# r2_appendix.py: Appendix A (Lemma A.1, Proposition A.2) for small N, k; ranks modulo p = 2147483647
## (A1) Lemma A.1
n=1 m=2: 1 noncrossing pairings, rank of the T^(n)_q = 1
n=1 m=4: 2 noncrossing pairings, rank of the T^(n)_q = 1
n=1 m=6: 5 noncrossing pairings, rank of the T^(n)_q = 1
n=1 m=8: 14 noncrossing pairings, rank of the T^(n)_q = 1
n=2 m=2: 1 noncrossing pairings, rank of the T^(n)_q = 1
n=2 m=4: 2 noncrossing pairings, rank of the T^(n)_q = 2
n=2 m=6: 5 noncrossing pairings, rank of the T^(n)_q = 5
n=2 m=8: 14 noncrossing pairings, rank of the T^(n)_q = 14
n=3 m=2: 1 noncrossing pairings, rank of the T^(n)_q = 1
n=3 m=4: 2 noncrossing pairings, rank of the T^(n)_q = 2
n=3 m=6: 5 noncrossing pairings, rank of the T^(n)_q = 5
n=3 m=8: 14 noncrossing pairings, rank of the T^(n)_q = 14
n=4 m=2: 1 noncrossing pairings, rank of the T^(n)_q = 1
n=4 m=4: 2 noncrossing pairings, rank of the T^(n)_q = 2
n=4 m=6: 5 noncrossing pairings, rank of the T^(n)_q = 5
n=4 m=8: 14 noncrossing pairings, rank of the T^(n)_q = 14
## (A2)-(A4) Proposition A.2
N=3 k=0: |NC_{2,1}(k)| = 1; rank T^1_p = 1, rank A_p = 1, rank of both = 1; x_p (s(p)>=1): 0 vectors, rank 0; lambda with invariant sum: nullity 1, |NC_2(k)| = 1  OK  [0.0s]
N=3 k=1: |NC_{2,1}(k)| = 1; rank T^1_p = 1, rank A_p = 1, rank of both = 1; x_p (s(p)>=1): 1 vectors, rank 1; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=3 k=2: |NC_{2,1}(k)| = 2; rank T^1_p = 2, rank A_p = 2, rank of both = 2; x_p (s(p)>=1): 1 vectors, rank 1; lambda with invariant sum: nullity 1, |NC_2(k)| = 1  OK  [0.0s]
N=3 k=3: |NC_{2,1}(k)| = 4; rank T^1_p = 4, rank A_p = 4, rank of both = 4; x_p (s(p)>=1): 4 vectors, rank 4; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=3 k=4: |NC_{2,1}(k)| = 9; rank T^1_p = 9, rank A_p = 9, rank of both = 9; x_p (s(p)>=1): 7 vectors, rank 7; lambda with invariant sum: nullity 2, |NC_2(k)| = 2  OK  [0.0s]
N=3 k=5: |NC_{2,1}(k)| = 21; rank T^1_p = 21, rank A_p = 21, rank of both = 21; x_p (s(p)>=1): 21 vectors, rank 21; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=3 k=6: |NC_{2,1}(k)| = 51; rank T^1_p = 51, rank A_p = 51, rank of both = 51; x_p (s(p)>=1): 46 vectors, rank 46; lambda with invariant sum: nullity 5, |NC_2(k)| = 5  OK  [0.0s]
N=3 k=7: |NC_{2,1}(k)| = 127; rank T^1_p = 127, rank A_p = 127, rank of both = 127; x_p (s(p)>=1): 127 vectors, rank 127; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.1s]
N=4 k=0: |NC_{2,1}(k)| = 1; rank T^1_p = 1, rank A_p = 1, rank of both = 1; x_p (s(p)>=1): 0 vectors, rank 0; y_{p,j}: 1 vectors, rank 1; lambda with invariant sum: nullity 1, |NC_2(k)| = 1  OK  [0.0s]
N=4 k=1: |NC_{2,1}(k)| = 1; rank T^1_p = 1, rank A_p = 1, rank of both = 1; x_p (s(p)>=1): 1 vectors, rank 1; y_{p,j}: 2 vectors, rank 2; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=4 k=2: |NC_{2,1}(k)| = 2; rank T^1_p = 2, rank A_p = 2, rank of both = 2; x_p (s(p)>=1): 1 vectors, rank 1; y_{p,j}: 5 vectors, rank 5; lambda with invariant sum: nullity 1, |NC_2(k)| = 1  OK  [0.0s]
N=4 k=3: |NC_{2,1}(k)| = 4; rank T^1_p = 4, rank A_p = 4, rank of both = 4; x_p (s(p)>=1): 4 vectors, rank 4; y_{p,j}: 14 vectors, rank 14; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=4 k=4: |NC_{2,1}(k)| = 9; rank T^1_p = 9, rank A_p = 9, rank of both = 9; x_p (s(p)>=1): 7 vectors, rank 7; y_{p,j}: 42 vectors, rank 42; lambda with invariant sum: nullity 2, |NC_2(k)| = 2  OK  [0.0s]
N=4 k=5: |NC_{2,1}(k)| = 21; rank T^1_p = 21, rank A_p = 21, rank of both = 21; x_p (s(p)>=1): 21 vectors, rank 21; y_{p,j}: 132 vectors, rank 132; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=4 k=6: |NC_{2,1}(k)| = 51; rank T^1_p = 51, rank A_p = 51, rank of both = 51; x_p (s(p)>=1): 46 vectors, rank 46; y_{p,j}: 429 vectors, rank 429; lambda with invariant sum: nullity 5, |NC_2(k)| = 5  OK  [0.1s]
