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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. Itô [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$$(S_{10}; P_ 5)$$.

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