A186366 -id:A186366 - cp's OEIS frontend

A344855 Number T(n,k) of permutations of [n] having k cycles of the form (c1, c2, ..., c_m) where c1 = min_{i>=1} c_i and c_j = min_{i>=j} c_i or c_j = max_{i>=j} c_i; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 4, 11, 6, 1, 0, 8, 40, 35, 10, 1, 0, 16, 148, 195, 85, 15, 1, 0, 32, 560, 1078, 665, 175, 21, 1, 0, 64, 2160, 5992, 5033, 1820, 322, 28, 1, 0, 128, 8448, 33632, 37632, 17913, 4284, 546, 36, 1, 0, 256, 33344, 190800, 280760, 171465, 52941, 9030, 870, 45, 1

Offset: 0

Alois P. Heinz, May 30 2021

The sequence of column k satisfies a linear recurrence with constant coefficients of order k*(k+1)/2 = A000217(k).

			T(4,1) = 4: (1234), (1243), (1423), (1432).
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    1;
  0,  2,    3,    1;
  0,  4,   11,    6,    1;
  0,  8,   40,   35,   10,    1;
  0, 16,  148,  195,   85,   15,   1;
  0, 32,  560, 1078,  665,  175,  21,  1;
  0, 64, 2160, 5992, 5033, 1820, 322, 28, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Permutation

Columns k=0-10 give: A000007, A166444, A346317, A346318, A346319, A346320, A346321, A346322, A346323, A346324, A346325.
Row sums give A187251.
Main diagonal gives A000012, lower diagonal gives A000217, second lower diagonal gives A000914.
T(n+1,n) gives A000217.
T(n+2,n) gives A000914.
T(2n,n) gives A345342.
Cf. A060281, A132393, A186366, A345341.

b:= proc(n) option remember; `if`(n=0, 1, add(expand(x*
      b(n-j)*binomial(n-1, j-1)*ceil(2^(j-2))), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..12);

b[n_] := b[n] = If[n == 0, 1, Sum[Expand[x*b[n-j]*
     Binomial[n-1, j-1]*Ceiling[2^(j-2)]], {j, n}]];
T[n_] := CoefficientList[b[n], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A345341(n).

For fixed k, T(n,k) ~ (2*k)^n / (4^k * k!). - Vaclav Kotesovec, Jul 15 2021

A211602 Number of binary increasing trees with n nodes and "min-path" of length 3.

Original entry on oeis.org

0, 1, 3, 7, 20, 70, 287, 1356, 7248

Offset: 2

N. J. A. Sloane, May 11 2012

Filippo Disanto, André permutations of the second kind associated to strictly binary increasing trees and left to right minima in their sub-permutations, arXiv preprint arXiv:1202.1139 [math.CO], 2012-2014.

A diagonal of A186366.

A344445 Number of cycle-up-down permutations of [2n] having n cycles.

Original entry on oeis.org

1, 1, 7, 105, 2345, 69405, 2559667, 113073961, 5820788545, 342176336073, 22616620648895, 1660292619682697, 134029227728536985, 11800452870718122325, 1125324001129006580475, 115551341953019187183225, 12711056625162235880359425, 1491325482312555276046069905

Offset: 0

Alois P. Heinz, May 19 2021

nonn

For the definition of cycle-up-down permutations see A186366.

			a(2) = 7: (1)(243), (143)(2), (142)(3), (132)(4), (12)(34), (13)(24), (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..338
Wikipedia, Permutation

Cf. A007820, A186366, A344532.

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
g:= proc(n) option remember; expand(`if`(n=0, 1,
      add(g(n-j)*binomial(n-1, j-1)*x*b(j-1, 0), j=1..n)))
    end:
a:= n-> coeff(g(2*n), x, n):
seq(a(n), n=0..18);

Join[{1}, Table[Sum[2^(2*n - 2*j + 1) * StirlingS1[2*j,n] * Sum[(-1)^k * k^(2*n) / ((j+k)!*(j-k)!), {k, 0, j}], {j, Floor[n/2], n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 22 2021 *)

a(n) = (2n)! * [x^(2n) y^n] 1/(1-sin(x))^y.
a(n) = A186366(2n,n).
a(n) ~ c * d^n * (n-1)!, where d = 7.3270710411718682766548233722838416956334898839746535623751... and c = 0.14278148012337362269164226210064788025688590260058738... - Vaclav Kotesovec, May 22 2021

A344532 Number of cycle-up-down permutations of [n^2] having n cycles.

Original entry on oeis.org

1, 1, 7, 14698, 51629528080, 914192102910317528125, 199979553262025879510473132453855232, 1131253316618666789979709230473744963049785439771172168, 309491168658231587025767619097898747214052900521443034546657433273562730332160

Offset: 0

Alois P. Heinz, May 22 2021

nonn

For the definition of cycle-up-down permutations see A186366.

			a(2) = 7: (1)(243), (143)(2), (142)(3), (132)(4), (12)(34), (13)(24), (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..22
Wikipedia, Permutation

Cf. A186366, A218141, A344445.

