A186762 -id:A186762 - cp's OEIS frontend

A186763 Number of increasing odd cycles in all permutations of {1,2,...,n}.

0, 1, 2, 7, 28, 141, 846, 5923, 47384, 426457, 4264570, 46910271, 562923252, 7318002277, 102452031878, 1536780478171, 24588487650736, 418004290062513, 7524077221125234, 142957467201379447, 2859149344027588940, 60042136224579367741

Offset: 0

Emeric Deutsch, Feb 27 2011

nonn

A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

A cycle is said to be odd if it has an odd number of entries.

			a(3)=7 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), and (132) we have a total of 3+1+1+1+1+0=7 increasing odd cycles.

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000166, A000522, A009628, A186761, A186762, A186764, A186765, A186766, A186768, A186769, A184958.

g := sinh(z)/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
# Alternatively:
A186763 := n -> (exp(1)*GAMMA(1+n,1) - exp(-1)*GAMMA(1+n,-1))/2:
seq(simplify(A186763(n)), n=0..21); # Peter Luschny, Dec 18 2017

a=0;Table[a=n*a+(1/2)(1-(-1)^n),{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
CoefficientList[Series[Sinh[x]/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *)

a(n) = Sum_{k>=0} k*A186761(n,k).
E.g.f.: (sinh z)/(1-z).
a(n) ~ n! * (exp(2)-1)*exp(-1)/2. - Vaclav Kotesovec, Oct 05 2013
a(n) = (exp(1)*Gamma(1+n,1) - exp(-1)*Gamma(1+n,-1))/2 = (A000522(n) - A000166(n))/2. - Peter Luschny, Dec 18 2017
a(n) = n! * Sum_{k=0..floor((n-1)/2)} 1 / (2*k+1)!. - Ilya Gutkovskiy, Jul 16 2021
D-finite with recurrence a(n) -n*a(n-1) -a(n-2) +(n-2)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

A186761 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing odd cycles (0<=k<=n). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 9, 4, 10, 0, 1, 33, 56, 10, 20, 0, 1, 235, 218, 211, 20, 35, 0, 1, 1517, 1982, 833, 616, 35, 56, 0, 1, 12593, 14040, 9612, 2408, 1526, 56, 84, 0, 1, 111465, 134248, 72588, 35176, 5838, 3360, 84, 120, 0, 1, 1122819, 1305126, 797461, 276120, 107710, 12516, 6762, 120, 165, 0, 1

Offset: 0

Emeric Deutsch, Feb 27 2011

Sum of entries in row n is n!.
T(n,0) = A186762(n).
Sum_{k=0..n} k*T(n,k) = A186763(n).

			T(3,1)=4 because we have (1)(23), (12)(3), (13)(2), and (123).
T(4,1)=4 because we have (1)(243), (143)(2), (142)(3), and (132)(4).
Triangle starts:
   1;
   0,  1;
   1,  0,  1;
   1,  4,  0,  1;
   9,  4, 10,  0, 1;
  33, 56, 10, 20, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..170, flattened

Cf. A186762, A186763, A186764, A186766, A186769.

g := exp((t-1)*sinh(z))/(1-z): gser := simplify(series(g, z = 0, 13)): for n from 0 to 10 do P[n] := sort(expand(factorial(n)*coeff(gser, z, n))) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-j)*(
      `if`(j::odd, x-1, 0)+(j-1)!)*binomial(n-1, j-1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..14);  # Alois P. Heinz, May 12 2017

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - j]*(If[OddQ[j], x - 1, 0] + (j - 1)!)*Binomial[n - 1, j - 1], {j, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)

E.g.f.: G(t,z) = exp((t-1)*sinh z)/(1-z).
The 5-variate e.g.f. H(x,y,u,v,z) of permutations with respect to size (marked by z), number of increasing odd cycles (marked by x), number of increasing even cycles (marked by y), number of nonincreasing odd cycles (marked by u), and number of nonincreasing even cycles (marked by v), is given by
H(x,y,u,v,z)=exp(((x-u)sinh z + (y-v)(cosh z - 1))*(1+z)^{(u-v)/2}/(1-z)^{(u+v)/2}.

