(1) Diaconis-Hanlon formulas (target proportional to theta^dist)
   Q_2: 1-(1+theta)/d                       max |beta_2 - formula| over 7 theta = 1.34e-51
   Q_3: 1-(1+theta)/d                       max |beta_2 - formula| over 7 theta = 8.02e-51
   Q_4: 1-(1+theta)/d                       max |beta_2 - formula| over 7 theta = 5.35e-51
   Q_5: 1-(1+theta)/d                       max |beta_2 - formula| over 7 theta = 8.02e-51
   S_3 transp.: 1-2theta/n-2/(n(n-1))       max |beta_2 - formula| over 7 theta = 6.68e-51
   S_4 transp.: 1-2theta/n-2/(n(n-1))       max |beta_2 - formula| over 7 theta = 6.68e-51
   S_5 transp.: 1-2theta/n-2/(n(n-1))       max |beta_2 - formula| over 7 theta = 2.0e-50
   -> OK

(2) K_n with arbitrary target: tau = (n-1) max pi  (corollary of Liu's theorem)
   exact check on 42 random rational targets (n=2..8): beta_2 = 1 - 1/((n-1) max pi): OK
   and (n-1) mu_p(e) = (n-1)(1+(n-2)p)/(n-p) exactly (n=2..8, three p): OK

(3)+(4) shapes on the grid p in {1e-6,1e-4,1e-3} + {j/200, j=1..199} + {1-1e-3,1-1e-4,1-1e-6}
   graph                              | n  | bip | tau shape                       | tau_abs(0)   | tau_abs shape
   K_2                                |  2 | B | non-decreasing (strictly increasing) | inf          | non-increasing (strictly decreasing)
   K_3                                |  3 | - | non-decreasing (strictly increasing) | 2.0          | constant
   K_4                                |  4 | - | non-decreasing (strictly increasing) | 1.5          | non-decreasing (with flat part)
   K_5                                |  5 | - | non-decreasing (strictly increasing) | 1.3333333    | non-decreasing (with flat part)
   K_6                                |  6 | - | non-decreasing (strictly increasing) | 1.25         | non-decreasing (with flat part)
   K_7                                |  7 | - | non-decreasing (strictly increasing) | 1.2          | non-decreasing (with flat part)
   K_8                                |  8 | - | non-decreasing (strictly increasing) | 1.1666667    | non-decreasing (with flat part)
   Z_3                                |  3 | - | non-decreasing (strictly increasing) | 2.0          | constant
   Z_4                                |  4 | B | non-decreasing (strictly increasing) | inf          | down-then-up
   Z_5                                |  5 | - | up-then-down                    | 5.236068     | down-then-up (2 turns)
   Z_6                                |  6 | B | up-then-down                    | inf          | down-then-up (2 turns)
   Z_7                                |  7 | - | up-then-down                    | 10.097835    | down-then-up (2 turns)
   Z_8                                |  8 | B | up-then-down                    | inf          | down-then-up (2 turns)
   Z_9                                |  9 | - | up-then-down                    | 16.581719    | down-then-up (2 turns)
   Z_10                               | 10 | B | up-then-down                    | inf          | down-then-up (2 turns)
   Z_11                               | 11 | - | up-then-down                    | 24.687075    | down-then-up (2 turns)
   Z_12                               | 12 | B | up-then-down                    | inf          | down-then-up (2 turns)
   Z_13                               | 13 | - | up-then-down                    | 34.413715    | down-then-up (2 turns)
   Z_14                               | 14 | B | up-then-down                    | inf          | down-then-up (2 turns)
   Q_1                                |  2 | B | non-decreasing (strictly increasing) | inf          | non-increasing (strictly decreasing)
   Q_2                                |  4 | B | non-decreasing (strictly increasing) | inf          | down-then-up
   Q_3                                |  8 | B | up-then-down                    | inf          | down-then-up (2 turns)
   Q_4                                | 16 | B | up-then-down                    | inf          | down-then-up (2 turns)
   Q_5                                | 32 | B | up-then-down                    | inf          | down-then-up (2 turns)
   S_3,all transpositions             |  6 | B | non-decreasing (strictly increasing) | inf          | down-then-up
   S_4,all transpositions             | 24 | B | up-then-down                    | inf          | down-then-up (2 turns)
   D_4 {r,r^-1,s}                     |  8 | B | up-then-down                    | inf          | down-then-up (2 turns)
   D_4 {s,sr}                         |  8 | B | up-then-down                    | inf          | down-then-up (2 turns)
   D_5 {r,r^-1,s}                     | 10 | - | up-then-down                    | 7.854102     | down-then-up (2 turns)
   D_5 {s,sr}                         | 10 | B | up-then-down                    | inf          | down-then-up (2 turns)
   Z_2xZ_4 {(1,0),(0,+-1)}            |  8 | B | up-then-down                    | inf          | down-then-up (2 turns)
   Z_2xZ_4 {(1,+-1),(0,+-1)}          |  8 | B | non-decreasing (strictly increasing) | inf          | down-then-up
   S_4,adjacent transpositions        | 24 | B | up-then-down                    | inf          | down-then-up (2 turns)
   D_4 {r,r^-1,s,sr}                  |  8 | - | non-decreasing (strictly increasing) | 2.0          | down-then-up
   D_5 {r,r^-1,s,sr}                  | 10 | - | up-then-down                    | 2.2676611    | up-then-down
   Z_2xZ_4 {(1,0),(0,+-1),(1,2)}      |  8 | B | non-decreasing (strictly increasing) | inf          | down-then-up
   Q8 {+-i,+-j}                       |  8 | B | non-decreasing (strictly increasing) | inf          | down-then-up
   A_4 {(012)^+-,(013)^+-}            | 12 | - | up-then-down                    | 2.0          | up-then-down
   S_4 {(012)^+-,(23)}                | 24 | - | up-then-down                    | 6.8423292    | up-then-down
   Z_3xZ_3 {(+-1,0),(0,+-1)}          |  9 | - | non-decreasing (strictly increasing) | 2.0          | non-decreasing (with flat part)
   Z_9{+-1,2}                         |  9 | - | non-decreasing (strictly increasing) | 2.0          | down-then-up
   Z_12{+-1,5}                        | 12 | B | up-then-down                    | inf          | down-then-up (2 turns)
   Z_13{+-1,5}                        | 13 | - | up-then-down                    | 2.9653647    | down-then-up (2 turns)
   D_7 {r,r^-1,s}                     | 14 | - | up-then-down                    | 15.146752    | down-then-up (2 turns)
   Petersen (vertex-transitive, not C | 10 | - | up-then-down                    | 3.0          | up-then-down
   counts of tau shapes: {'non-decreasing (strictly increasing)': 18, 'up-then-down': 27}
done
