  This paper explores the mixing time of the random transposition walk on
permutations with one-sided interval restrictions. In particular, we're
interested in the notion of cutoff, a phenomenon which occurs when mixing
occurs in a window of order smaller than the mixing time. One of the main tools
of the paper is the diagonalization obtained by Hanlon; the use of the spectral
information is inspired by the famous paper of Diaconis and Shahshahani on the
mixing time of the random transposition walk on the entire symmetric group. The
diagonalization allows us to prove chi-squared cutoff for a broad class of
one-sided restriction matrices. Furthermore, under an extra condition, the walk
also undergoes total variation cutoff. Finally, a large collection of examples
which undergo chi-squared cutoff but in which total variation mixing occurs
substantially earlier and without cutoff is produced. These results resolve a
conjecture of Diaconis and Hanlon from Section 5 of Hanlon's paper.
