Integer Sequences and Matrices Over Finite Fields

Kent E. Morrison Department of Mathematics California Polytechnic State University San Luis Obispo, CA 93407 USA

                 kmorriso@calpoly.edu

Abstract

In this expository article we collect the integer sequences that count several different types of matrices over finite fields and provide references to the Online Encyclopedia of Integer Sequences. Section 1 contains the sequences, their generating functions, and examples. Section 2 contains the proofs of the formulas for the coefficients and the generating functions of those sequences if the proofs are not easily available in the literature. The cycle index for matrices is an essential ingredient in most of the derivations.

Notation

q a prime power

F_{q} field with q elements

GL_{n}(q) group of n × n invertible matrices over F_{q}

M_{n}(q) algebra of n × n matrices over F_{q}

γ_{n}(q) order of the group GL_{n}(q)

γ_{n} order of GL_{n}(q) with q understood

ν_{d} number of irreducible monic polynomials of degree d over F_{q}

Sequences  Section  OEIS
All matrices  1.1  A002416
Invertible matrices  1.2  A002884
Subspaces, q-binomial coefficients  1.3  A022166–188

A006116–122 A015195–217

Splittings (direct sum decompositions)  1.4
q-Stirling numbers, q-Bell numbers  1.4
Flags of subspaces  1.5  A005329, A069777
Linear binary codes  1.6  A022166, A076831
Matrices by rank  1.7
Linear derangements 1.8, 2.2 A002820
Projective derangements  1.8, 2.2
Diagonalizable matrices  1.9, 2.3
Projections 1.10, 2.4 A053846
Solutions of A^{k} = I  1.11, 2.5 A053718, 722, 725

A053770–777 A053846–849 A053851–857 A053859–863

Nilpotent matrices 1.12 A053763
Cyclic(regular) matrices  1.13, 2.6
Semi-simple matrices  1.14, 2.7
Separable matrices  1.15, 2.8
Conjugacy classes  1.16, 2.9 A070933, A082877

1 The Sequences

References to the Online Encyclopedia of Integer Sequences (OEIS) are accurate as of February 1, 2006. Sequences mentioned in this article may have been added since then and entries in the OEIS may have been modified.

1.1 n × n matrices over F_{q}

Over the field F_{q} the number of n × n matrices is q^{n^{2}}. For q = 2 this sequence is A002416 of the OEIS, indexed from n = 0. The terms for 0 ≤ n ≤ 4 are

1, 2, 16, 512, 65536.

1.2 Invertible matrices

The number of invertible n × n matrices is given by

γ_{n}(q) := |GL_{n}(q)| = (q^{n} − 1)(q^{n} − q)(q^{n} − q^{2}) ⋯ (q^{n} − q^{n−1}).

We will use γ_{n} in place of γ_{n}(q) unless there is a need to be explicit about the base field. It is convenient to define γ_{0} to be 1. For q = 2 the sequence is A002884, and from γ_{0} to γ_{5} the terms are

1, 1, 6, 168, 20160, 9999360.

A second formula for γ_{n} is

γ_{n} = (q − 1)^{n}q^{\binom{n}{2}}[n]_{q}!,

where [n]_{q}! := [n]_{q}[n − 1]_{q} ⋯ [2]_{q}[1]_{q} and [i]_{q} := 1 + q + q^{2} + ⋯ + q^{i−1} are the q-analogs of n! and i. Also, [i]_{q} is the number of points in the projective space of dimension i − 1 over F_{q}.

A third formula is

γ_{n} = (−1)^{n}q^{\binom{n}{2}}(q; q)_{n},

where (a; q)_{n} is defined for n > 0 by

(a; q)_{n} := ∏_{j=0}^{n−1}(1 − aq^{j}).

Construct a random n × n matrix over F_{q} by choosing the entries independently and uniformly from F_{q}. Then γ_{n}/q^{n^{2}} is the probability that the matrix is invertible, and this probability has a limit as n → ∞

lim_{n→∞} \frac{γ_{n}}{q^{n^{2}}} = ∏_{r≥1} (1 − \frac{1}{q^{r}} ) .

For q = 2 the limit is 0.28878 . . .. For q = 3 the limit is 0.56012 . . .. As q → ∞ the probability of being invertible goes to 1.

1.3 Subspaces

The number of k-dimensional subspaces of a vector space of dimension n over F_{q} is given by the Gaussian binomial coefficient (also called the q-binomial coefficient)

\binom{n}{k}_{q} = \frac{(q^{n} − 1)(q^{n} − q)(q^{n} − q^{2}) ⋯ (q^{n} − q^{k−1})}{(q^{k} − 1)(q^{k} − q)(q^{k} − q^{2}) ⋯ (q^{k} − q^{k−1})}.

