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%I #39 Feb 19 2022 18:46:55
%S 1,1,1,3,2,2,13,9,6,6,73,52,36,24,24,501,365,260,180,120,120,4051,
%T 3006,2190,1560,1080,720,720,37633,28357,21042,15330,10920,7560,5040,
%U 5040,394353,301064,226856,168336,122640,87360,60480,40320,40320
%N 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.
%D 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.
%D 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.
%H Alois P. Heinz, <a href="/A206703/b206703.txt">Rows n = 0..140, flattened</a>
%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 132.
%F E.g.f.: exp(x/(1-x))/(1-y*x).
%F From _Alois P. Heinz_, Feb 19 2022: (Start)
%F Sum_{k=1..n} T(n,k) = A052852.
%F Sum_{k=0..n} k * T(n,k) = A103194(n).
%F Sum_{k=0..n} (n-k) * T(n,k) = A105219(n).
%F Sum_{k=0..n} (-1)^k * T(n,k) = A331725(n). (End)
%e 1;
%e 1, 1;
%e 3, 2, 2;
%e 13, 9, 6, 6;
%e 73, 52, 36, 24, 24;
%e 501, 365, 260, 180, 120, 120;
%e 4051, 3006, 2190, 1560, 1080, 720, 720;
%e ...
%p b:= proc(n) option remember; `if`(n=0, 1, add((p-> p+x^j*
%p coeff(p, x, 0))(b(n-j)*binomial(n-1, j-1)*j!), j=1..n))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
%p seq(T(n), n=0..10); # _Alois P. Heinz_, Feb 19 2022
%t 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
%Y Columns k = 0..1 give: A000262, A006152.
%Y Main diagonal gives A000142.
%Y Row sums give A002720.
%Y T(2n,n) gives A088026.
%Y Cf. A002720, A052852, A103194, A105219, A331725.
%K nonn,tabl
%O 0,4
%A _Geoffrey Critzer_, Feb 11 2012
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