Unitary Toric Classes, the Reality and Desire Diagram, and Sorting by Transpositions

• Authors: Rodrigo de A. Hausen;Luerbio Faria;Celina M. H. de Figueiredo;Luis Antonio B. Kowada
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2010

H. Eriksson et al. made a breakthrough to the problem of sorting by transpositions by proposing a quotient structure named toric graph, which allowed the reduction of the search space, establishing the transposition diameter $D_t(n)=\lfloor\frac{n+1}{2}\rfloor+1$, for the cases $n=13$ and $n=15$, and invalidating a conjecture by J. Meidanis, M. E. M. T. Walter, and Z. Dias that the transposition diameter would be equal to the transposition distance of the reverse permutation $\lfloor n/2\rfloor+1$. I. Elias and T. Hartman extended the lower bound $D_t(n)\geq\lfloor\frac{n+1}{2}\rfloor+1$, to all odd values of $n$, $n\geq13$. The value $n=15$ is the largest for which $D_t(n)$ is known. The goal of the present paper is to further study the toric graph, focusing on the case when $n+1$ is prime, providing positive evidence that J. Meidanis, M. E. M. T. Walter, and Z. Dias's conjecture is still valid when $n$ is even. We show that, when $n+1$ is prime, the properties of the reverse permutation are shared by permutations that fall into unitary toric classes; we prove that their reality and desire diagrams have just one cycle, consequently proving that those permutations are separated by at least $n/2$ transpositions among themselves, and we show that there are at least two permutations whose transposition distance is $n/2$ and two permutations, other than the reverse, whose distance is at least $n/2+1$, with respect to the identity.