Transpositional sequences and multigraphs

Abstract

When ${\bf t} := \langle t_1,t_2,\ldots,t_k\rangle$ is a sequence of transpositions on the finite set $n:=\{0,1,\ldots,n-1\}$, then $\bigcirc{\bf t}:= t_1\circ t_2\circ\cdots\circ t_k$ denotes the compositional product of the sequence. Our paper treats the set ${\rm Prod}({\bf t})$ of all $\bigcirc{\bf s}$, where ${\bf s}$ is a sequence obtained by rearranging the terms of ${\bf t}$. The paper characterizes the set of all transpositional sequences ${\bf t}$ for which ${\rm Prod}({\bf t})$ is the subset of a single congugacy class in the symmetric group ${\rm Sym}(n)$; we call such ${\bf t}$ {\it conjugacy invariant}. At the opposite extreme, the paper studies conditions under which ${\bf t}$ is {\it permutationally complete}, which is to say, those ${\bf t}$ for which either ${\rm Prod}({\bf t}) = {\rm Alt}(n)$ or ${\rm Prod}({\bf t}) = {\rm Sym}(n)\setminus{\rm Alt}(n)$.