Also, we define \binom{0}{0}_{q} = 1. Other useful expressions for the Gaussian binomial coefficients are

\binom{n}{k}_{q} = \frac{[n]_{q}!}{[k]_{q}![n − k]_{q}!},

which shows the q-analog nature of the Gaussian coefficients, and

\binom{n}{k}_{q} = \frac{γ_{n}}{γ_{k}γ_{n−k}q^{k(n−k)}} ,

which comes from the transitive action of GL_{n}(q) on the set of subspaces of dimension k, the denominator being the order of the subgroup stabilizing one of the subspaces.

Entries A022166–A022188 are the triangles of Gaussian binomial coefficients for q = 2 to q = 24. Note that when q is not a prime power the formula does not count subspaces. For q = 2 the rows from n = 0 to 6 are

1

1 1
1 3 1
1 7 7  1
1 15 35 15 1
1 31 155 155 31 1
1 63 651 1395 651 63 1

For the Gaussian binomial coefficients we have the q-binomial theorem [2]

∏_{1≤i≤n}(1 + q^{i}t) = ∑_{0≤k≤n} q^{\binom{k}{2}}\binom{n}{k}_{q}t^{k}.

Summing over k from 0 to n for a fixed n gives the total number of subspaces. Sequences A006116–A006122 and A015195–A015217 correspond to the values q = 2, …, 8 and q = 9, …, 24. (Same warning applies to those q that are not prime powers.) For q = 2 and n = 0, …, 8 the sequence begins

1, 2, 5, 16, 67, 374, 2825, 29212, 417199.

1.4 Splittings

Let \left\{{n \atop k}\right\}_{q} be the number of direct sum splittings of an n-dimensional vector space into k non-trivial subspaces without regard to the order among the subspaces. These numbers are q-analogs of the Stirling numbers of the second kind, which count the number of partitions of an n-set into k non-empty subsets, and the notation follows that of Knuth for the Stirling numbers. Then it is easy to see that

\left\{{n \atop k}\right\}_{q} = \frac{1}{k!} ∑_{n_{i}≥1^{n_{1}+···+n_{k}=n}} \frac{γ_{n}}{γ_{n_{1}} ⋯ γ_{n_{k}}} .

It is not difficult to verify that the triangle \left\{{n \atop k}\right\}_{q} satisfies the two variable generating function identity

1 + ∑_{n≥1} ∑_{k=1}^{n} \left\{{n \atop k}\right\}_{q} \frac{u^{n}}{γ_{n}} t^{k} = exp (t ∑_{r≥1} \frac{u^{r}}{γ_{r}} ) .

For q = 2 the entries for 1 ≤ n ≤ 6 and 1 ≤ k ≤ n are

1

1  3
1  28  28
1  400  1680  840
1 10416 168640  277760  83328
1 525792 36053248 159989760 139991040 27998208

Let b_{n} be the total number of non-trivial splittings of an n-dimensional vector space:

b_{n} = ∑_{k=1}^{n} \left\{{n \atop k}\right\}_{q}.

The b_{n} are analogs of the Bell numbers counting the number of partitions of finite sets. Then we have the formula for the generating function for the b_{n},

1 + ∑_{n≥1} b_{n} \frac{u^{n}}{γ_{n}} = exp (∑_{r≥1} \frac{u^{r}}{γ_{r}} ) .

For q = 2 the values of b_{n} for n = 1, …, 6 are

1, 4, 57, 2921, 540145, 364558049.

Bender and Goldman [1] first wrote down the generating function for the q-Bell numbers, which, along with the q-Stirling numbers, can be put into the context of q-exponential families [10].

1.5 Flags

A flag of length k in a vector space V is an increasing sequence of subspaces

{0} = V_{0} ⊂ V_{1} ⊂ V_{2} ⊂ ⋯ V_{k} = V.

If dim V = n, then a complete flag is a flag of length n. Necessarily, dim V_{i} = i. The number of complete flags of an n-dimensional vector space over F_{q} is

\binom{n}{1}_{q}\binom{n − 1}{1}_{q} ⋯ \binom{1}{1}_{q} = [n]_{q}[n − 1]_{q} ⋯ [1]_{q} = [n]_{q}!.

The reason is that having selected V_{1}, …, V_{i}, the number of choices for V_{i+1} is \binom{n−i}{1}_{q}. Alternatively, the general linear group acts transitively on the set of complete flags. The stabilizer subgroup of the standard flag is the subgroup of upper triangular matrices, whose order is (q − 1)^{n}q^{\binom{n}{2}}. Hence, the number of complete flags is

\frac{γ_{n}}{(q − 1)^{n}q^{\binom{n}{k}}} = [n]_{q}!.

