Section 16.1.3 -- adjacent transpositions lower bound

For every natural number $n$ with $n \ge 2$ (this holds for all such $n$, with no asymptotic threshold),

where the right-hand side is a natural number (cast to a real) defined as follows, and on the left $n$ is cast to a real and $n-1$ is real subtraction. The state space is the group $S_n$ of permutations of $\{0,\dots,n-1\}$. A weight function $\mu_n : S_n \to \mathbb{R}$ is defined by: $\mu_n(g) = \tfrac12$ if $g$ is the identity; otherwise $\mu_n(g) = \dfrac{1}{2(n-1)}$ if $g$ equals the transposition swapping $i$ and $i+1$ for some index $i$ with $i+1 < n$ (an adjacent transposition; the second point is written $(i+1)\bmod n$ in the code but the guard $i+1<n$ makes it literally $i+1$); and $\mu_n(g) = 0$ for all other permutations. The transition matrix is the random-walk matrix $P_n(x,y) = \mu_n(y\,x^{-1})$. The reference distribution is the constant function $\pi_n(g) = 1/n!$ on $S_n$ (the uniform distribution). The distance between two functions $\mu,\nu : S_n \to \mathbb{R}$ is $d(\mu,\nu) = \sup_{A \subseteq S_n}\bigl|\sum_{g\in A}\mu(g) - \sum_{g\in A}\nu(g)\bigr|$, the supremum over all finite subsets of the absolute difference of sums (note: no factor $\tfrac12$). The worst-case distance to $\pi_n$ at time $t$ is $d_n(t) = \max_{x\in S_n} d\bigl(P_n^{t}(x,\cdot),\,\pi_n\bigr)$, using row $x$ of the $t$-th matrix power. Finally, the mixing quantity is taken at the fixed threshold $\tfrac14$:

which by the natural-number infimum convention equals $0$ if no such $t$ exists (in which case the claimed inequality would force $n^2(n-1)/16 \le 0$, impossible for $n \ge 2$, so the statement implicitly requires the set to be nonempty with a large infimum). The statement carries no hypothesis that $P_n$ is stochastic or that $\pi_n$ is stationary for it; the only hypothesis is $n \ge 2$. The inequality is non-strict and is a lower bound on the mixing quantity.