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A060281 Triangle T(n,k) read by rows giving number of labeled mappings (or functional digraphs) from n points to themselves (endofunctions) with exactly k cycles, k=1..n. 24

%I #55 Dec 17 2021 12:09:10

%S 1,3,1,17,9,1,142,95,18,1,1569,1220,305,30,1,21576,18694,5595,745,45,

%T 1,355081,334369,113974,18515,1540,63,1,6805296,6852460,2581964,

%U 484729,49840,2842,84,1,148869153,158479488,64727522,13591116,1632099,116172,4830,108,1

%N Triangle T(n,k) read by rows giving number of labeled mappings (or functional digraphs) from n points to themselves (endofunctions) with exactly k cycles, k=1..n.

%C Also called sagittal graphs.

%C T(n,k)=1 iff n=k (counts the identity mapping of [n]). - _Len Smiley_, Apr 03 2006

%C Also the coefficients of the tree polynomials t_{n}(y) defined by (1-T(z))^(-y) = Sum_{n>=0} t_{n}(y) (z^n/n!) where T(z) is Cayley's tree function T(z) = Sum_{n>=1} n^(n-1) (z^n/n!) giving the number of labeled trees A000169. - _Peter Luschny_, Mar 03 2009

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

%D W. Szpankowski. Average case analysis of algorithms on sequences. John Wiley & Sons, 2001. - _Peter Luschny_, Mar 03 2009

%H Alois P. Heinz, <a href="/A060281/b060281.txt">Rows n = 1..141, flattened</a>

%H Julia Handl and Joshua Knowles, <a href="http://dx.doi.org/10.1007/11844297_85">An Investigation of Representations and Operators for Evolutionary Data Clustering with a Variable Number of Clusters</a>, in Parallel Problem Solving from Nature-PPSN IX, Lecture Notes in Computer Science, Volume 4193/2006, Springer-Verlag. [From _N. J. A. Sloane_, Jul 09 2009]

%H D. E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, The Mathematica J., 2 (1992), 67-78.

%H D. E. Knuth and B. Pittel, <a href="http://www.jstor.org/stable/2046947">A recurrence related to trees</a>, Proceedings of the American Mathematical Society, 105(2):335-349, 1989. [From _Peter Luschny_, Mar 03 2009]

%H J. Riordan, <a href="http://dx.doi.org/10.1214/aoms/1177704722">Enumeration of Linear Graphs for Mappings of Finite Sets</a>, Ann. Math. Stat., 33, No. 1, Mar. 1962, pp. 178-185.

%H David M. Smith and Geoffrey Smith, <a href="https://doi.org/10.1109/CSF.2017.18">Tight Bounds on Information Leakage from Repeated Independent Runs</a>, 2017 IEEE 30th Computer Security Foundations Symposium (CSF).

%F E.g.f.: 1/(1 + LambertW(-x))^y.

%F T(n,k) = Sum_{j=0..n-1} C(n-1,j)*n^(n-1-j)*(-1)^(k+j+1)*A008275(j+1,k) = Sum_{j=0..n-1} binomial(n-1,j)*n^(n-1-j)*s(j+1,k). [Riordan] (Note: s(m,p) denotes signless Stirling cycle number (first kind), A008275 is the signed triangle.) - _Len Smiley_, Apr 03 2006

%F From _Alois P. Heinz_, Dec 17 2021: (Start)

%F Sum_{k=1..n} k * T(n,k) = A190314(n).

%F Sum_{k=1..n} (-1)^(k+1) * T(n,k) = A000169(n) for n>=1. (End)

%e Triangle T(n,k) begins:

%e : 1;

%e : 3, 1;

%e : 17, 9, 1;

%e : 142, 95, 18, 1;

%e : 1569, 1220, 305, 30, 1;

%e : 21576, 18694, 5595, 745, 45, 1;

%e : 355081, 334369, 113974, 18515, 1540, 63, 1;

%e : 6805296, 6852460, 2581964, 484729, 49840, 2842, 84, 1;

%e : ...

%e T(3,2)=9: (1,2,3)--> [(2,1,3),(3,2,1),(1,3,2),(1,1,3),(1,2,1), (1,2,2),(2,2,3),(3,2,3),(1,3,3)].

%e From _Peter Luschny_, Mar 03 2009: (Start)

%e Tree polynomials (with offset 0):

%e t_0(y) = 1;

%e t_1(y) = y;

%e t_2(y) = 3y + y^2;

%e t_3(y) = 17y + 9y^2 + y^3; (End)

%p with(combinat):T:=array(1..8,1..8):for m from 1 to 8 do for p from 1 to m do T[m,p]:=sum(binomial(m-1,k)*m^(m-1-k)*(-1)^(p+k+1)*stirling1(k+1,p),k=0..m-1); print(T[m,p]) od od; # _Len Smiley_, Apr 03 2006

%p From _Peter Luschny_, Mar 03 2009: (Start)

%p T := z -> sum(n^(n-1)*z^n/n!,n=1..16):

%p p := convert(simplify(series((1-T(z))^(-y),z,12)),'polynom'):

%p seq(print(coeff(p,z,i)*i!),i=0..8); (End)

%t t=Sum[n^(n-1) x^n/n!,{n,1,10}];

%t Transpose[Table[Rest[Range[0, 10]! CoefficientList[Series[Log[1/(1 - t)]^n/n!, {x, 0, 10}], x]], {n,1,10}]]//Grid (* _Geoffrey Critzer_, Mar 13 2011*)

%t Table[k! SeriesCoefficient[1/(1 + ProductLog[-t])^x, {t, 0, k}, {x, 0, j}], {k, 10}, {j, k}] (* _Jan Mangaldan_, Mar 02 2013 *)

%Y Columns k=1-10 give: A001865, A065456, A273434, A273435, A273436, A273437, A273438, A273439, A273440, A273441.

%Y Row sums: A000312.

%Y Main diagonal and first lower diagonal give: A000012, A045943.

%Y T(2n,n) gives A273442.

%Y Cf. A000169, A190314, A242027, A209324, A347999.

%K easy,nonn,tabl

%O 1,2

%A _Vladeta Jovovic_, Apr 09 2001

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