A186770 -id:A186770 - cp's OEIS frontend

A186769 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k nonincreasing even cycles (0<=k<=floor(n/4)). 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)

1, 1, 2, 6, 19, 5, 95, 25, 451, 269, 3157, 1883, 21092, 18353, 875, 189828, 165177, 7875, 1660351, 1764749, 203700, 18263861, 19412239, 2240700, 197541565, 237001478, 43736682, 721875, 2568040345, 3081019214, 568576866, 9384375, 33029787974, 43065489284, 10638945317, 444068625

Offset: 0

Emeric Deutsch, Feb 27 2011

Row n has 1+floor(n/4) entries.
Sum of entries in row n is n!.
T(n,0) = A186770(n).
Sum(k*T(n,k),k>=0) = A184958(n).

			T(4,1)=5 because we have (1243), (1324), (1342), (1423), and (1432).
Triangle starts:
1;
1;
2;
6;
19,5;
95,25;
451,269;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A186761, A186764, A186766, A186769, A186770, A184958.

g := exp((1-t)*(cosh(z)-1))*(1+z)^((1-t)*1/2)/(1-z)^((1+t)*1/2): gser := simplify(series(g, z = 0, 18)): for n from 0 to 14 do P[n] := sort(expand(factorial(n)*coeff(gser, z, n))) end do; for n from 0 to 14 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*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)*binomial(n-1, j-1)*
      `if`(j::odd, (j-1)!, 1+x*((j-1)!-1)), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..14);  # Alois P. Heinz, Apr 13 2017

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n-j]*Binomial[n - 1, j - 1]*If[ OddQ[j], (j - 1)!, 1 + x*((j - 1)! - 1)], {j, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, May 03 2017, after Alois P. Heinz *)

E.g.f.: G(t,z)=exp((1-t)(cosh z - 1))*(1+z)^{(1-t)/2}/(1-z)^{(1+t)/2}.
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}.
We have: G(t,z) = H(1,1,1,t,z).

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).

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A186769 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k nonincreasing even cycles (0<=k<=floor(n/4)). 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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Maple

Mathematica

Formula