A337161 Square array read by antidiagonals: T(n,k) is the number of simple labeled graphs G with vertex set V(G) = {v_1,...,v_n} along with a (coloring) function C:V(G) ->[k] such that v_i adjacent to v_j implies C(v_i) != C(v_j) and i=0, k>=0.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 10, 1, 0, 1, 5, 16, 35, 34, 1, 0, 1, 6, 25, 84, 195, 162, 1, 0, 1, 7, 36, 165, 644, 1635, 1090, 1, 0, 1, 8, 49, 286, 1605, 7620, 21187, 10370, 1, 0, 1, 9, 64, 455, 3366, 24389, 143748, 430467, 139522, 1, 0, 1, 10, 81, 680, 6279, 62310, 599685, 4412164, 13812483, 2654722, 1, 0, 1, 11, 100, 969, 10760, 136871, 1882054, 24413445, 223233540, 702219779, 71435266, 1, 0

Offset: 0

Author

Geoffrey Critzer, Jan 28 2021

Keywords

• nonn
• tabl

Examples

			  1, 1,    1,     1,      1,      1,       1, ...
  0, 1,    2,     3,      4,      5,       6, ...
  0, 1,    4,     9,     16,     25,      36, ...
  0, 1,   10,    35,     84,    165,     286, ...
  0, 1,   34,   195,    644,   1605,    3366, ...
  0, 1,  162,  1635,   7620,  24389,   62310, ...
  0, 1, 1090, 21187, 143748, 599685, 1882054, ...
		

References

• R. P. Stanley, Enumerative Combinatorics, Vol I, Second Edition, Section 3.18.

Crossrefs

Cf. A322280, A117402 (column k=2).

Programs

Mathematica

nn = 6; e[x_] := Sum[x^n/(2^Binomial[n, 2]), {n, 0, nn}];
Table[Table[2^Binomial[n, 2], {n, 0, nn}] PadRight[CoefficientList[Series[e[x]^k, {x, 0, nn}], x], nn + 1], {k, 0, nn}] // Transpose // Grid

Formula

Let e(x) = Sum_{n>=0} x^n/2^binomial(n,2). Then e(x)^k = Sum_{n>=0} Z_n(k)*x^n/2^biomial(n,2) and T(n,k) = Z_n(k). Z_n(k) is the zeta polynomial of the class of posets described in A117402.