A384038 Number of 2n X 2n matrices M over GF(2) such that the column space of M is equal to the null space of M.

1, 3, 210, 234360, 4047865920, 1092146608143360, 4650098142288472473600, 314462403262051153026062745600, 338960040818652280796119613717033779200, 5834618256563872511581456247120956565738854809600, 1605370810586153268821245248112723240374305354675084328960000

Offset: 0

Author

Geoffrey Critzer, May 17 2025

Keywords

• nonn

Comments

Let M be a 2n X 2n matrix over GF(2) such that the column space of M is equal to the null space of M. Then M is idempotent and nullity(M) = n and index(M) = 2. If M' is similar to M then the column space of M' equals the null space of M'. Moreover, all such matrices are in the same similarity class (see Hoffman link).

Examples

			a(1) = 3 because there are 3 matrices of size 2 X 2 over GF(2) with the desired property: {{0, 0}, {1, 0}}, {{0, 1}, {0, 0}}, {{1, 1}, {1, 1}}.
		

Links

• D. G. Hoffman, Digraphs of finite linear transformations, Australasion Journal of Combinatorics,12: 225-238 (1995).

Crossrefs

Cf. A002884, A006098, A346214, A053763.

Programs

Mathematica

q = 2; b[p_, i_] := Count[p, i]; d[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}]; aut[deg_, p_] := Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; Table[Product[2^(2 k) - 2^i, {i, 0, (2 k) - 1}]/aut[1, Table[2, {k}]], {k,0, 10}]

Formula

a(n) = A002884(n)*A006098(n).