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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 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). 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. 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

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Last modified December 2 18:40 EST 2022. Contains 358510 sequences. (Running on oeis4.)