Relating tournaments and permutations with xrays

In a 2005 paper, Bebeacua et al. investigated the xrays of permutations, and conjectured a correspondence between binary xrays and score sequences of tournaments. In 2014, Brualdi and Fritscher conjectured a possible correspondence between score sequences of $2$-tournaments and (not necessarily binary) xrays of permutations. In this paper, we first introduce the concept of a transitive tournament decomposition of $k$-tournaments, then present a construction by which a permutation is used to build $1$- and $2$-tournaments whose score sequences agree with the xray of the permutation in the manner outlined by Bebeacua et al. and Brualdi and Fritscher. We close with an investigation of xrays with restricted terms, including binary xrays, and show that the recent conjectures by Bebeacua et al. and Brualdi and Fritscher are special cases of a more general statement, which we conjecture and for which we provide supporting evidence.