A206703 Triangular array read by rows. T(n,k) is the number of partial permutations (injective partial functions) of {1,2,...,n} that have exactly k elements in a cycle. The k elements are not necessarily in the same cycle. A fixed point is considered to be in a cycle.

1, 1, 1, 3, 2, 2, 13, 9, 6, 6, 73, 52, 36, 24, 24, 501, 365, 260, 180, 120, 120, 4051, 3006, 2190, 1560, 1080, 720, 720, 37633, 28357, 21042, 15330, 10920, 7560, 5040, 5040, 394353, 301064, 226856, 168336, 122640, 87360, 60480, 40320, 40320

Offset: 0

Author

Geoffrey Critzer, Feb 11 2012

Keywords

• nonn
• tabl

Examples

			     1;
     1,     1;
     3,     2,     2;
    13,     9,     6,     6;
    73,    52,    36,    24,    24;
   501,   365,   260,   180,   120,  120;
  4051,  3006,  2190,  1560,  1080,  720,   720;
  ...
		

References

• Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
• Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60. Solution published in Vol. 39, No. 1, January 2008, pp. 66-67.

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132.

Crossrefs

Columns k = 0..1 give: A000262, A006152.

Main diagonal gives A000142.

Row sums give A002720.

T(2n,n) gives A088026.

Cf. A002720, A052852, A103194, A105219, A331725.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add((p-> p+x^j*
      coeff(p, x, 0))(b(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 19 2022

Mathematica

nn = 7; a = 1/(1 - x); ay = 1/(1 - y x); f[list_] := Select[list, # > 0 &]; Map[f, Range[0, nn]! CoefficientList[Series[Exp[a x] ay, {x, 0, nn}], {x, y}]] // Flatten

Formula

E.g.f.: exp(x/(1-x))/(1-y*x).

From Alois P. Heinz, Feb 19 2022: (Start)

Sum_{k=1..n} T(n,k) = A052852.

Sum_{k=0..n} k * T(n,k) = A103194(n).

Sum_{k=0..n} (n-k) * T(n,k) = A105219(n).

Sum_{k=0..n} (-1)^k * T(n,k) = A331725(n). (End)