A212580 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb where a<b<c.

1, 1, 2, 5, 20, 102, 626, 4458, 36144, 328794, 3316944, 36755520, 443828184, 5800823880, 81591320880, 1228888215960, 19733475278880, 336551479543440, 6075437671458000, 115733952138747600, 2320138519554562560, 48827468196234035280, 1076310620915575933440

Offset

0,3

Comments

Also the number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> bac where a<b<c.

Also the number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> cba where a<b<c.

Also the number of permutations of [n] avoiding consecutive triples j, j+1, j-1. a(4) = 20 = 4! - 4 counts all permutations of [4] except 1342, 2314, 3421, 4231. - Alois P. Heinz, Apr 14 2021

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, 2011 arXiv:1111.3920 [math.CO], J. Int. Seq. 15 (2012) #12.9.1

Formula

From Seiichi Manyama, Feb 20 2024: (Start)

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

a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (n-2*k)! * binomial(n-2*k,k). (End)

Example

From Alois P. Heinz, May 22 2012: (Start)
a(3) = 5: {123, 132}, {213}, {231}, {312}, {321}.
a(4) = 20: {1234, 1243, 1324}, {1342}, {1423}, {1432}, {2134}, {2143}, {2314}, {2341, 2431}, {2413}, {3124}, {3142}, {3214}, {3241}, {3412}, {3421}, {4123, 4132}, {4213}, {4231}, {4312}, {4321}. (End)

Maple

b:= proc(s, x, y) option remember; `if`(s={}, 1, add(
      `if`(x=0 or x-y<>1 or j-x<>1, b(s minus {j}, y, j), 0), j=s))
    end:
a:= n-> b({$1..n}, 0$2):
seq(a(n), n=0..14);  # Alois P. Heinz, Apr 14 2021
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [1$2, 2, 5, 20][n+1],
       n*a(n-1)+3*a(n-2)-(2*n-2)*a(n-3)+(n-2)*a(n-5))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 14 2021

Mathematica

a[n_] := a[n] = If[n < 5, {1, 1, 2, 5, 20}[[n+1]],
     n*a[n-1] + 3*a[n-2] - (2n - 2)*a[n-3] + (n-2)*a[n-5]];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Apr 20 2022, after Alois P. Heinz *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*(1-x^2))^k)) \\ Seiichi Manyama, Feb 20 2024

Crossrefs

Cf. A174072, A210667, A210668, A210669, A210671, A212417, A212581.

Column k=0 of A343535.

Sequence in context: A009551 A323274 A006924 * A370669 A261779 A227096

Adjacent sequences: A212577 A212578 A212579 * A212581 A212582 A212583

Keyword

nonn

Author

Tom Roby, May 21 2012

Extensions

a(9)-a(22) from Alois P. Heinz, Apr 14 2021

Status

approved