cp's OEIS Frontend

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.

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