Number of classes

(a) (b) (c)
indices and values adjacent only indices adjacent neither adjacent ("classical")
(1) 123 ↔ 132 [ 5, 20, 102, 626, 4458, 36144, 328794, 3316944 ] [ 5, 16, 62, 284, 1507, 9104, 61766, 465208 ] [ 5, 14, 42, 132, 429, 1430 ]
Catalan
(2) 123 ↔ 213
(4) 123 ↔ 321 [ 5, 20, 102, 626, 4458, 36144, 328794, 3316944 ] [ 5, 16, 60, 260, 1260, 6744, 39303, 247892 ] [ 5, 10, 3, 1, 1, 1 ]
trivial
(3) 123 ↔ 132 ↔ 213 [ 4, 17, 89, 556, 4011, 32843, 301210, 3059625 ] [ 4, 10, 26, 76, 232, 764, 2620, 9496 ]
involutions		 [ 4, 8, 16, 32, 64, 128 ]
powers of 2
(5) 123 ↔ 132 ↔ 321 [ 4, 16, 84, 536, 3912, 32256, 297072, 3026112 ] [ 4, 8, 14, 27, 68, 159, 496, 1337 ] [ 4, 2, 1, 1, 1, 1 ]
trivial
(6) 123 ↔ 213 ↔ 321
(7) 123 ↔ 132 ↔ 213 ↔ 321 [ 3, 13, 71, 470, 3497, 29203, 271500, 2786711 ] [ 3, 4, 5, 8, 11, 20, 29, 57 ] [ 3, 2, 1, 1, 1, 1 ]
trivial

Size of class containing identity

(a) (b) (c)
indices and values adjacent only indices adjacent neither adjacent ("classical")
(1) 123 ↔ 132 [ 2, 3, 5, 8, 13, 21, 34, 55 ]
Fibonacci numbers		 [ 2, 4, 12, 36, 144, 576, 2880, 14400 ]
product of two factorials A010551		 [ 2, 6, 24, 120, 720, 5040 ]
(n-1)!
(2) 123 ↔ 213
(4) 123 ↔ 321 [ 2, 3, 4, 6, 9, 13, 19, 28 ]
A000930		 [ 2, 3, 6, 10, 20, 35, 70, 126 ]
central binomial coefficients		 [ 2, 4, 24, 720, 5040, 40320 ]
trivial
(3) 123 ↔ 132 ↔ 213 [ 3, 4, 8, 12, 21, 33, 55, 88 ]
A052952		 [ 3, 7, 35, 135, 945, 5193, 46737, 336825 ]
"many factors of 3"		 [ 3, 13, 71, 461, 3447, 29093 ]
connected permutations A003319
(5) 123 ↔ 132 ↔ 321 [ 3, 5, 9, 17, 31, 57, 105, 193 ]
A000213 (tribonacci numbers)		 [ 3, 9, 54, 285, 2160, 15825, 151200, 1411095 ]
proven for odd terms: (3/2) * ((n+1)/2)((n-1)/2) * (n-2)!		 [ 3, 23, 120, 720, 5040, 40320 ]
trivial
(6) 123 ↔ 213 ↔ 321
(7) 123 ↔ 132 ↔ 213 ↔ 321 [ 4, 6, 13, 23, 44, 80, 149, 273 ]
A000073 (tribonacci) - [ 2|n ]		 [ 4, 21, 116, 713, 5030, 40301, 362852, 3628744 ] [ 3, 23, 120, 720, 5040, 40320 ]
trivial