OFFSET
0,3
COMMENTS
LINKS
Alois P. Heinz, Rows n = 0..58, flattened
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Wikipedia, Acyclic orientation
Wikipedia, Chromatic Polynomial
Wikipedia, Multipartite graph
FORMULA
T(n,k) = [q^(3*n-k)] Sum_{k,m=0..n} S2(n,k) * S2(n,m) * (q-k-m)^n * Product_{i=0..k+m-1} (q-i) with S2 = A008277.
Sum_{k=0..3n} (-1)^k * T(n,k) = A370961(n). - Alois P. Heinz, May 02 2024
EXAMPLE
2 example graphs: +-------------+
. | +-------+ |
. +-o---o---o |
. \ / \ / \ /
. X X X
. / \ / \ / \
. o---o---o +-o---o---o |
. +-------+ | +-------+ |
. +-------------+
Graph: K_(1,1,1) K_(2,2,2)
Vertices: 3 6
Edges: 3 12
The complete tripartite graph K_(1,1,1) is the cycle graph C_3 with chromatic polynomial q*(q-1)*(q-2) = q^3 -3*q^2 +2*q => [1, -3, 2, 0].
Triangle T(n,k) begins:
1;
1, -3, 2, 0;
1, -12, 58, -137, 154, -64, 0;
1, -27, 324, -2223, 9414, -24879, 39528, ...
1, -48, 1064, -14244, 126936, -784788, 3409590, ...
1, -75, 2650, -58100, 878200, -9632440, 78681510, ...
1, -108, 5562, -180585, 4123350, -70008186, 912054348, ...
...
MAPLE
P:= proc(n) option remember;
expand(add(add(Stirling2(n, k) *Stirling2(n, m)
*mul(q-i, i=0..k+m-1) *(q-k-m)^n, m=0..n), k=0..n))
end:
T:= n-> seq(coeff(P(n), q, 3*n-k), k=0..3*n):
seq(T(n), n=0..6);
MATHEMATICA
P[n_] := P[n] = Expand[Sum[Sum[StirlingS2[n, k] *StirlingS2[n, m]*Product[q - i, {i, 0, k + m - 1}]*(q - k - m)^n, {m, 1, n}], {k, 1, n}]];
T[n_] := Table[Coefficient[P[n], q, 3*n - k], {k, 0, 3*n}];
Array[T, 6] // Flatten (* Jean-François Alcover, May 29 2018, from Maple *)
KEYWORD
sign,tabf
AUTHOR
Alois P. Heinz, May 06 2012
EXTENSIONS
T(0,0)=1 prepended by Alois P. Heinz, May 02 2024
STATUS
approved