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

0,2

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 nilpotent 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).

Links

Table of n, a(n) for n=0..10.

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

Formula

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

Example

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}}.

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}]

Crossrefs

Cf. A002884, A006098, A346214, A053763.

Sequence in context: A251864 A349073 A063407 * A303214 A258110 A072196

Adjacent sequences: A384035 A384036 A384037 * A384039 A384040 A384041

Keyword

nonn

Author

Geoffrey Critzer, May 17 2025

Status

approved