Verification run 2 -- J: the families of Diaconis and Hanlon; full chains and lumped chains
   N = 3 (n = 6), theta = 1.0: formula 0.0000000000; full chain beta_2 = 0.0000000000; lumped chain: 3 eigenvalues; further eigenvalues of the full chain: 0 distinct, the largest []
   N = 3 (n = 6), theta = 0.5: formula 0.3333333333; full chain beta_2 = 0.3333333333; lumped chain: 3 eigenvalues; further eigenvalues of the full chain: 1 distinct, the largest [0.0]
   N = 3 (n = 6), theta = 0.1: formula 0.6000000000; full chain beta_2 = 0.6000000000; lumped chain: 3 eigenvalues; further eigenvalues of the full chain: 1 distinct, the largest [-0.0]
   N = 4 (n = 24), theta = 1.0: formula 0.3333333333; full chain beta_2 = 0.3333333333; lumped chain: 5 eigenvalues; further eigenvalues of the full chain: 0 distinct, the largest []
   N = 4 (n = 24), theta = 0.5: formula 0.5833333333; full chain beta_2 = 0.5833333333; lumped chain: 5 eigenvalues; further eigenvalues of the full chain: 6 distinct, the largest [0.56704263, 0.39179684, 0.26112433]
   N = 4 (n = 24), theta = 0.1: formula 0.7833333333; full chain beta_2 = 0.7833333333; lumped chain: 5 eigenvalues; further eigenvalues of the full chain: 6 distinct, the largest [0.7818014, 0.57742844, 0.47346753]
   N = 5 (n = 120), theta = 1.0: formula 0.5000000000; full chain beta_2 = 0.5000000000; lumped chain: 7 eigenvalues; further eigenvalues of the full chain: 0 distinct, the largest []
   N = 5 (n = 120), theta = 0.5: formula 0.7000000000; full chain beta_2 = 0.7000000000; lumped chain: 7 eigenvalues; further eigenvalues of the full chain: 17 distinct, the largest [0.68196136, 0.56986182, 0.45607196]
   N = 5 (n = 120), theta = 0.1: formula 0.8600000000; full chain beta_2 = 0.8600000000; lumped chain: 7 eigenvalues; further eigenvalues of the full chain: 18 distinct, the largest [0.8582051, 0.72644919, 0.72551334]
   N = 6 (n = 720), theta = 1.0: formula 0.6000000000; full chain beta_2 = 0.6000000000; lumped chain: 11 eigenvalues; further eigenvalues of the full chain: 0 distinct, the largest []
   N = 6 (n = 720), theta = 0.5: formula 0.7666666667; full chain beta_2 = 0.7666666667; lumped chain: 11 eigenvalues; further eigenvalues of the full chain: 66 distinct, the largest [0.74812925, 0.6696242, 0.59330372]
   N = 6 (n = 720), theta = 0.1: formula 0.9000000000; full chain beta_2 = 0.9000000000; lumped chain: 11 eigenvalues; further eigenvalues of the full chain: 66 distinct, the largest [0.89805142, 0.81269488, 0.80413851]
   N = 7 (n = 5040), theta = 1.0: formula 0.6666666667; full chain beta_2 = 0.6666666667; lumped chain: 15 eigenvalues; further eigenvalues of the full chain: 0 distinct, the largest []
   N = 7 (n = 5040), theta = 0.5: formula 0.8095238095; full chain beta_2 = 0.8095238095; lumped chain: 15 eigenvalues; further eigenvalues of the full chain: 225 distinct, the largest [0.79108801, 0.73278909, 0.66035343]
   N = 7 (n = 5040), theta = 0.1: formula 0.9238095238; full chain beta_2 = 0.9238095238; lumped chain: 15 eigenvalues; further eigenvalues of the full chain: 225 distinct, the largest [0.92176828, 0.85648802, 0.85578735]
PASS  S_N, N = 3..7: the chain lumped to conjugacy classes has the eigenvalues (1 - theta) + (theta n(lambda') - n(lambda))/C(N,2), lambda a partition of N (lumped transition matrix exact; eigenvalues in double precision)
PASS  S_N: the second largest of these is the one of lambda = (N-1,1), equal to 1 - 2 theta/N - 2/(N(N-1))
PASS  S_N, N = 3..7: the second largest eigenvalue of the full Metropolis chain on S_N equals 1 - 2 theta/N - 2/(N(N-1)) (theta = 1, 1/2, 1/10; double precision)
   (the full chain has further eigenvalues, which are not eigenvalues of the lumped chain; in these cases none of them exceeds the second eigenvalue of the lumped chain)
PASS  Q_d, d = 2..8: the full Metropolis chain for theta^|x| is the product chain (a uniform coordinate is set to 0 if it is 1, and to 1 with probability theta if it is 0) and has the eigenvalues 1 - (j/d)(1+theta) with multiplicity C(d,j)

checks passed: 4   TOTAL failures: 0