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
g:= proc(n) option remember; expand(`if`(n=0, 1,
      add(g(n-j)*binomial(n-1, j-1)*x*b(j-1, 0), j=1..n)))
    end:
a:= n-> coeff(g(n^2), x, n):
seq(a(n), n=0..9);

b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]];
g[n_] := g[n] = Expand[If[n == 0, 1,
     Sum[g[n-j]*Binomial[n-1, j-1]*x*b[j-1, 0], {j, 1, n}]]];
a[n_] := Coefficient[g[n^2], x, n];
a /@ Range[0, 9] (* Jean-François Alcover, Jun 10 2021, after Alois P. Heinz *)

a(n) = (n^2)! * [x^(n^2) y^n] 1/(1-sin(x))^y.

a(n) = A186366(n^2,n).

A186367 Number of cycles in all cycle-up-down permutations of {1,2,...,n}. A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)b(3)<... .

Original entry on oeis.org

1, 3, 10, 38, 165, 812, 4478, 27408, 184529, 1356256, 10809786, 92892928, 856329253, 8430600960, 88292571934, 980197173248, 11499036105537, 142147625652224, 1846872283846922, 25161923756064768, 358706981125488581, 5340498034862030848

Offset: 1

Emeric Deutsch, Feb 28 2011

nonn

			a(3) = 10 because the cycle-up-down permutations (1)(2)(3), (12)(3), (13)(2), (1)(23), and (132), have a total of 3+2+2+2+1=10 cycles.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
E. Deutsch and S. Elizalde, Cycle up-down permutations, arXiv:0909.5199 [math.CO], 2009.

Cf. A186366.

g := -ln(1-sin(z))/(1-sin(z)): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 1 .. 22);

Rest[CoefficientList[Series[-Log[1-Sin[x]]/(1-Sin[x]), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 02 2013 *)

x='x+O('x^30); Vec(serlaplace(-log(1-sin(x))/(1-sin(x)))) \\ G. C. Greubel, Aug 30 2018

E.g.f.: -log(1-sin(z)) / (1-sin(z)).
a(n) = Sum_{k=1..n} k*A186366(n,k).
a(n) ~ n!*n*2^(n+3)/Pi^(n+2)*(2*log(n/Pi) + 2*gamma + 3*log(2) - 2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 02 2013

A212258 Number of binary increasing trees with n nodes and "min-path" of length 4.

Original entry on oeis.org

0, 0, 1, 6, 25, 105, 490, 2548, 14698, 93420, 649715, 4912776, 40154387, 352937312, 3320636540, 33305992320, 354819046132, 4001699525376, 47637151241125, 596958623741440, 7855611484697773, 108314507544748032, 1561635447992241230, 23498865431367684096

Offset: 2

N. J. A. Sloane, May 11 2012

nonn

Filippo Disanto, André permutations of the second kind associated to strictly binary increasing trees and left to right minima in their sub-permutations, arXiv preprint arXiv:1202.1139 [math.CO], 2012-2014.

A diagonal of A186366.

More terms from Alois P. Heinz, Apr 03 2014

A212259 Number of binary increasing trees with n nodes and "min-path" of length 5.

Original entry on oeis.org

0, 0, 0, 1, 10, 65, 385, 2345, 15204, 105880, 793210, 6382860, 55020966, 506505272, 4963812035, 51629528080, 568303728360, 6602266433920, 80751432154868, 1037402030622720, 13968636570706370, 196748236140538368, 2893482720437769317, 44355269272024284160

Offset: 2

N. J. A. Sloane, May 11 2012

nonn

Filippo Disanto, André permutations of the second kind associated to strictly binary increasing trees and left to right minima in their sub-permutations, arXiv preprint arXiv:1202.1139 [math.CO], 2012-2014.

A diagonal of A186366.

More terms from Alois P. Heinz, Apr 03 2014

cp's OEIS Frontend

A344855 Number T(n,k) of permutations of [n] having k cycles of the form (c1, c2, ..., c_m) where c1 = min_{i>=1} c_i and c_j = min_{i>=j} c_i or c_j = max_{i>=j} c_i; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A211602 Number of binary increasing trees with n nodes and "min-path" of length 3.

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A344445 Number of cycle-up-down permutations of [2n] having n cycles.

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A344532 Number of cycle-up-down permutations of [n^2] having n cycles.

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A186367 Number of cycles in all cycle-up-down permutations of {1,2,...,n}. A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)b(3)<... .

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A212258 Number of binary increasing trees with n nodes and "min-path" of length 4.

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A212259 Number of binary increasing trees with n nodes and "min-path" of length 5.

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