A372230 Triangular array read by rows. T(n,k) is the number of size k circuits in the linear matroid M[A] where A is the n X 2^n-1 matrix whose columns are the nonzero vectors in GF(2)^n, n>=2, 3<=k<=n+1.

1, 7, 7, 35, 105, 168, 155, 1085, 5208, 13888, 651, 9765, 109368, 874944, 3999744, 2667, 82677, 1984248, 37039296, 507967488, 4063739904, 10795, 680085, 33732216, 1349288640, 43177236480, 1036253675520, 14737830051840

Offset: 2

Author

Geoffrey Critzer, Apr 28 2024

Keywords

• nonn
• tabl

Comments

For n>=2 and 3<=k<=n, to construct a size k circuit of M[A]: Choose a basis b_1,b_2,...,b_{k-1} of a k-1 dimensional subspace of GF(2)^n. Append the vector b_1 + b_2 + ... + b_{k-1}.

Examples

			Triangle begins ...
    1;
    7,    7;
   35,   105,     168;
  155,  1085,    5208,    13888;
  651,  9765,  109368,   874944,   3999744;
 2667, 82677, 1984248, 37039296, 507967488, 4063739904;
...
		

References

• J. Oxley, Matroid Theory, Oxford Graduate Texts in Mathematics, 1992, page 8.

Crossrefs

Cf. A022166, A053601, A006095, A372350 (row sums).

Programs

Mathematica

nn = 8; Map[Select[#, # > 0 &] &, Table[Table[PadRight[Table[Product[(2^n - 2^i)/(2^k - 2^i), {i, 0, k - 1}], {k, 2, n}], nn], {n, 2, nn}][[All, j]]*    Table[Product[2^n - 2^i, {i, 0, n - 1}]/(n + 1)!, {n, 2, nn}][[j]], {j, 1, nn - 1}] // Transpose] // Grid

Formula

T(n,k) = A022166(n,k-1)*A053601(k-1)/k.

T(n,3) = A006095.

T(n,n+1) = A053601(n)/(n+1).