by is also non-reversible, stationary, and strand-symmetric, and yields the same joint probability distribution J $^{ab}$. Both models satisfy all constraints of Lemma 3.1 except assumption b. Generating such examples is straightforward, and a Python script for generating random examples is included in the ancillary material. The above example was generated by that script and rounded to two decimal places. Proof of Lemma 3.1. Single out the permutation matrix which has the effect of reversing the order of rows or columns, depending on the direction of multiplication. First note that if a matrix P is strand-symmetric, then P = SP S . Also, P ⊺ = ( SP S ) ⊺ = SP ⊺ S and P − 1 = ( SP S ) − 1 = SP − $^{1}$S . and 10 π m = ( 0 . 20 0 . 30 0 . 30 0 . 20 ) , P ma =    $^{0}$$^{.}$$^{35}$ 0 . 34 0 . 12 0 . 19 0 . 15 0 . 46 0 . 24 0 . 15 0 . 15 0 . 24 0 . 46 0 . 15 0 . 19 0 . 12 0 . 34 0 . 35    $^{,}$ P mb =    $^{0}$$^{.}$$^{43}$ 0 . 27 0 . 19 0 . 11 0 . 07 0 . 39 0 . 31 0 . 23 0 . 23 0 . 31 0 . 39 0 . 07 0 . 11 0 . 19 0 . 27 0 . 43     S =    $^{0}$ 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0    $^{,}$