Single Length Cycle Research Articles

Permutations as minimal powers of a single-cycle class-sum

The structure-constant [( p)] n · [(1) l 1 (2) l 2 ⋯ ( n) l n ] n | [(1) l 1− k 1 (2) l 2 − k 2 ⋯( u) l u +1⋯( n) l n ] n that corresponds to the merging of the p cycles (1) k 1 (2) k 2 ⋯ ( u − 1) k u−1 , (Σ i=1 u−1 k i = p) into a single cycle of length u = Σ i=1 u−1 ik l is derived. The result is used to evaluate the structure constant [( p)] n k | [(1) l 1 ( p) l 2 ⋯(( i−1)( p−1)+1) l i ⋯] n , where k is the minimal power of [( P)] n generating the class-sum [(1) l 1 ( p) l 2 ⋯(( i − 1)( p − 1) + 1) l i ⋯] n (i.e. Σ i=1 n ( i − 1) l i = k).

Cyclic antiautomorphisms of Mendelsohn triple systems

A cyclic triple ( a, b, c) is defined to be set {( a, b),( b, c),( c, a)} of ordered pairs. A Mendelsohn triple system of order v, M(2, 3, v), is a pair ( M, β), where M is a set of v points and β is a collection of cyclic triples of pairwise distinct points of M such that any ordered pair of distinct points of M is contained in precisely one cyclic triple of β. An antiautomorphism of a Mendelsohn triple system ( M, β) is a permutation of M which maps β to β -1, where β -1={( c, b, a); ( a, b, c) ϵβ}. In this paper we give necessary and sufficient conditions for the existence of a Mendelsohn triple system of order v admitting an antiautomorphism consisting of a single cycle of length d and having v - d fixed points. Further, we give a more general result for partial Mendelsohn triple systems in which the missing ordered pairs are precisely those containing two fixed points.