N=4 k=7: |NC_{2,1}(k)| = 127; rank T^1_p = 127, rank A_p = 127, rank of both = 127; x_p (s(p)>=1): 127 vectors, rank 127; y_{p,j}: 1430 vectors, rank 1430; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [1.3s]
N=4 k=8: |NC_{2,1}(k)| = 323; rank T^1_p = 323, rank A_p = 323, rank of both = 323; x_p (s(p)>=1): 309 vectors, rank 309; y_{p,j}: 4862 vectors, rank not computed; lambda with invariant sum: nullity 14, |NC_2(k)| = 14  OK  [20.5s]
N=5 k=0: |NC_{2,1}(k)| = 1; rank T^1_p = 1, rank A_p = 1, rank of both = 1; x_p (s(p)>=1): 0 vectors, rank 0; y_{p,j}: 1 vectors, rank 1; lambda with invariant sum: nullity 1, |NC_2(k)| = 1  OK  [0.0s]
N=5 k=1: |NC_{2,1}(k)| = 1; rank T^1_p = 1, rank A_p = 1, rank of both = 1; x_p (s(p)>=1): 1 vectors, rank 1; y_{p,j}: 2 vectors, rank 2; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=5 k=2: |NC_{2,1}(k)| = 2; rank T^1_p = 2, rank A_p = 2, rank of both = 2; x_p (s(p)>=1): 1 vectors, rank 1; y_{p,j}: 5 vectors, rank 5; lambda with invariant sum: nullity 1, |NC_2(k)| = 1  OK  [0.0s]
N=5 k=3: |NC_{2,1}(k)| = 4; rank T^1_p = 4, rank A_p = 4, rank of both = 4; x_p (s(p)>=1): 4 vectors, rank 4; y_{p,j}: 14 vectors, rank 14; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=5 k=4: |NC_{2,1}(k)| = 9; rank T^1_p = 9, rank A_p = 9, rank of both = 9; x_p (s(p)>=1): 7 vectors, rank 7; y_{p,j}: 42 vectors, rank 42; lambda with invariant sum: nullity 2, |NC_2(k)| = 2  OK  [0.0s]
N=5 k=5: |NC_{2,1}(k)| = 21; rank T^1_p = 21, rank A_p = 21, rank of both = 21; x_p (s(p)>=1): 21 vectors, rank 21; y_{p,j}: 132 vectors, rank 132; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=5 k=6: |NC_{2,1}(k)| = 51; rank T^1_p = 51, rank A_p = 51, rank of both = 51; x_p (s(p)>=1): 46 vectors, rank 46; y_{p,j}: 429 vectors, rank 429; lambda with invariant sum: nullity 5, |NC_2(k)| = 5  OK  [0.5s]
N=5 k=7: |NC_{2,1}(k)| = 127; rank T^1_p = 127, rank A_p = 127, rank of both = 127; x_p (s(p)>=1): 127 vectors, rank 127; y_{p,j}: 1430 vectors, rank not computed; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [8.1s]
N=6 k=0: |NC_{2,1}(k)| = 1; rank T^1_p = 1, rank A_p = 1, rank of both = 1; x_p (s(p)>=1): 0 vectors, rank 0; y_{p,j}: 1 vectors, rank 1; lambda with invariant sum: nullity 1, |NC_2(k)| = 1  OK  [0.0s]
N=6 k=1: |NC_{2,1}(k)| = 1; rank T^1_p = 1, rank A_p = 1, rank of both = 1; x_p (s(p)>=1): 1 vectors, rank 1; y_{p,j}: 2 vectors, rank 2; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=6 k=2: |NC_{2,1}(k)| = 2; rank T^1_p = 2, rank A_p = 2, rank of both = 2; x_p (s(p)>=1): 1 vectors, rank 1; y_{p,j}: 5 vectors, rank 5; lambda with invariant sum: nullity 1, |NC_2(k)| = 1  OK  [0.0s]
N=6 k=3: |NC_{2,1}(k)| = 4; rank T^1_p = 4, rank A_p = 4, rank of both = 4; x_p (s(p)>=1): 4 vectors, rank 4; y_{p,j}: 14 vectors, rank 14; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.0s]
N=6 k=4: |NC_{2,1}(k)| = 9; rank T^1_p = 9, rank A_p = 9, rank of both = 9; x_p (s(p)>=1): 7 vectors, rank 7; y_{p,j}: 42 vectors, rank 42; lambda with invariant sum: nullity 2, |NC_2(k)| = 2  OK  [0.0s]
N=6 k=5: |NC_{2,1}(k)| = 21; rank T^1_p = 21, rank A_p = 21, rank of both = 21; x_p (s(p)>=1): 21 vectors, rank 21; y_{p,j}: 132 vectors, rank 132; lambda with invariant sum: nullity 0, |NC_2(k)| = 0  OK  [0.1s]
N=6 k=6: |NC_{2,1}(k)| = 51; rank T^1_p = 51, rank A_p = 51, rank of both = 51; x_p (s(p)>=1): 46 vectors, rank 46; y_{p,j}: 429 vectors, rank 429; lambda with invariant sum: nullity 5, |NC_2(k)| = 5  OK  [2.1s]
