Read-back: riffle_mixing_upper

For every natural number $n$ with $n \ge 2$, the following inequality holds:

where the left-hand side is a natural number cast to a real, $\log_2$ is the base-two logarithm of a real number, and the objects are as follows. $R_n$ is a real matrix indexed by the permutations of the $n$-element set $\{0,\dots,n-1\}$, defined as the transpose of the following "inverse riffle" matrix $Q_n$: for permutations $x, y$,

where, for each of the $2^n$ assignments $b$ of a bit to each element, $\sigma_b$ denotes the permutation that stably sorts the bit sequence $q \mapsto b(x(q))$ in increasing order with $\mathrm{false} < \mathrm{true}$ (i.e. $\sigma_b$ lists first all positions carrying bit false, in increasing position order, then all positions carrying bit true, in increasing position order); so $Q_n(x,y)$ is the fraction of bit assignments $b$ for which $y = x \circ \sigma_b$ as permutations. The matrix in the theorem is $R_n(x,y) = Q_n(y,x)$. The quantity $t_{\mathrm{mix}}(R_n)$ is the mixing time of $R_n$ at threshold $\tfrac14$ toward the uniform distribution $u$ on the permutation group (each of the $n!$ permutations having mass $1/n!$): it is the least natural number $t$ (an infimum over naturals, equal to $0$ if no $t$ qualifies) such that

the supremum ranging over all permutations $x$ as starting states, where $R_n^{\,t}$ is the $t$-th matrix power and $d_{TV}(\mu,\nu)$ is defined as the supremum, over all finite subsets $A$ of the state space, of $\left|\sum_{x\in A}\mu(x) - \sum_{x\in A}\nu(x)\right|$ (the sup-over-events form of total-variation distance, with no factor $\tfrac12$; these real suprema default to $0$ when the value set is empty or unbounded above). The statement is a concrete, non-asymptotic upper bound of the form $2\log_2 n + O(1)$, asserted for every individual $n \ge 2$; it gives no lower bound and says nothing about $n \le 1$.