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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums = A002720.

Column for k = 0, A000262.

Column for k = 1, A006152.

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.

FORMULA

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

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;

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

Cf. A002720.

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

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Last modified January 26 20:11 EST 2020. Contains 331288 sequences. (Running on oeis4.)