%I #37 Feb 21 2024 08:27:46

%S 1,1,2,4,17,89,556,4011,32843,301210,3059625,34104275,413919214,

%T 5434093341,76734218273,1159776006262,18681894258591,319512224705645,

%U 5782488507020050,110407313135273127,2218005876646727423,46767874983437110354,1032732727339665789981

%N Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> bac where a<b<c.

%H Alois P. Heinz, <a href="/A212581/b212581.txt">Table of n, a(n) for n = 0..450</a>

%H Anders Claesson, <a href="https://akc.is/papers/036-From-Hertzsprungs-problem-to-pattern-rewriting-systems.pdf">From Hertzsprung's problem to pattern-rewriting systems</a>, University of Iceland (2020).

%H S. Linton, J. Propp, T. Roby, and J. West, <a href="http://arxiv.org/abs/1111.3920">Equivalence classes of permutations under various relations generated by constrained transpositions, 2011</a> arXiv:1111.3920 [math.CO] <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Roby/roby4.html">J. Int. Seq. 15 (2012) #12.9.1</a>

%F G.f.: Sum_{k>=0} k! * ( x * (1-x^2)^2/(1-x^3) )^k. - _Seiichi Manyama_, Feb 20 2024

%e From _Alois P. Heinz_, May 22 2012: (Start)

%e a(3) = 4: {123, 132, 213}, {231}, {312}, {321}.

%e a(4) = 17: {1234, 1243, 1324, 2134}, {1342}, {1423}, {1432}, {2143}, {2314}, {2341, 2431, 3241}, {2413}, {3124}, {3142}, {3214}, {3412}, {3421}, {4123, 4132, 4213}, {4231}, {4312}, {4321}. (End)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*(1-x^2)^2/(1-x^3))^k)) \\ _Seiichi Manyama_, Feb 20 2024

%Y Cf. A210667, A210668, A210669, A210671, A212417, A212580.

%K nonn

%O 0,3

%A _Tom Roby_, May 21 2012

%E a(9) from _Alois P. Heinz_, May 22 2012

%E a(10)-a(22) from _Alois P. Heinz_, Apr 14 2021