Sensitivity of Mixing Times of the Eulerian Functional Digraphs of the Full Transformation Semigroup T_n

Let $X_n=\left\{1,2,？,n\right\}$ and let $T_n$ denote the full transformation semigroup of all $n^n$ maps $\alpha :X_n\to X_n$ . Each $\alpha \in T_n$ induces a functional digraph ${\Gamma }_{\alpha }$ whose vertex set is $X_n$ and whose arc set is $\left\{\left(x,\alpha \left(x\right)\right):x\in X_n\right\}$ . Because every vertex has out-degree exactly one, ${\Gamma }_{\alpha }$ belongs to the class of functional (mapping) digraphs. The Eulerian members of this class are precisely the functional digraphs induced by permutations $\alpha \in S_n？T_n$ : there are $n!$ Eulerian functional digraphs on $X_n$ , and their isomorphism classes are in bijection with the integer partitions of $n$ . A functional