Define [0]_{q}! = 1. The sequence [n]_{q}! for q = 2 is A005329. The terms for n = 0, 1, 2, …, 8 are

1, 1, 3, 21, 315, 9765, 615195, 78129765, 19923090075.

By looking up the phrase “q-factorial numbers” in the OEIS one can find the sequences [n]_{q}! for q ≤ 13. The triangle of q-factorial numbers is A069777.

The q-multinomial coefficient

\binom{n}{n_{1} n_{2} ⋯ n_{k}}_{q} := \frac{[n]_{q}!}{[n_{1}]_{q}![n_{2}]_{q}! ⋯ [n_{k}]_{q}!},

where n = n_{1} + ⋯ + n_{k}, is the number of flags of length k in an n-dimensional space such that dim V_{i} = n_{1} + ⋯ + n_{i}.

1.6 Linear binary codes

An [n, k] linear binary code is a k-dimensional subspace of the space of F_{2}^{n}. Thus, the number of [n, k] linear binary codes is the Gaussian binomial coefficient \binom{n}{k}_{2} from section 1.3 and is given by the triangle A022166.

Two linear binary codes are equivalent (or isometric) if there is a permutation matrix P mapping one subspace to the other. The number of equivalence classes of [n, k] codes is given by the triangle A076831. The early entries are identical with the corresponding binomial coefficients, but that does not hold in general. The rows for n = 0 to 6 are

1

1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 16 22 16 6 1

For linear codes over other finite fields, the notion of equivalence uses matrices P having exactly one non-zero entry in each row and column. These matrices, as linear maps, preserve the Hamming distance between vectors. They form a group isomorphic to the wreath product of the multiplicative group of F_{q} with S_{n}.

There are more than a dozen sequences and triangles associated to linear binary codes in the OEIS. They are listed in the short index of the OEIS under “Codes.”

1.7 Matrices by rank

The number of m × n matrices of rank k over F_{q} is

\binom{m}{k}_{q}(q^{n} − 1)(q^{n} − q) ⋯ (q^{n} − q^{k−1})

= \binom{n}{k}_{q}(q^{m} − 1)(q^{m} − q) ⋯ (q^{m} − q^{k−1})

= \frac{(q^{m} − 1)(q^{m} − q) ⋯ (q^{m} − q^{k−1}) (q^{n} − 1)(q^{n} − q) ⋯ (q^{n} − q^{k−1})}{(q^{k} − 1)(q^{k} − q) ⋯ (q^{k} − q^{k−1})}.

(To justify the first line of the formula note that the number of k-dimensional subspaces of F_{q}^{m} to serve as the column space of a rank k matrix is \binom{m}{k}_{q}. Identify the column space with the image of the associated linear map from F_{q}^{n} to F_{q}^{m}. There are (q^{n} − 1)(q^{n} − q) ⋯ (q^{n} − q^{k−1}) surjective linear maps from F_{q}^{n} to that k-dimensional image. The second line follows by transposing.)

Define the triangle r(n, k) to be the number of n × n matrices of rank k over F_{q} and r(0, 0) := 1. Thus,

r(n, k) = \frac{((q^{n} − 1)(q^{n} − q) ⋯ (q^{n} − q^{k−1}))^{2}}{(q^{k} − 1)(q^{k} − q) ⋯ (q^{k} − q^{k−1})} .

For q = 2 the entries from r(0, 0) to r(5, 5) are

1

1 1
1 9  6
1 49 294 168
1 225 7350 37800  20160
1 961 144150 4036200 19373760 9999360

1.8 Linear and projective derangements

A matrix is a linear derangement if it is invertible and does not fix any non-zero vector. Such a matrix is characterized as not having 0 or 1 as an eigenvalue. Let e_{n} be the number of linear derangements and define e_{0} = 1. Then e_{n} satisfies the recursion

e_{n} = e_{n−1}(q^{n} − 1)q^{n−1} + (−1)^{n}q^{n(n−1)/2}.

For q = 2, the sequence is A002820 (with offset 2) and the first few terms beginning with e_{0} are 1, 0, 2, 48, 5824, 2887680, . . .. The sequence can be obtained from the generating function

1 + ∑_{n≥1} \frac{e_{n}}{γ_{n}} u^{n} = \frac{1}{1 − u} ∏_{r≥1} (1 − \frac{u}{q^{r}} ) .

