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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. 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 (list; table; graph; refs; listen; history; text; internal format)
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. 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
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132.
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)
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(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
CROSSREFS
Columns k = 0..1 give: A000262, A006152.
Main diagonal gives A000142.
Row sums give A002720.
T(2n,n) gives A088026.
Sequence in context: A368557 A074248 A266004 * A122101 A108032 A053370
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Feb 11 2012
STATUS
approved

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Last modified April 14 17:14 EDT 2024. Contains 371666 sequences. (Running on oeis4.)