A248090 Square array read by antidiagonals: T(n,k) is the number of k-edge colored trees on vertex set [n] (n>=2, k>=2).

2, 3, 6, 4, 18, 24, 5, 36, 168, 120, 6, 60, 528, 2160, 720, 7, 90, 1200, 10920, 35640, 5040, 8, 126, 2280, 34200, 293760, 720720, 40320, 9, 168, 3864, 82800, 1275120, 9767520, 17297280, 362880, 10, 216, 6048, 170520, 3946320, 58968000, 387636480, 481178880, 3628800

Offset

2,1

Comments

T(n,2) = n! = A000142(n).

References

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999, Exercise 5.28, pp. 81, 124.

Links

Table of n, a(n) for n=2..46.

M. Cho, D. Kim, S. Seo, and H. Shin, Colored Prufer codes for k-edge colored trees, The Electronic Journal of Combinatorics, 11 (2004), #N10.

Formula

T(n,k) = k(n-2)!*binomial((k-1)n, n-2).

Example

T(3,3) = 18; indeed, a 3-vertex tree ABC can be labeled in 6 ways and for each labeled tree the 2 edges can be colored in 3 ways (a and b, a and c, b and c).
The antidiagonals start:
  2;
  3,  6;
  4, 18,  24;
  5, 36, 168,  120;
  6, 60, 528, 2160, 720;

Maple

T := proc(n, k) options operator, arrow: k*factorial(n-2)*binomial((k-1)*n, n-2) end proc: seq(seq(T(i, j-i), i = 2 .. j-2), j = 4 .. 12); # the command T(n, k) yields T(n, k).

Mathematica

T[n_, k_] := k (n - 2)! * Binomial[(k - 1) n, n - 2];
Table[T[n - k + 2, k], {n, 2, 10}, {k, n, 2, -1}] // Flatten (* Jean-François Alcover, Aug 26 2024 *)

Crossrefs

Cf. A000142.

Sequence in context: A231451 A126063 A214352 * A229774 A137524 A282507

Adjacent sequences: A248087 A248088 A248089 * A248091 A248092 A248093

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 27 2014

Status

approved