recursive QS_n == brute-force sum over S_n (checked for n<=4 on N=3, n<=3 on N=4)
=== flip3 (involutive)  N=3  involutive=True idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 3, 6, 10, 15, 21, 28, 36, 45]
  H_B (rank QS_n, n=0..5)    : [1, 3, 3, 1, 0, 0]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 3, 3, 3), (2, 3, 3, 3), (3, 1, 1, 1), (4, 0, 0, 0), (5, 0, 0, 0)]
  H_K (dim K_n = dim A^!_n)   : [1, 3, 3, 1, 0, 0]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, True, True]
  time 0.1s
=== identity3 (involutive+idempotent)  N=3  involutive=True idempotent=True
  H_A (dim A_n, n=0..8)      : [1, 3, 9, 27, 81, 243, 729, 2187, 6561]
  H_B (rank QS_n, n=0..5)    : [1, 3, 0, 0, 0, 0]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 3, 3, 3), (2, 0, 0, 0), (3, 0, 0, 0), (4, 0, 0, 0), (5, 0, 0, 0)]
  H_K (dim K_n = dim A^!_n)   : [1, 3, 0, 0, 0, 0]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, True, True]
  time 0.0s
=== Lyubashenko3 f=(012) (involutive)  N=3  involutive=True idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 3, 6, 10, 15, 21, 28, 36, 45]
  H_B (rank QS_n, n=0..5)    : [1, 3, 3, 1, 0, 0]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 3, 3, 3), (2, 3, 3, 3), (3, 1, 1, 1), (4, 0, 0, 0), (5, 0, 0, 0)]
  H_K (dim K_n = dim A^!_n)   : [1, 3, 3, 1, 0, 0]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, True, True]
  time 0.0s
=== Lyubashenko3 f=(01) (involutive)  N=3  involutive=True idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 3, 6, 10, 15, 21, 28, 36, 45]
  H_B (rank QS_n, n=0..5)    : [1, 3, 3, 1, 0, 0]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 3, 3, 3), (2, 3, 3, 3), (3, 1, 1, 1), (4, 0, 0, 0), (5, 0, 0, 0)]
  H_K (dim K_n = dim A^!_n)   : [1, 3, 3, 1, 0, 0]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, True, True]
  time 0.0s
=== Lyubashenko4 f=(01)(23) (involutive)  N=4  involutive=True idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 4, 10, 20, 35, 56, 84, 120, 165]
  H_B (rank QS_n, n=0..4)    : [1, 4, 6, 4, 1]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 4, 4, 4), (2, 6, 6, 6), (3, 4, 4, 4), (4, 1, 1, 1)]
  H_K (dim K_n = dim A^!_n)   : [1, 4, 6, 4, 1]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, True]
  time 0.4s
=== dihedral quandle R3 (-sigma -> FK_3)  N=3  involutive=False idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 3, 5, 6, 6, 6, 6, 6, 6]
  H_B (rank QS_n, n=0..5)    : [1, 3, 4, 3, 1, 0]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 3, 3, 3), (2, 4, 4, 4), (3, 3, 3, 3), (4, 1, 1, 1), (5, 0, 0, 0)]
  H_K (dim K_n = dim A^!_n)   : [1, 3, 4, 3, 1, 0]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, True, True]
  time 0.1s
=== idempotent3 (min,max) [free comm. monoid]  N=3  involutive=False idempotent=True
  H_A (dim A_n, n=0..8)      : [1, 3, 6, 10, 15, 21, 28, 36, 45]
  H_B (rank QS_n, n=0..5)    : [1, 3, 3, 1, 0, 0]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 3, 3, 3), (2, 3, 3, 3), (3, 1, 1, 1), (4, 0, 0, 0), (5, 0, 0, 0)]
  H_K (dim K_n = dim A^!_n)   : [1, 3]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True]
  time 0.0s
=== idempotent3 (x,x)  N=3  involutive=False idempotent=True
  H_A (dim A_n, n=0..8)      : [1, 3, 3, 3, 3, 3, 3, 3, 3]
  H_B (rank QS_n, n=0..5)    : [1, 3, 6, 12, 24, 48]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 3, 3, 3), (2, 6, 6, 6), (3, 12, 12, 12), (4, 24, 24, 24), (5, 48, 48, 48)]
  H_K (dim K_n = dim A^!_n)   : [1, 3]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True]
  time 0.1s
=== idempotent3 (y,y)  N=3  involutive=False idempotent=True
  H_A (dim A_n, n=0..8)      : [1, 3, 3, 3, 3, 3, 3, 3, 3]
  H_B (rank QS_n, n=0..5)    : [1, 3, 6, 12, 24, 48]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 3, 3, 3), (2, 6, 6, 6), (3, 12, 12, 12), (4, 24, 24, 24), (5, 48, 48, 48)]
  H_K (dim K_n = dim A^!_n)   : [1, 3]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True]
  time 0.1s
=== permrack2_10  N=2  involutive=False idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 2, 1, 1, 1, 1, 1, 1, 1]
  H_B (rank QS_n, n=0..7)    : [1, 2, 3, 5, 8, 13, 21, 34]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 2, 2, 2), (2, 3, 3, 3), (3, 5, 5, 5), (4, 8, 8, 8), (5, 13, 13, 13), (6, 21, 21, 21), (7, 34, 34, 34)]
  H_K (dim K_n = dim A^!_n)   : [1, 2, 3, 5, 8, 13]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, True, True]
  time 0.0s
=== permrack3_120  N=3  involutive=False idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 3, 2, 1, 1, 1, 1, 1, 1]
  H_B (rank QS_n, n=0..6)    : [1, 3, 7, 16, 33, 64, 113]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 3, 3, 3), (2, 7, 7, 7), (3, 16, 16, 16), (4, 33, 33, 33), (5, 64, 64, 64), (6, 113, 113, None)]
  H_K (dim K_n = dim A^!_n)   : [1, 3, 7, 16, 36, 81]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, -3, 8, -24]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, False, False]
  time 0.4s
=== permrack3_102  N=3  involutive=False idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 3, 3, 4, 5, 6, 7, 8, 9]
  H_B (rank QS_n, n=0..6)    : [1, 3, 6, 13, 28, 60, 129]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 3, 3, 3), (2, 6, 6, 6), (3, 13, 13, 13), (4, 28, 28, 28), (5, 60, 60, 60), (6, 129, 129, None)]
  H_K (dim K_n = dim A^!_n)   : [1, 3, 6, 13, 28, 60]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, 0, 0, 0]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, True, True]
  time 0.2s
=== permrack4_1230  N=4  involutive=False idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 4, 2, 1, 1, 1, 1, 1, 1]
  H_B (rank QS_n, n=0..5)    : [1, 4, 14, 47, 152, 489]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 4, 4, 4), (2, 14, 14, 14), (3, 47, 47, 47), (4, 152, 152, 152), (5, 489, 489, None)]
  H_K (dim K_n = dim A^!_n)   : [1, 4, 14, 49, 171, 597]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 2, -11, 36]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, False, False, False]
  time 3.6s
=== permrack4_1032  N=4  involutive=False idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 4, 4, 4, 5, 6, 7, 8, 9]
  H_B (rank QS_n, n=0..5)    : [1, 4, 12, 36, 104, 298]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 4, 4, 4), (2, 12, 12, 12), (3, 36, 36, 36), (4, 104, 104, 104), (5, 298, 298, None)]
  H_K (dim K_n = dim A^!_n)   : [1, 4, 12, 36, 107, 318]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, -3, 8]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, False, False]
  time 1.0s
=== permrack4_1203  N=4  involutive=False idempotent=False
  H_A (dim A_n, n=0..8)      : [1, 4, 4, 4, 5, 6, 7, 8, 9]
  H_B (rank QS_n, n=0..5)    : [1, 4, 12, 36, 98, 252]
     ranks (mod P1, mod P2, exact Bareiss or None): [(1, 4, 4, 4), (2, 12, 12, 12), (3, 36, 36, 36), (4, 98, 98, 98), (5, 252, 252, None)]
  H_K (dim K_n = dim A^!_n)   : [1, 4, 12, 36, 107, 318]
  H_A(t)H_B(-t) coefficients  : [1, 0, 0, 0, -9, 30]  <-- must be 1,0,0,... if Q44 holds
  H_A(t)H_K(-t) coefficients  : [1, 0, 0, 0, 0, 0]  (numerical Koszulity test)
  B_n == K_n ?                : [True, True, True, True, False, False]
  time 1.0s
