A212417 -id:A212417 - cp's OEIS frontend

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

Tom Roby, May 21 2012

nonn

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

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

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

			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)

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

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

Column k=0 of A343535.

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

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 *)

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

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)

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

A210667 Number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> acb where a

Original entry on oeis.org

1, 1, 2, 5, 16, 62, 284, 1507, 9104, 61766

Offset: 0

Tom Roby, May 08 2012

nonn

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

			From _Alois P. Heinz_, May 16 2012: (Start)
a(3) = 5: {123, 132}, {213}, {231}, {312}, {321}.
a(4) = 16: {1234, 1243, 1324, 1423}, {1342, 1432}, {2134, 2143}, {2314}, {2341, 2431}, {2413}, {3124, 3142}, {3214}, {3241}, {3412}, {3421}, {4123, 4132}, {4213}, {4231}, {4312}, {4321}. (End)

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]

Cf. A210668, A210669, A210671, A212417.

Definition improved by Tom Roby, May 15 2012

a(0)-a(2), a(9) from Alois P. Heinz, May 16 2012

A210668 Number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> cba where a

Original entry on oeis.org

1, 1, 2, 5, 16, 60, 260, 1260, 6744, 39303

Offset: 0

Tom Roby, May 08 2012

nonn

			From _Alois P. Heinz_, May 16 2012: (Start)
a(3) = 5: {123, 321}, {132}, {213}, {231}, {312}.
a(4) = 16: {1234, 1432, 3214}, {1243, 4213}, {1324}, {1342, 4312}, {1423}, {2134, 2431}, {2143}, {2314}, {2341, 4123, 4321}, {2413}, {3124, 3421}, {3142}, {3241}, {3412}, {4132}, {4231}. (End)

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], J. Int. Seq. 15 (2012) #12.9.1

Cf. A210667, A210669, A210671, A212417.

Definition improved by Tom Roby, May 15 2012

a(0)-a(2), a(9) from Alois P. Heinz, May 16 2012

A210669 Number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> acb <--> cba where a

Original entry on oeis.org

1, 1, 2, 4, 8, 14, 27, 68, 159, 496

Offset: 0

Tom Roby, May 08 2012

nonn

Also number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> bac <--> cba where a

			From _Alois P. Heinz_, May 19 2012: (Start)
a(3) = 4: {123, 132, 321}, {213}, {231}, {312}.
a(4) = 8: {1234, 1243, 1324, 1342, 1423, 1432, 3214, 4213, 4312}, {2134, 2143, 2341, 2431, 4123, 4132, 4321}, {2314}, {2413}, {3124, 3142, 3421}, {3241}, {3412}, {4231}. (End)

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], J. Int. Seq. 15 (2012) #12.9.1

Cf. A210667, A210668, A210671, A212417, A212418.

Definition improved and comment added by Tom Roby, May 15 2012

A212581 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> bac where a

Original entry on oeis.org

1, 1, 2, 4, 17, 89, 556, 4011, 32843, 301210, 3059625, 34104275, 413919214, 5434093341, 76734218273, 1159776006262, 18681894258591, 319512224705645, 5782488507020050, 110407313135273127, 2218005876646727423, 46767874983437110354, 1032732727339665789981

Offset: 0

Tom Roby, May 21 2012

nonn

			From _Alois P. Heinz_, May 22 2012: (Start)
a(3) = 4: {123, 132, 213}, {231}, {312}, {321}.
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)

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

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

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

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

a(9) from Alois P. Heinz, May 22 2012

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

A278589 Number of equivalence classes under the fixed-point-free braid relation B_{FPF}(w_0).

Original entry on oeis.org

1, 2, 16, 320, 12448, 809792

Offset: 0

N. J. A. Sloane, Dec 02 2016

Z. Hamaker, E. Marberg, B. Pawlowski, Involution words II: braid relations and atomic structures, arXiv preprint arXiv:1601.02269, 2016. See Remark following Th. 6.18.

Cf. A212417, A278590.

A278590 a(n) = A278589(n)/2^n.

Original entry on oeis.org

1, 1, 4, 40, 778, 25306

Offset: 0

N. J. A. Sloane, Dec 02 2016

Z. Hamaker, E. Marberg, B. Pawlowski, Involution words II: braid relations and atomic structures, arXiv preprint arXiv:1601.02269, 2016. See Remark following Th. 6.18.

Cf. A212417, A278589.

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cp's OEIS Frontend

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

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A210667 Number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> acb where a

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A210668 Number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> cba where a

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A210669 Number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> acb <--> cba where a

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A212581 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> bac where a

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A278589 Number of equivalence classes under the fixed-point-free braid relation B_{FPF}(w_0).

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A278590 a(n) = A278589(n)/2^n.

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