A346222 Number of semi-simple n X n matrices over GF(2).

1, 2, 10, 218, 25426, 11979362, 24071588290, 195647202043778, 6352629358366433026, 829377572450912758955522, 434523953108209440907114707970, 911402584183760891982341170891585538, 7638756947617134519287879000741815013863426, 256253116935172010151547980961815772566257949204482

Offset: 0

Author

Geoffrey Critzer, Jul 11 2021

Keywords

• nonn

Comments

Equivalently, number of n X n matrices over GF(2) that are diagonalizable over the algebraic closure of GF(2).

Equivalently, the number of n X n matrices over GF(2) whose minimal polynomial is a product of distinct irreducible factors, i.e., the minimal polynomial is squarefree.

Links

• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Crossrefs

Cf. A001037, A002884, A132186.

Programs

Mathematica

nn = 13; q = 2; A001037 =Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; \[Gamma]q[j_, d_] :=Table[Product[(q^d)^n - (q^d)^i, {i, 0, n - 1}], {n, 1, nn}][[j]];Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[ Series[Product[(1 + Sum[u^(j d)/\[Gamma]q[j, d], {j, 1, nn}])^
     A001037[[d]], {d, 1, nn}], {u, 0, nn}], u]

Formula

Sum_{n>=0} a(n)x^n/A002884(n) = Product_{d>=1} Sum_{j>=1} x^(j*d)/|GL_j(F_2^d)|)^A001037(d) where |GL_j(F_2^d)| is the order of the general linear group of degree j over the field with 2^d elements.