

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.


6



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
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OFFSET

0,4


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. 3744. 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. 6667.


LINKS

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


FORMULA

E.g.f.: exp(x/(1x))/(1y*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} (nk) * T(n,k) = A105219(n).
Sum_{k=0..n} (1)^k * T(n,k) = A331725(n). (End)


EXAMPLE

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;
...


MAPLE

b:= proc(n) option remember; `if`(n=0, 1, add((p> p+x^j*
coeff(p, x, 0))(b(nj)*binomial(n1, j1)*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


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.
Sequence in context: A143175 A074248 A266004 * A122101 A108032 A053370
Adjacent sequences: A206700 A206701 A206702 * A206704 A206705 A206706


KEYWORD

nonn,tabl


AUTHOR

Geoffrey Critzer, Feb 11 2012


STATUS

approved



