A212085 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the complete bipartite graph K_(k,k).

0, 0, 2, 0, 2, 6, 0, 2, 18, 12, 0, 2, 42, 84, 20, 0, 2, 90, 420, 260, 30, 0, 2, 186, 1812, 2420, 630, 42, 0, 2, 378, 7332, 18500, 9750, 1302, 56, 0, 2, 762, 28884, 127220, 121590, 30702, 2408, 72, 0, 2, 1530, 112740, 825860, 1324470, 583422, 81032, 4104, 90

Offset: 1

Author

Alois P. Heinz, Apr 30 2012

Keywords

• nonn
• tabl

Comments

The complete bipartite graph K_(n,n) has 2*n vertices and n^2 = A000290(n) edges. The chromatic polynomial of K_(n,n) has 2*n+1 coefficients.

A(n,k) is the number of pairs of strings of length k over an alphabet of size n such that the strings do not share any letter. - Lin Zhangruiyu, Aug 19 2022

Examples

			A(3,1) = 6 because there are 6 3-colorings of the complete bipartite graph K_(1,1): 1-2, 1-3, 2-1, 2-3, 3-1, 3-2.
Square array A(n,k) begins:
   0,   0,    0,      0,       0,        0,         0, ...
   2,   2,    2,      2,       2,        2,         2, ...
   6,  18,   42,     90,     186,      378,       762, ...
  12,  84,  420,   1812,    7332,    28884,    112740, ...
  20, 260, 2420,  18500,  127220,   825860,   5191220, ...
  30, 630, 9750, 121590, 1324470, 13284630, 126657750, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..141, flattened
• Eric Weisstein's World of Mathematics, Complete Bipartite Graph
• Wikipedia, Chromatic Polynomial

Crossrefs

Rows n=1-3 give: A000004, A007395, A068293(k+1).

Columns k=1-2 give: A002378(n-1), A091940.

Cf. A008277, A212084, A266695, A282245.

Programs

Maple

A:= (n, k)-> add(Stirling2(k, j) *mul(n-i, i=0..j-1) *(n-j)^k, j=1..k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

Mathematica

a[n_, k_] := Sum[(-1)^j*(n-j)^k*Pochhammer[-n, j]*StirlingS2[k, j], {j, 1, k}]; Table[a[n-k, k], {n, 1, 11}, {k, n-1, 1, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)

Formula

A(n,k) = Sum_{j=1..k} (n-j)^k * S2(k,j) * Product_{i=0..j-1} (n-i).

A(n,n)/n = A282245(n).