PASS Lemma 2.1(a) on all permutations of length <= 8: the components at the cuts are indecomposable and reassemble pi
PASS Lemma 2.1(a) uniqueness, by exhaustive search of all decompositions into indecomposables (length <= 6)
PASS Lemma 2.1(b): no permutation of length 2..8 is both sum- and skew-decomposable
PASS Lemma 2.2 (both forms) on all pi of length <= 7 and all rho of length <= 4 contained in pi  [219534 pairs]
PASS separable counts n <= 9 are the large Schroeder numbers (control)
PASS control: the tree agrees with a direct test of all permutations for n <= 8
PASS Lemma 2.3: every separable permutation of length 2..9 has a sum cut or a skew cut
PASS the closure of {1} under direct and skew sums equals Av(2413,3142) for n <= 9
PASS 2413 and 3142 are sum- and skew-indecomposable (no cut)
PASS the sets {pi(1..c)}, c=1,2,3, are {2},{2,4},{1,2,4} for 2413 and {3},{1,3},{1,3,4} for 3142 (as printed)
PASS the sets {pi(1..c)} for 2341 are {2},{2,3},{2,3,4}; 2341, 2413, 3142 are sum-indecomposable
PASS Section 5.3: the simple separable permutations of length <= 9 are 1, 12, 21
PASS |C_n| for n <= 9 (separable perms avoiding 2341)
PASS Theorem 1.1(a): 2341 is separable and not in C
PASS Theorem 1.1(a): 231 occurs in 2341, 2413, 3142 at the entries 2,3,1 / 2,4,1 / 3,4,2 (as printed)
PASS Av(231) is contained in C for n <= 8
PASS Lemma 3.1: sigma + tau in C for all sigma, tau in C with |sigma|+|tau| <= 9
PASS Lemma 3.2 on all 13143 skew-decomposable permutations of length <= 8
PASS Lemma 3.2 on all 26232 skew-decomposable separable permutations of length <= 9
PASS 2341 = 123 skew-sum 1, and the skew components of 2341 are 123 and 1
PASS Lemma 3.3: separable skew-indecomposable 123-avoiders are 1 and delta_a + delta_b, distinct, in C (n <= 9)
PASS members of C of length >= 2: skew-indecomposable iff sum-decomposable, sum-indecomposable iff skew-decomposable
    (1-z-sqrt(1-6z+z^2))/(2z) = ['1', '2', '6', '22', '90', '394', '1806', '8558', '41586', '206098']
PASS Section 1.1: (1-z-sqrt(1-6z+z^2))/(2z) = 1+2z+6z^2+22z^3+90z^4+..., coefficient of z^n = #separable of length n+1 (n <= 8)
PASS Section 5.1: 2/(1+z+sqrt(1-6z+5z^2)) = 1,1,2,6,21,79,311,1265,5275,22431,96900,... and equals the count of Av(2143,2413,3142) for n <= 9
PASS Section 5.1: z(2-z) g^2 - (1+z) g + 1 = 0 on the series
PASS Section 5.1: 213 occurs in 2143, 2413, 3142 at the entries 2,1,4 / 2,1,3 / 3,1,4 (as printed)
PASS Section 5.4: each of 132, 213, 231, 312 occurs in 2413 and 3142 at the printed entries
PASS Section 5.4: 123 and 321 avoid 132, 213, 231, 312
PASS Section 5.4: Av(2413,3142,beta) = Av(beta) is counted by the Catalan numbers for beta in {132,213,231,312}
PASS the symmetries of 231 are exactly 132, 213, 231, 312
    |Av_n(123,2413,3142)| = [1, 2, 5, 12, 28, 65, 151, 351, 816]
PASS Av(2413,3142,beta) has at most one permutation of each length n >= 1 for beta in {1, 12, 21}
PASS Section 5.2: each of 1324, 2413, 3142 contains 132, so Av(132) is contained in Av(1324,2413,3142)
PASS Section 5.2: CMS Table 1 case 221 formula 1+(1-2x)/(2(1-x))(1/sqrt(1-4x)-1) counts Av(1324,2413,3142) for n <= 9  [[1, 1, 2, 6, 21, 77, 287, 1079, 4082, 15522]]

33 checks, 0 failed
ALL CHECKS PASSED
