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

1, 1, 2, 4, 16, 84, 536, 3912, 32256, 297072, 3026112, 33798720, 410826624, 5399704320, 76317546240, 1154312486400, 18604815528960, 318348065548800, 5763746405053440, 110086912964367360, 2212209395234979840, 46657233031296706560, 1030510550216174469120

Offset: 0

Author

Tom Roby, Jun 21 2012

Keywords

• nonn

Comments

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.

Examples

			From _Alois P. Heinz_, Jun 22 2012: (Start)
a(3) = 4: {123, 132, 321}, {213}, {231}, {312}.
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}.
a(5) = 84: {12345, 12354, 12435, 12543, 13245, 13254, 14325, 32145, 32154}, {12453}, ..., {53421}, {54213}, {54231}.
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450
• Anders Claesson, From Hertzsprung's problem to pattern-rewriting systems, University of Iceland (2020).
• S. Linton, J. Propp, T. Roby, and J. West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, arXiv:1111.3920, 2011 [math.CO], J. Int. Seq. 15 (2012) #12.9.1

Crossrefs

Cf. A212580, A212581.

Formula

From Seiichi Manyama, Feb 21 2024: (Start)

G.f.: Sum_{k>=0} k! * ( x * (1-2*x^2) )^k.

a(n) = Sum_{k=0..floor(n/3)} (-2)^k * (n-2*k)! * binomial(n-2*k,k). (End)

Extensions

a(9) from Alois P. Heinz, Jun 23 2012

a(10)-a(22) from Alois P. Heinz, Apr 14 2021