A186759 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k cycles that are either nonincreasing or of length 1 (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, 1, 0, 1, 1, 4, 0, 1, 4, 9, 10, 0, 1, 11, 53, 35, 20, 0, 1, 41, 280, 268, 95, 35, 0, 1, 162, 1804, 1904, 903, 210, 56, 0, 1, 715, 12971, 15727, 8008, 2408, 406, 84, 0, 1, 3425, 104600, 142533, 80323, 25662, 5502, 714, 120, 0, 1, 17722, 936370, 1418444, 871575, 303385, 68712, 11256, 1170, 165, 0, 1

Offset: 0

Author

Emeric Deutsch, Feb 26 2011

Keywords

• nonn
• tabl

Examples

			T(3,1) = 4 because we have (1)(23), (12)(3), (13)(2), and (132).
T(4,4) = 1 because we have (1)(2)(3)(4).
Triangle starts:
   1;
   0,  1;
   1,  0,  1;
   1,  4,  0,  1;
   4,  9, 10,  0, 1;
  11, 53, 35, 20, 0, 1;
		

Links

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

Crossrefs

Cf. A000296, A186754, A186755, A186756, A186757, A186758, A186760.

Programs

Maple

G := exp((1-t)*(exp(z)-1-z))/(1-z)^t: 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, 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)*
      `if`(i=1, x, 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]*If[i == 1, x, 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, Nov 28 2016 after Alois P. Heinz *)

Formula

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

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,1,t,z).

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

T(n,0) = A000296(n).

Sum_{k=0..n} k*T(n,k) = A186760(n).