OFFSET
1,3
COMMENTS
Specialization of Tutte polynomials for complete graphs. See the Gessel and Sagan paper. - Tom Copeland, Jan 17 2017
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
J. W. Moon, Counting labelled trees, Canad. Math. Monographs No 1 (1970) Section 4.5.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.48.
LINKS
Alois P. Heinz, Rows n = 1..50, flattened
I. Gessel and B. Sagan, The Tutte polynomial of a graph, depth-first search, and simplicial complex partitions, The Elect. Jrn. of Comb., Vol. 3, Issue 2, 1996.
I. M. Gessel, B. E. Sagan, Y.-N. Yeh, Enumeration of trees by inversions, J. Graph Theory 19 (4) (1995) 435-459
C. L. Mallows, J. Riordan, The inversion enumerator for labeled trees, Bull. Am. Math. Soc. 74 (1) (1968) 92-94, eq. (5)
FORMULA
Sum_{k=0..binomial(n-1,2)} T(n,k) = A000272(n).
Sum_{k=0..binomial(n-1,2)} (-1)^k*T(n,k) = A000111(n-1).
E.g.f.: (y-1)*log(Sum_{n>=0} (y-1)^(-n)*y^binomial(n, 2)*x^n/n!).
Sum_{k=0..binomial(n-1,2)} k*T(n,k) = A057500(n). - Alois P. Heinz, Nov 29 2015
Equals the coefficient [x^t] of the polynomial J_n(x) which satisfies sum_{>=0} J_{n+1}(x)*y^n/n! = exp[ sum_{n>=1} J_n(x) (x^n-1)/(x-1)*y^n/n!]. - R. J. Mathar, Jul 02 2018
EXAMPLE
1 : 1;
2 : 1;
3 : 2, 1;
4 : 6, 6, 3, 1;
5 : 24, 36, 30, 20, 10, 4, 1;
6 : 120, 240, 270, 240, 180, 120, 70, 35, 15, 5, 1;
7 : 720, 1800, 2520, 2730, 2520, 2100, 1610, 1140, 750, 455, 252, 126, 56, 21, 6, 1;
...
MAPLE
for n from 2 to 10 do
add( J[i]*(x^i-1)/(x-1)*y^i/i! , i=1..n-1) ;
exp(%) ;
coeftayl(%, y=0, n-1)*(n-1)! ;
expand(%) ;
J[n] := factor(convert(%, polynom)) ;
for t from 0 to (n-1)*(n-2)/2 do
printf("%d, ", coeff(J[n], x, t)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, Jul 02 2018
MATHEMATICA
rows = 8; egf = (y - 1)*Log[Sum[(y^Binomial[n, 2]*(x^n/n!))/(y - 1)^n, {n, 0, rows + 1}]]; t = CoefficientList[ Series[egf, {x, 0, rows}, {y, 0, 3*rows}], {x, y}] ; Table[(n - 1)!*t[[n, k]], {n, 2, rows + 1}, {k, 1, Binomial[n - 2, 2] + 1}] // Flatten (* Jean-François Alcover, Dec 10 2012, after Vladeta Jovovic *)
CROSSREFS
KEYWORD
nonn,easy,nice,tabf
AUTHOR
N. J. A. Sloane, Jan 23 2000
EXTENSIONS
Formulae and more terms from Vladeta Jovovic, Apr 06 2001
STATUS
approved