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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.

%I A372230 #38 May 05 2024 20:03:31
%S A372230 1,7,7,35,105,168,155,1085,5208,13888,651,9765,109368,874944,3999744,
%T A372230 2667,82677,1984248,37039296,507967488,4063739904,10795,680085,
%U A372230 33732216,1349288640,43177236480,1036253675520,14737830051840
%N 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.
%C A372230 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 A372230 J. Oxley, Matroid Theory, Oxford Graduate Texts in Mathematics, 1992, page 8.
%F A372230 T(n,k) = A022166(n,k-1)*A053601(k-1)/k.
%F A372230 T(n,3) = A006095.
%F A372230 T(n,n+1) = A053601(n)/(n+1).
%e A372230 Triangle begins ...
%e A372230     1;
%e A372230     7,    7;
%e A372230    35,   105,     168;
%e A372230   155,  1085,    5208,    13888;
%e A372230   651,  9765,  109368,   874944,   3999744;
%e A372230  2667, 82677, 1984248, 37039296, 507967488, 4063739904;
%e A372230 ...
%t A372230 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 A372230 Cf. A022166, A053601, A006095, A372350 (row sums).
%K A372230 nonn,tabl
%O A372230 2,2
%A A372230 _Geoffrey Critzer_, Apr 28 2024