A186754 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing 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) < b(2) < b(3) < ... .

1, 0, 1, 0, 1, 1, 1, 1, 3, 1, 5, 5, 7, 6, 1, 23, 36, 25, 25, 10, 1, 129, 234, 166, 110, 65, 15, 1, 894, 1597, 1316, 686, 385, 140, 21, 1, 7202, 12459, 10893, 5754, 2611, 1106, 266, 28, 1, 65085, 111451, 97287, 54559, 22428, 8841, 2730, 462, 36, 1, 651263, 1116277, 963121, 554670, 229405, 80871, 26397, 6000, 750, 45, 1

Offset: 0

Author

Emeric Deutsch, Feb 26 2011

Keywords

• nonn
• tabl

Examples

			T(3,0) = 1 because we have (132).
T(4,2) = 7 because we have (1)(234), (13)(24), (12)(34), (123)(4), (124)(3), (134)(2), and (14)(23).
Triangle starts:
   1;
   0,  1;
   0,  1,  1;
   1,  1,  3,  1;
   5,  5,  7,  6,  1;
  23, 36, 25, 25, 10, 1;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-1 give: A186755, A349788.

Cf. A000142, A002627, A186755, A186756, A186757, A186758, A186759, A186760.

Programs

Maple

G := exp((t-1)*(exp(z)-1))/(1-z); Gser := simplify(series(G, z = 0, 16)): 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, j), j = 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-i)*binomial(n-1, i-1)*(x+(i-1)!-1), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Sep 25 2016

Mathematica

b[n_] := b[n] = Expand[If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-1]*(x + (i-1)! - 1), {i, 1, n}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 04 2016, after Alois P. Heinz *)

Formula

E.g.f.: G(t,z) = exp((t-1)*(exp(z)-1))/(1-z).

The 4-variate e.g.f. H(u,v,w,z) of the permutations of {1,2,...,n} with respect to size (marked by z), number of fixed points (marked by u), number of increasing cycles of length >=2 (marked by v), and number of nonincreasing cycles (marked by w) is given by H(u,v,w,z) = exp(u*z+v*(exp(z)-1-z)+w*(1-exp(z)))/(1-z)^w. Remark: the nonincreasing cycles are necessarily of length >=3. We have: G(t,z) = H(t,t,1,z).

Sum_{k=0..n} T(n,k) = n!.

T(n,0) = A186755(n).

Sum_{k=0..n} k*T(n,k) = A002627(n).