|
| |
|
|
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
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
2,2
|
|
|
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}.
|
|
|
REFERENCES
|
J. Oxley, Matroid Theory, Oxford Graduate Texts in Mathematics, 1992, page 8.
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
Triangle begins ...
1;
7, 7;
35, 105, 168;
155, 1085, 5208, 13888;
651, 9765, 109368, 874944, 3999744;
2667, 82677, 1984248, 37039296, 507967488, 4063739904;
...
|
|
|
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
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
