A212580 -id:A212580 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

Tom Roby, Jun 21 2012

nonn

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.

			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)

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

Cf. A212580, A212581.

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)

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

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

A370508 Expansion of Sum_{k>=0} k! * ( x * (1-x^3) )^k.

Original entry on oeis.org

1, 1, 2, 6, 23, 116, 702, 4944, 39722, 358578, 3593664, 39595440, 475746474, 6190838544, 86740334160, 1301939398080, 20842001737224, 354469125185880, 6382790173842480, 121310821042966800, 2426863248540057480, 50975836645480342560, 1121691979824460425360

Offset: 0

Seiichi Manyama, Feb 20 2024

Cf. A013999, A212580.

Cf. A370510.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*(1-x^3))^k))

a(n) = sum(k=0, n\4, (-1)^k*(n-3*k)!*binomial(n-3*k, k));

a(n) = Sum_{k=0..floor(n/4)} (-1)^k * (n-3*k)! * binomial(n-3*k,k).

A370509 Expansion of Sum_{k>=0} k! * ( x * (1+x^2) )^k.

Original entry on oeis.org

1, 1, 2, 7, 28, 138, 818, 5658, 44784, 399366, 3962256, 43289760, 516432984, 6679346280, 93091875120, 1390851720840, 22175338353120, 375794883339120, 6745177713093840, 127830886641354960, 2550687440585679360, 53451172032327664560, 1173650135526055272960

Offset: 0

Seiichi Manyama, Feb 20 2024

Cf. A240172, A370510.

Cf. A212580.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*(1+x^2))^k))

a(n) = sum(k=0, n\3, (n-2*k)!*binomial(n-2*k, k));

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

A212433 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> bac <--> cba, where a

Original entry on oeis.org

1, 1, 2, 3, 13, 71, 470, 3497, 29203, 271500, 2786711, 31322803, 382794114, 5054810585, 71735226535, 1088920362030, 17607174571553, 302143065676513, 5484510055766118, 104999034898520903, 2114467256458136473, 44682676397748896010, 988663144904696100347

Offset: 0

Tom Roby, Jun 21 2012

nonn

			From _Alois P. Heinz_, Jun 23 2012: (Start)
a(3) = 3: {123, 132, 213, 321}, {231}, {312}.
a(4) = 13: {1234, 1243, 1324, 1432, 2134, 3214}, {1342}, {1423}, {2143}, {2314}, {2341, 2431, 3241, 4123, 4132, 4213, 4321}, {2413}, {3124}, {3142}, {3412}, {3421}, {4231}, {4312}.
a(5) = 71: {12345, 12354, 12435, 12543, 13245, 13254, 14325, 21345, 21354, 21435, 21543, 32145, 32154}, {12453}, ..., {53412}, {53421}, {54231}.
(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, arXiv:1111.3920, 2011 [math.CO]

Cf. A212432, A212580, A212581.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*((1-x^2)^2/(1-x^3)-x^2))^k)) \\ Seiichi Manyama, Feb 25 2024

G.f.: Sum_{k>=0} k! * ( x * ((1-x^2)^2/(1-x^3) - x^2) )^k. - Seiichi Manyama, Feb 25 2024

a(8)-a(9) from Alois P. Heinz, Jun 23 2012

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

A343535 Number T(n,k) of permutations of [n] having exactly k consecutive triples j, j+1, j-1; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 20, 4, 102, 18, 626, 92, 2, 4458, 564, 18, 36144, 4032, 144, 328794, 32898, 1182, 6, 3316944, 301248, 10512, 96, 36755520, 3057840, 102240, 1200, 443828184, 34073184, 1085904, 14304, 24, 5800823880, 413484240, 12538080, 174000, 600, 81591320880

Offset: 0

Alois P. Heinz, Apr 18 2021

Terms in column k are multiples of k!.

			T(4,1) = 4: 1342, 2314, 3421, 4231.
Triangle T(n,k) begins:
              1;
              1;
              2;
              5,           1;
             20,           4;
            102,          18;
            626,          92,          2;
           4458,         564,         18;
          36144,        4032,        144;
         328794,       32898,       1182,        6;
        3316944,      301248,      10512,       96;
       36755520,     3057840,     102240,     1200;
      443828184,    34073184,    1085904,    14304,     24;
     5800823880,   413484240,   12538080,   174000,    600;
    81591320880,  5428157760,  156587040,  2214720,  10800;
  1228888215960, 76651163160, 2105035440, 29777520, 175800, 120;
  ...

Anders Claesson, From Hertzsprung's problem to pattern-rewriting systems, University of Iceland (2020).
Wikipedia, Permutation

Column k=0 gives A212580.
Row sums give A000142.
Cf. A047921, A123513, A197365, A216716.

b:= proc(s, l, t) option remember; `if`(s={}, 1, add((h->
      expand(b(s minus {j}, j, `if`(h=1, 2, 1))*
     `if`(t=2 and h=-2, x, 1)))(j-l), j=s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
               b({$1..n}, -1, 1)):
seq(T(n), n=0..13);

b[s_, l_, t_] := b[s, l, t] = If[s == {}, 1, Sum[Function[h,
     Expand[b[s ~Complement~ {j}, j, If[h == 1, 2, 1]]*
     If[t == 2 && h == -2, x, 1]]][j - l], {j, s}]];
T[n_] := CoefficientList[b[Range[n], -1, 1], x];
T /@ Range[0, 13] // Flatten (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)

T(3n,n) = n!.

A370669 Expansion of Sum_{k>=0} k! * ( x/(1+x^2) )^k.

Original entry on oeis.org

1, 1, 2, 5, 20, 103, 630, 4475, 36232, 329341, 3320890, 36787889, 444125628, 5803850515, 81625106990, 1229298774647, 19738870726160, 336627732586105, 6076590994501938, 115752541255203869, 2320456607696181220, 48833227436258924671, 1076420625931284514342

Offset: 0

Seiichi Manyama, Feb 25 2024

nonn

Cf. A000255, A370670.

Cf. A177249, A212580, A276080.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x/(1+x^2))^k))

a(n) = sum(k=0, n\2, (-1)^k*(n-2*k)!*binomial(n-k-1, k));

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

a(n) = n*a(n-1) + (n-4)*a(n-3) + a(n-4) for n > 4.

cp's OEIS Frontend

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

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A370508 Expansion of Sum_{k>=0} k! * ( x * (1-x^3) )^k.

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A370509 Expansion of Sum_{k>=0} k! * ( x * (1+x^2) )^k.

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A212433 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> bac <--> cba, where a

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A343535 Number T(n,k) of permutations of [n] having exactly k consecutive triples j, j+1, j-1; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.

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A370669 Expansion of Sum_{k>=0} k! * ( x/(1+x^2) )^k.

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PARI

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