%I #38 May 05 2024 20:03:31

%S 1,7,7,35,105,168,155,1085,5208,13888,651,9765,109368,874944,3999744,

%T 2667,82677,1984248,37039296,507967488,4063739904,10795,680085,

%U 33732216,1349288640,43177236480,1036253675520,14737830051840

%N 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.

%C 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}.

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

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

%F T(n,3) = A006095.

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

%e Triangle begins ...

%e 1;

%e 7, 7;

%e 35, 105, 168;

%e 155, 1085, 5208, 13888;

%e 651, 9765, 109368, 874944, 3999744;

%e 2667, 82677, 1984248, 37039296, 507967488, 4063739904;

%e ...

%t 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

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

%K nonn,tabl,new

%O 2,2

%A _Geoffrey Critzer_, Apr 28 2024