A212418 Size of the equivalence class of S_n containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> acb <--> cba where a<b<c.

1, 1, 1, 3, 9, 54, 285, 2160, 15825, 151200, 1411095, 16329600, 185067855, 2514758400, 33530101605, 523069747200, 8020402655265, 141228831744000, 2447966414868975, 48017802792960000, 928344187296100575, 20071441567457280000, 428190753438433910925

Offset

0,4

Comments

Also size of the equivalence class of S_n containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> bac <--> cba where a<b<c.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Steven Linton, James Propp, Tom Roby, and Julian West, Equivalence classes of permutations under various relations generated by constrained transpositions, 2011 arXiv:1111.3920 [math.CO]

Formula

a(n) = 1 for n<3, otherwise: a(2k+1) = (3/2)*k*(k+1)*(2k-1)!, a(2k) = (3/2)*k*(k-1/3)*(2k-2)!-(2k-3)!!.

Example

From Alois P. Heinz, May 21 2012: (Start)
a(3) = 3: {123, 132, 321}.
a(4) = 9: {1234, 1243, 1324, 1342, 1423, 1432, 3214, 4213, 4312}. (End)

Maple

a:= proc(n) local k;
       k:= iquo(n, 2, 'r');
      `if`(n<3, 1, `if`(r=0, (3/2)*k*(k-1/3)*(2*k-2)!
       -doublefactorial(2*k-3), (3/2)*k*(k+1)*(2*k-1)!))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 20 2012

Mathematica

a[n_ /; n < 3] = 1;
a[n_?OddQ] := With[{k = (n - 1)/2}, (3/2)*k*(k + 1)*(2 k - 1)!];
a[n_?EvenQ] := With[{k = n/2}, (3/2)*k*(k - 1/3)*(2 k - 2)! - (2 k - 3)!!];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 07 2017 *)

Crossrefs

Cf. A210669.

Sequence in context: A261397 A238906 A363442 * A337039 A025226 A001194

Adjacent sequences: A212415 A212416 A212417 * A212419 A212420 A212421

Keyword

nonn

Author

Tom Roby, May 15 2012

Status

approved