A186765 Number of permutations of {1,2,...,n} having no increasing even cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

Original entry on oeis.org

1, 1, 1, 3, 14, 70, 419, 2933, 23421, 210789, 2108144, 23189584, 278279165, 3617629145, 50646737049, 759701055735, 12155215581362, 206638664883154, 3719496008830391, 70670424167777429, 1413408484443295197, 29681578173309199137, 652994719769134284068

Offset: 0

Emeric Deutsch, Feb 27 2011

nonn

			a(3)=3 because we have (1)(2)(3), (132), and (123).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=0 of A186764.

Cf. A186761, A186762, A186766, A186767, A186769, A186770

g := exp(1-cosh(z))/(1-z); gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*((j-1)!+irem(j, 2)-1), j=1..n))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Feb 05 2025

CoefficientList[Series[E^(1-Cosh[x])/(1-x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 24 2014 *)

a(n):=((sum(sum(((-1)^k*sum(((sum((j-2*i)^m*binomial(j,i),i,0,j))*(-1)^(j-k)*binomial(k,j))/2^j,j,0,k))/k!,k,1,m)/m!,m,1,n))+1)*n!; /* Vladimir Kruchinin, Apr 25 2011 */

my(x='x+O('x^66)); Vec(serlaplace(exp(1-cosh(x))/(1-x))) /* Joerg Arndt, Apr 26 2011 */

a(n) = A186764(n,0).
E.g.f.: exp(1-cosh(z))/(1-z).
a(n) = ((sum(m=1..n,sum(k=1..m,((-1)^k*sum(j=0..k,((sum(i=0..j,(j-2*i)^m*binomial(j, i)))*(-1)^(j-k)*binomial(k, j))/2^j))/k!)/m!))+1)*n!. [Vladimir Kruchinin, Apr 25 2011]
a(n) ~ n! * exp(1-cosh(1)). - Vaclav Kotesovec, Feb 24 2014

A186767 Number of permutations of {1,2,...,n} having no nonincreasing odd cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

Original entry on oeis.org

1, 1, 2, 5, 20, 77, 472, 2585, 21968, 157113, 1724064, 15229645, 204738624, 2151199429, 34194201472, 416221515169, 7631627843840, 105565890206193, 2192501224174080, 33962775502534165, 787900686999286784, 13509825183288167869

Offset: 0

Emeric Deutsch, Feb 27 2011

nonn

a(n) = A186766(n,0).

			a(3)=5 because we have (1)(2)(3), (1)(23), (12)(3), (13)(2), and (123).

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

Cf. A186761, A186762, A186763, A186764, A186765, A186766, A186769, A186770.

g := exp(sinh(z))/sqrt(1-z^2): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*`if`(j::even, (j-1)!, 1), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 13 2017

a[n_] := a[n] = If[n==0, 1, Sum[a[n-j]*Binomial[n-1, j-1]*If[EvenQ[j], (j-1)!, 1], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 18 2017, after Alois P. Heinz *)

E.g.f.: g(z) = exp(sinh z)/sqrt(1-z^2).

A186770 Number of permutations of {1,2,...,n} having no nonincreasing even cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

Original entry on oeis.org

1, 1, 2, 6, 19, 95, 451, 3157, 21092, 189828, 1660351, 18263861, 197541565, 2568040345, 33029787974, 495446819610, 7377279473779, 125413751054243, 2120559951767503, 40290639083582557, 762353357154540584, 16009420500245352264

Offset: 0

Emeric Deutsch, Feb 27 2011

nonn

a(n) = A186769(n,0).

			a(4)=19 because among the 24 permutations of {1,2,3,4} only (1243), (1324), (1342), (1423), and (1432) have nonincreasing even cycles.

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

Cf. A186761, A186762, A186764, A186765, A186766, A186767, A186769.

g := exp(cosh(z)-1)*sqrt((1+z)/(1-z)): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*`if`(j::odd, (j-1)!, 1), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 13 2017

a[n_] := a[n] = If[n==0, 1, Sum[a[n-j]*Binomial[n-1, j-1]*If[OddQ[j], (j-1)!, 1], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 18 2017, after Alois P. Heinz *)

E.g.f.: g(z) = exp(cosh z - 1)*sqrt((1+z)/(1-z)).

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A186763 Number of increasing odd cycles in all permutations of {1,2,...,n}.

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A186761 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing odd cycles (0<=k<=n). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

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A186765 Number of permutations of {1,2,...,n} having no increasing even cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

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A186767 Number of permutations of {1,2,...,n} having no nonincreasing odd cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

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A186770 Number of permutations of {1,2,...,n} having no nonincreasing even cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

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