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<j implies C(v_i) <= C(v_j); n>=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

0,8

References

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

Links

Table of n, a(n) for n=0..90.

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.

Example

  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, ...

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

Crossrefs

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

Sequence in context: A210391 A071921 A003992 * A246118 A171882 A214075

Adjacent sequences: A337158 A337159 A337160 * A337162 A337163 A337164

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jan 28 2021

Status

approved