0817.05080
Let \((G; X)\) be a transitive permutation group on a finite set \(X\) and \(R_ 0,\dots, R_ d\) the 2-orbits of \((G; X)\), i.e. the orbits of the induced action of \(G\) on \(X\times X\). Then the pair \((X, \{R_ i\}^ n_{i= 0})\) is an association scheme (see E. Bannai and T. ItĂ´ [Algebraic combinatorics. I: Association schemes (1984; Zbl 0555.05019)]). This scheme is called 2-orbit scheme of \((G; X)\) and is denoted by 2-orb\((G;X)\). For example, the Johnson scheme \(J(m,n)\) is introduced as a 2-orbit scheme of the symmetric group \(S_ n\) acting on the \(m\)-element subsets of \(\{1,2,\dots,n\}\). The structure of fusion schemes of \(J(m,n)\) was studied by many authors. Since \(J(m,n)\) is a 2- orbit scheme of the primitive permutation representation of \(S_ n\), it is natural to study other primitive representations of these groups. Let \(S_{2n}\) be the symmetric group on the set \(\{1,2,\dots, 2n\}\). Consider the induced action of \(S_{2n}\) on the set \(P_ n\) of all partitions of \(\{1,2,\dots, 2n\}\) into \(n\) equal parts of size 2. In this paper, the author investigates the schemes 2-orb\((S_{2n}; P_ n)\). In particular, he proves that there is no non-trivial fusion scheme of 2- orb\((S_{10}; P_ 5)\).

05E30 Association schemes, strongly regular graphs
20B30 Symmetric groups
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures