##########################################################################################
N = 2: 5 bijective YBE solutions, 5 isomorphism classes (0.0s)
------------------------------------------------------------------------------------------
class 0: sigma = [(0, 0), (0, 1), (1, 0), (1, 1)]
   involutive=True rack=False non-degenerate=False
   H_A=[1, 2, 4, 8, 16, 32, 64, 128, 256]  H_B=[1, 2, 0, 0, 0, 0, 0, 0, 0]  H_K=[1, 2, 0, 0, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 8
   eps=1: H_n(B,del) n<5 = [1, 2, 0, 0, 0] ; H_n(K,del) = [1, 2, 0, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [4, 2], ranks = [2, 2]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [12, 8, 0], ranks = [4, 8, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [32, 24, 0, 0], ranks = [8, 24, 0, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [80, 64, 0, 0, 0], ranks = [16, 64, 0, 0, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
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class 1: sigma = [(0, 0), (1, 0), (0, 1), (1, 1)]
   involutive=True rack=True non-degenerate=True
   H_A=[1, 2, 3, 4, 5, 6, 7, 8, 9]  H_B=[1, 2, 1, 0, 0, 0, 0, 0, 0]  H_K=[1, 2, 1, 0, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 8
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [4, 2], ranks = [2, 2]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [10, 8, 1], ranks = [3, 7, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [20, 20, 4, 0], ranks = [4, 16, 4, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [35, 40, 10, 0, 0], ranks = [5, 30, 10, 0, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
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class 2: sigma = [(0, 1), (1, 1), (0, 0), (1, 0)]
   involutive=False rack=True non-degenerate=True
   H_A=[1, 2, 1, 1, 1, 1, 1, 1, 1]  H_B=[1, 2, 3, 5, 8, 13, 21, 34, 55]  H_K=[1, 2, 3, 5, 8, 13]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 8
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 0, 0] ; H_n(K,del) = [1, 1, 0, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [4, 2], ranks = [2, 2]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [6, 8, 3], ranks = [1, 5, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [6, 12, 12, 5], ranks = [1, 5, 7, 5]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [7, 12, 18, 20, 8], ranks = [1, 6, 6, 12, 8]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
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class 3: sigma = [(1, 0), (0, 0), (1, 1), (0, 1)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 2, 1, 1, 1, 1, 1, 1, 1]  H_B=[1, 2, 3, 5, 8, 13, 21, 34, 55]  H_K=[1, 2, 3, 5, 8, 13]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 8
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 0, 0] ; H_n(K,del) = [1, 1, 0, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [4, 2], ranks = [2, 2]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [6, 8, 3], ranks = [1, 5, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [6, 12, 12, 5], ranks = [1, 5, 7, 5]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [7, 12, 18, 20, 8], ranks = [1, 6, 6, 12, 8]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 4: sigma = [(1, 1), (0, 1), (1, 0), (0, 0)]
   involutive=True rack=False non-degenerate=True
   H_A=[1, 2, 3, 4, 5, 6, 7, 8, 9]  H_B=[1, 2, 1, 0, 0, 0, 0, 0, 0]  H_K=[1, 2, 1, 0, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 8
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 0, 0] ; H_n(K,del) = [1, 1, 0, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [4, 2], ranks = [2, 2]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [10, 8, 1], ranks = [3, 7, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [20, 20, 4, 0], ranks = [4, 16, 4, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [35, 40, 10, 0, 0], ranks = [5, 30, 10, 0, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
N = 2: 0 of 5 classes fail the Euler test
##########################################################################################
N = 3: 73 bijective YBE solutions, 29 isomorphism classes (0.3s)
------------------------------------------------------------------------------------------
class 0: sigma = [(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
   involutive=True rack=False non-degenerate=False
   H_A=[1, 3, 9, 27, 81, 243, 729]  H_B=[1, 3, 0, 0, 0, 0, 0]  H_K=[1, 3, 0, 0, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 3, 0, 0, 0] ; H_n(K,del) = [1, 3, 0, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [27, 18, 0], ranks = [9, 18, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [108, 81, 0, 0], ranks = [27, 81, 0, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [405, 324, 0, 0, 0], ranks = [81, 324, 0, 0, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 1: sigma = [(0, 0), (0, 1), (2, 0), (1, 0), (1, 1), (2, 1), (0, 2), (1, 2), (2, 2)]
   involutive=True rack=False non-degenerate=False
   H_A=[1, 3, 7, 15, 31, 63, 127]  H_B=[1, 3, 2, 0, 0, 0, 0]  H_K=[1, 3, 2, 0, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 3, 2, 0, 0] ; H_n(K,del) = [1, 3, 2, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [23, 18, 2], ranks = [7, 16, 2]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [72, 69, 12, 0], ranks = [15, 57, 12, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [201, 216, 46, 0, 0], ranks = [31, 170, 46, 0, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 2: sigma = [(0, 0), (0, 1), (2, 1), (1, 0), (1, 1), (2, 0), (1, 2), (0, 2), (2, 2)]
   involutive=True rack=False non-degenerate=False
   H_A=[1, 3, 7, 15, 31, 63, 127]  H_B=[1, 3, 2, 0, 0, 0, 0]  H_K=[1, 3, 2, 0, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [23, 18, 2], ranks = [7, 16, 2]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [72, 69, 12, 0], ranks = [15, 57, 12, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [201, 216, 46, 0, 0], ranks = [31, 170, 46, 0, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 3: sigma = [(0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (2, 1), (0, 2), (1, 2), (2, 2)]
   involutive=True rack=True non-degenerate=True
   H_A=[1, 3, 6, 10, 15, 21, 28]  H_B=[1, 3, 3, 1, 0, 0, 0]  H_K=[1, 3, 3, 1, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 3, 3, 1, 0] ; H_n(K,del) = [1, 3, 3, 1, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [21, 18, 3], ranks = [6, 15, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [56, 63, 18, 1], ranks = [10, 46, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [126, 168, 63, 6, 0], ranks = [15, 111, 57, 6, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 4: sigma = [(0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (2, 1), (1, 2), (0, 2), (2, 2)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 5, 7, 9, 11, 13]  H_B=[1, 3, 4, 4, 3, 1, 0]  H_K=[1, 3, 4, 4, 4, 4]
   H_A(t)H_B(-t) = [1, 0, 0, 0, -1, 0, 0]  -> FAILS(Euler, weight 4)
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 1, 2] ; H_n(K,del) = [1, 2, 1, 0, 0]   <-- DIFFER
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [44, 57, 24, 4], ranks = [7, 37, 20, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [85, 132, 76, 24, 3], ranks = [9, 76, 56, 20, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 1}
------------------------------------------------------------------------------------------
class 5: sigma = [(0, 0), (1, 0), (2, 0), (0, 1), (1, 2), (2, 2), (0, 2), (1, 1), (2, 1)]
   involutive=False rack=True non-degenerate=True
   H_A=[1, 3, 4, 5, 6, 7, 8]  H_B=[1, 3, 5, 8, 13, 21, 34]  H_K=[1, 3, 5, 8, 13, 21]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [17, 18, 5], ranks = [4, 13, 5]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [34, 51, 30, 8], ranks = [5, 29, 22, 8]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [58, 102, 85, 48, 13], ranks = [6, 52, 50, 35, 13]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 6: sigma = [(0, 0), (1, 0), (2, 0), (0, 1), (2, 1), (1, 1), (0, 2), (2, 2), (1, 2)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 4, 5, 6, 7, 8]  H_B=[1, 3, 5, 8, 13, 21, 34]  H_K=[1, 3, 5, 8, 13, 21]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [17, 18, 5], ranks = [4, 13, 5]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [34, 51, 30, 8], ranks = [5, 29, 22, 8]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [58, 102, 85, 48, 13], ranks = [6, 52, 50, 35, 13]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 7: sigma = [(0, 0), (1, 0), (2, 0), (0, 1), (2, 2), (1, 2), (0, 2), (2, 1), (1, 1)]
   involutive=True rack=False non-degenerate=True
   H_A=[1, 3, 6, 10, 15, 21, 28]  H_B=[1, 3, 3, 1, 0, 0, 0]  H_K=[1, 3, 3, 1, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [21, 18, 3], ranks = [6, 15, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [56, 63, 18, 1], ranks = [10, 46, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [126, 168, 63, 6, 0], ranks = [15, 111, 57, 6, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 8: sigma = [(0, 0), (1, 0), (2, 0), (0, 2), (1, 1), (2, 1), (0, 1), (1, 2), (2, 2)]
   involutive=False rack=True non-degenerate=True
   H_A=[1, 3, 5, 7, 9, 11, 13]  H_B=[1, 3, 4, 4, 3, 1, 0]  H_K=[1, 3, 4, 4, 4, 4]
   H_A(t)H_B(-t) = [1, 0, 0, 0, -1, 0, 0]  -> FAILS(Euler, weight 4)
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 1, 2] ; H_n(K,del) = [1, 2, 1, 0, 0]   <-- DIFFER
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [44, 57, 24, 4], ranks = [7, 37, 20, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [85, 132, 76, 24, 3], ranks = [9, 76, 56, 20, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 1}
------------------------------------------------------------------------------------------
class 9: sigma = [(0, 0), (1, 0), (2, 0), (0, 2), (1, 2), (2, 2), (0, 1), (1, 1), (2, 1)]
   involutive=False rack=True non-degenerate=True
   H_A=[1, 3, 3, 4, 5, 6, 7]  H_B=[1, 3, 6, 13, 28, 60, 129]  H_K=[1, 3, 6, 13, 28, 60]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [15, 18, 6], ranks = [3, 12, 6]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [26, 45, 36, 13], ranks = [4, 22, 23, 13]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [43, 78, 90, 78, 28], ranks = [5, 38, 40, 50, 28]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 10: sigma = [(0, 0), (1, 0), (2, 0), (0, 2), (2, 1), (1, 1), (0, 1), (2, 2), (1, 2)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 3, 4, 5, 6, 7]  H_B=[1, 3, 6, 13, 28, 60, 129]  H_K=[1, 3, 6, 13, 28, 60]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [15, 18, 6], ranks = [3, 12, 6]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [26, 45, 36, 13], ranks = [4, 22, 23, 13]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [43, 78, 90, 78, 28], ranks = [5, 38, 40, 50, 28]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 11: sigma = [(0, 0), (1, 0), (2, 0), (0, 2), (2, 2), (1, 2), (0, 1), (2, 1), (1, 1)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 5, 7, 9, 11, 13]  H_B=[1, 3, 4, 4, 3, 1, 0]  H_K=[1, 3, 4, 4, 4, 4]
   H_A(t)H_B(-t) = [1, 0, 0, 0, -1, 0, 0]  -> FAILS(Euler, weight 4)
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 1, 2] ; H_n(K,del) = [1, 2, 1, 0, 0]   <-- DIFFER
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [44, 57, 24, 4], ranks = [7, 37, 20, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [85, 132, 76, 24, 3], ranks = [9, 76, 56, 20, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 1}
------------------------------------------------------------------------------------------
class 12: sigma = [(0, 0), (1, 0), (2, 0), (2, 2), (0, 2), (1, 2), (1, 1), (2, 1), (0, 1)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 5, 6, 6, 6, 6]  H_B=[1, 3, 4, 3, 1, 0, 0]  H_K=[1, 3, 4, 3, 1, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, -1]  -> FAILS(Euler, weight 6)
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 1, 1] ; H_n(K,del) = [1, 1, 0, 1, 1] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [42, 57, 24, 3], ranks = [6, 36, 21, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [73, 126, 76, 18, 1], ranks = [6, 67, 59, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 13: sigma = [(0, 0), (1, 0), (2, 1), (0, 1), (1, 1), (2, 0), (1, 2), (0, 2), (2, 2)]
   involutive=True rack=False non-degenerate=True
   H_A=[1, 3, 6, 10, 15, 21, 28]  H_B=[1, 3, 3, 1, 0, 0, 0]  H_K=[1, 3, 3, 1, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [21, 18, 3], ranks = [6, 15, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [56, 63, 18, 1], ranks = [10, 46, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [126, 168, 63, 6, 0], ranks = [15, 111, 57, 6, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 14: sigma = [(0, 0), (1, 2), (2, 1), (0, 2), (1, 1), (2, 0), (0, 1), (1, 0), (2, 2)]
   involutive=False rack=True non-degenerate=True
   H_A=[1, 3, 5, 6, 6, 6, 6]  H_B=[1, 3, 4, 3, 1, 0, 0]  H_K=[1, 3, 4, 3, 1, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, -1]  -> FAILS(Euler, weight 6)
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 1, 1] ; H_n(K,del) = [1, 1, 0, 1, 1] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [42, 57, 24, 3], ranks = [6, 36, 21, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [73, 126, 76, 18, 1], ranks = [6, 67, 59, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 15: sigma = [(0, 0), (2, 0), (1, 0), (0, 1), (1, 2), (2, 2), (0, 2), (1, 1), (2, 1)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 3, 4, 5, 6, 7]  H_B=[1, 3, 6, 13, 28, 60, 129]  H_K=[1, 3, 6, 13, 28, 60]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [15, 18, 6], ranks = [3, 12, 6]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [26, 45, 36, 13], ranks = [4, 22, 23, 13]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [43, 78, 90, 78, 28], ranks = [5, 38, 40, 50, 28]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 16: sigma = [(0, 0), (2, 0), (1, 0), (0, 1), (2, 1), (1, 1), (0, 2), (2, 2), (1, 2)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 3, 4, 5, 6, 7]  H_B=[1, 3, 6, 13, 28, 60, 129]  H_K=[1, 3, 6, 13, 28, 60]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [15, 18, 6], ranks = [3, 12, 6]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [26, 45, 36, 13], ranks = [4, 22, 23, 13]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [43, 78, 90, 78, 28], ranks = [5, 38, 40, 50, 28]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 17: sigma = [(0, 0), (2, 0), (1, 0), (0, 1), (2, 2), (1, 2), (0, 2), (2, 1), (1, 1)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 5, 7, 9, 11, 13]  H_B=[1, 3, 4, 4, 3, 1, 0]  H_K=[1, 3, 4, 4, 4, 4]
   H_A(t)H_B(-t) = [1, 0, 0, 0, -1, 0, 0]  -> FAILS(Euler, weight 4)
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 1, 2] ; H_n(K,del) = [1, 2, 1, 0, 0]   <-- DIFFER
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [44, 57, 24, 4], ranks = [7, 37, 20, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [85, 132, 76, 24, 3], ranks = [9, 76, 56, 20, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 1}
------------------------------------------------------------------------------------------
class 18: sigma = [(0, 0), (2, 0), (1, 0), (0, 2), (1, 2), (2, 2), (0, 1), (1, 1), (2, 1)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 4, 5, 6, 7, 8]  H_B=[1, 3, 5, 8, 13, 21, 34]  H_K=[1, 3, 5, 8, 13, 21]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [17, 18, 5], ranks = [4, 13, 5]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [34, 51, 30, 8], ranks = [5, 29, 22, 8]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [58, 102, 85, 48, 13], ranks = [6, 52, 50, 35, 13]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 19: sigma = [(0, 0), (2, 0), (1, 0), (0, 2), (2, 1), (1, 1), (0, 1), (2, 2), (1, 2)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 4, 5, 6, 7, 8]  H_B=[1, 3, 5, 8, 13, 21, 34]  H_K=[1, 3, 5, 8, 13, 21]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [17, 18, 5], ranks = [4, 13, 5]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [34, 51, 30, 8], ranks = [5, 29, 22, 8]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [58, 102, 85, 48, 13], ranks = [6, 52, 50, 35, 13]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 20: sigma = [(0, 0), (2, 0), (1, 0), (0, 2), (2, 2), (1, 2), (0, 1), (2, 1), (1, 1)]
   involutive=True rack=False non-degenerate=True
   H_A=[1, 3, 6, 10, 15, 21, 28]  H_B=[1, 3, 3, 1, 0, 0, 0]  H_K=[1, 3, 3, 1, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 2, 1, 0, 0] ; H_n(K,del) = [1, 2, 1, 0, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [21, 18, 3], ranks = [6, 15, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [56, 63, 18, 1], ranks = [10, 46, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [126, 168, 63, 6, 0], ranks = [15, 111, 57, 6, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 21: sigma = [(0, 0), (2, 0), (1, 0), (2, 1), (1, 1), (0, 1), (1, 2), (0, 2), (2, 2)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 5, 6, 6, 6, 6]  H_B=[1, 3, 4, 3, 1, 0, 0]  H_K=[1, 3, 4, 3, 1, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, -1]  -> FAILS(Euler, weight 6)
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 1, 1] ; H_n(K,del) = [1, 1, 0, 1, 1] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [42, 57, 24, 3], ranks = [6, 36, 21, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [73, 126, 76, 18, 1], ranks = [6, 67, 59, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 22: sigma = [(0, 0), (2, 2), (1, 1), (0, 1), (2, 0), (1, 2), (0, 2), (2, 1), (1, 0)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 5, 6, 6, 6, 6]  H_B=[1, 3, 4, 3, 1, 0, 0]  H_K=[1, 3, 4, 3, 1, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, -1]  -> FAILS(Euler, weight 6)
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 1, 1] ; H_n(K,del) = [1, 1, 0, 1, 1] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [42, 57, 24, 3], ranks = [6, 36, 21, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [73, 126, 76, 18, 1], ranks = [6, 67, 59, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 23: sigma = [(0, 1), (1, 1), (2, 1), (0, 2), (1, 2), (2, 2), (0, 0), (1, 0), (2, 0)]
   involutive=False rack=True non-degenerate=True
   H_A=[1, 3, 2, 1, 1, 1, 1]  H_B=[1, 3, 7, 16, 33, 64, 113]  H_K=[1, 3, 7, 16, 36, 81]
   H_A(t)H_B(-t) = [1, 0, 0, 0, -3, 8, -24]  -> FAILS(Euler, weight 4)
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 1, 1] ; H_n(K,del) = [1, 1, 0, 0, 0]   <-- DIFFER
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [13, 18, 7], ranks = [2, 11, 7]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [14, 39, 42, 16], ranks = [1, 13, 26, 16]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [12, 42, 91, 96, 33], ranks = [1, 11, 31, 60, 33]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 3}
------------------------------------------------------------------------------------------
class 24: sigma = [(1, 0), (2, 0), (0, 0), (1, 1), (2, 1), (0, 1), (1, 2), (2, 2), (0, 2)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 2, 1, 1, 1, 1]  H_B=[1, 3, 7, 16, 33, 64, 113]  H_K=[1, 3, 7, 16, 36, 81]
   H_A(t)H_B(-t) = [1, 0, 0, 0, -3, 8, -24]  -> FAILS(Euler, weight 4)
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 1, 1] ; H_n(K,del) = [1, 1, 0, 0, 0]   <-- DIFFER
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [13, 18, 7], ranks = [2, 11, 7]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [14, 39, 42, 16], ranks = [1, 13, 26, 16]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [12, 42, 91, 96, 33], ranks = [1, 11, 31, 60, 33]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 3}
------------------------------------------------------------------------------------------
class 25: sigma = [(1, 1), (0, 1), (2, 1), (0, 2), (2, 2), (1, 2), (2, 0), (1, 0), (0, 0)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 5, 6, 6, 6, 6]  H_B=[1, 3, 4, 3, 1, 0, 0]  H_K=[1, 3, 4, 3, 1, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, -1]  -> FAILS(Euler, weight 6)
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 1, 1] ; H_n(K,del) = [1, 1, 0, 1, 1] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [42, 57, 24, 3], ranks = [6, 36, 21, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [73, 126, 76, 18, 1], ranks = [6, 67, 59, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 26: sigma = [(1, 1), (2, 0), (0, 2), (1, 0), (2, 2), (0, 1), (1, 2), (2, 1), (0, 0)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 5, 6, 6, 6, 6]  H_B=[1, 3, 4, 3, 1, 0, 0]  H_K=[1, 3, 4, 3, 1, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, -1]  -> FAILS(Euler, weight 6)
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 1, 1] ; H_n(K,del) = [1, 1, 0, 1, 1] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [19, 18, 4], ranks = [5, 14, 4]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [42, 57, 24, 3], ranks = [6, 36, 21, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [73, 126, 76, 18, 1], ranks = [6, 67, 59, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
------------------------------------------------------------------------------------------
class 27: sigma = [(1, 1), (2, 1), (0, 1), (1, 2), (2, 2), (0, 2), (1, 0), (2, 0), (0, 0)]
   involutive=False rack=False non-degenerate=True
   H_A=[1, 3, 2, 1, 1, 1, 1]  H_B=[1, 3, 7, 16, 33, 64, 113]  H_K=[1, 3, 7, 16, 36, 81]
   H_A(t)H_B(-t) = [1, 0, 0, 0, -3, 8, -24]  -> FAILS(Euler, weight 4)
   eps=1: H_n(B,del) n<5 = [1, 1, 0, 1, 1] ; H_n(K,del) = [1, 1, 0, 0, 0]   <-- DIFFER
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [13, 18, 7], ranks = [2, 11, 7]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [14, 39, 42, 16], ranks = [1, 13, 26, 16]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [12, 42, 91, 96, 33], ranks = [1, 11, 31, 60, 33]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): {3: 3}
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class 28: sigma = [(1, 2), (2, 2), (0, 2), (1, 0), (2, 0), (0, 0), (1, 1), (2, 1), (0, 1)]
   involutive=True rack=False non-degenerate=True
   H_A=[1, 3, 6, 10, 15, 21, 28]  H_B=[1, 3, 3, 1, 0, 0, 0]  H_K=[1, 3, 3, 1, 0, 0]
   H_A(t)H_B(-t) = [1, 0, 0, 0, 0, 0, 0]  -> passes Euler test to weight 6
   eps=1: H_n(B,del) n<5 = [1, 1, 1, 1, 0] ; H_n(K,del) = [1, 1, 1, 1, 0] 
   direct homology of Im f:
  weight D=1: dim C_n (n=0..D) = [6, 3], ranks = [3, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=2: dim C_n (n=0..D) = [21, 18, 3], ranks = [6, 15, 3]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=3: dim C_n (n=0..D) = [56, 63, 18, 1], ranks = [10, 46, 17, 1]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
  weight D=4: dim C_n (n=0..D) = [126, 168, 63, 6, 0], ranks = [15, 111, 57, 6, 0]
     homology of Im f (augmented; index -1 = coker of A(x)A->A): ACYCLIC
N = 3: 13 of 29 classes fail the Euler test
