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A212432 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> cba where a

%I A212432 #50 Feb 21 2024 08:18:25
%S A212432 1,1,2,4,16,84,536,3912,32256,297072,3026112,33798720,410826624,
%T A212432 5399704320,76317546240,1154312486400,18604815528960,318348065548800,
%U A212432 5763746405053440,110086912964367360,2212209395234979840,46657233031296706560,1030510550216174469120
%N A212432 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> cba where a<b<c.
%C A212432 Also number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> bac <--> cba where a < b < c.
%H A212432 Alois P. Heinz, <a href="/A212432/b212432.txt">Table of n, a(n) for n = 0..450</a>
%H A212432 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 A212432 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</a>, arXiv:1111.3920, 2011 [math.CO], <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Roby/roby4.html">J. Int. Seq. 15 (2012) #12.9.1</a>
%F A212432 From _Seiichi Manyama_, Feb 21 2024: (Start)
%F A212432 G.f.: Sum_{k>=0} k! * ( x * (1-2*x^2) )^k.
%F A212432 a(n) = Sum_{k=0..floor(n/3)} (-2)^k * (n-2*k)! * binomial(n-2*k,k). (End)
%e A212432 From _Alois P. Heinz_, Jun 22 2012: (Start)
%e A212432 a(3) = 4: {123, 132, 321}, {213}, {231}, {312}.
%e A212432 a(4) = 16: {1234, 1243, 1324, 1432, 3214}, {1342}, {1423}, {2134}, {2143}, {2314}, {2341, 2431, 4123, 4132, 4321}, {2413}, {3124}, {3142}, {3241}, {3412}, {3421}, {4213}, {4231}, {4312}.
%e A212432 a(5) = 84: {12345, 12354, 12435, 12543, 13245, 13254, 14325, 32145, 32154}, {12453}, ..., {53421}, {54213}, {54231}.
%e A212432 (End)
%Y A212432 Cf. A212580, A212581.
%K A212432 nonn
%O A212432 0,3
%A A212432 _Tom Roby_, Jun 21 2012
%E A212432 a(9) from _Alois P. Heinz_, Jun 23 2012
%E A212432 a(10)-a(22) from _Alois P. Heinz_, Apr 14 2021