The proof of this is in section 2. The asymptotic probability that an invertible matrix is a linear derangement is

lim_{n→∞} \frac{e_{n}}{γ_{n}} = ∏_{r≥1} (1 − \frac{1}{q^{r}} ) .

The asymptotic probability that a square matrix is a linear derangement is

lim_{n→∞} \frac{e_{n}}{q^{n^{2}}} = ∏_{r≥1} (1 − \frac{1}{q^{r}} )^{2}

A matrix over F_{q} is a projective derangement if the induced map on projective space has no fixed points. Equivalently, this means that the matrix has no eigenvalues in F_{q}. If d_{n} is the number of such n × n matrices, then

1 + ∑_{n≥1} \frac{d_{n}}{γ_{n}} u^{n} = \frac{1}{1 − u} ∏_{r≥1} (1 − \frac{u}{q^{r}} )^{q−1} .

For q = 2 the notions of projective and linear derangement are the same, and the corresponding sequence is given above. For q = 3 the sequence has initial terms from n = 1 to 5

0, 18, 3456, 7619508, 149200289280.

Since two matrices that differ by a scalar multiple induce the same map on projective space, the number of maps that are projective derangements is d_{n}/(q − 1). The asymptotic probability that a random invertible matrix is a projective derangement is the limit

lim_{n→∞} \frac{d_{n}}{γ_{n}} = ∏_{r≥1} (1 − \frac{1}{q^{r}} )^{q−1} .

The asymptotic probability that a random n × n matrix is a projective derangement is

lim_{n→∞} \frac{d_{n}}{q^{n^{2}}} = ∏_{r≥1} (1 − \frac{1}{q^{r}} )^{q} .

As q → ∞, this probability goes to 1/e, the same value as the asymptotic probability that a random permutation is a derangement.

1.9 Diagonalizable matrices

In this section let d_{n} be the number of diagonalizable n × n matrices over F_{q}. Then

1 + ∑_{n≥1} \frac{d_{n}}{γ_{n}} u^{n} = (∑_{m≥0} \frac{u^{m}}{γ_{m}} )^{q} .

It follows that

d_{n} = ∑_{n_{1}+···+n_{q}=n} \frac{γ_{n}}{γ_{n_{1}} ⋯ γ_{n_{q}}} .

For q = 2 this simplifies to

d_{n} = ∑_{i=0}^{n} \frac{γ_{n}}{γ_{i}γ_{n−i}} .

The sequence for d_{1} to d_{8} is

2, 8, 58, 802, 20834, 1051586, 102233986, 196144424834.

For q = 3 the five initial terms are

3, 39, 2109, 417153, 346720179.

For arbitrary q, we can easily find d_{2},

d_{2} = ∑_{n_{1}+···+n_{q}=2} \frac{γ_{n}}{γ_{n_{1}} ⋯ γ_{n_{q}}}

= q \frac{γ_{2}}{γ_{0}γ_{2}} + \binom{q}{2}\frac{γ_{2}}{γ_{1}γ_{1}}

= \frac{q^{4} − q^{2} + 2q}{2} .

1.10 Projections

A projection is a matrix P such that P^{2} = P. Let p_{n} be the number of n × n projections. Then

1 + ∑_{n≥1} \frac{p_{n}}{γ_{n}} u^{n} = (∑_{m≥0} \frac{u^{m}}{γ_{m}} )^{2} .

It follows that

p_{n} = ∑_{0≤i≤n} \frac{γ_{n}}{γ_{i}γ_{n−i}} .

In the sum the term γ_{n}/γ_{i}γ_{n−i} is the number of projections of rank i. Projections are also characterized as diagonalizable matrices having eigenvalues 0 or 1. Thus, for q = 2 the diagonalizable matrices are precisely the projections, and so we get the same sequence as in section 1.9. From p_{1} to p_{8} the sequence is

2, 8, 58, 802, 20834, 1051586, 102233986, 196144424834.

For q = 3 the map P ↦ P + I gives a bijection from the set of projections to the set of diagonalizable matrices with eigenvalues 1 or 2. Such matrices are precisely the solutions of X^{2} = I. The sequence is given in the OEIS by A053846. From p_{1} to p_{7} the sequence is

1, 2, 14, 236, 12692, 1783784, 811523288, 995733306992.

There is a bijection between the set of n × n projections and the direct sum splittings F_{q}^{n} = V ⊕ W. The projection P corresponds to the splitting ImP ⊕ KerP. Note that V ⊕ W is regarded as different from W ⊕ V. Hence,

p_{n} = 2 + 2\left\{{n \atop 2}\right\}_{q},

because \left\{{n \atop 2}\right\}_{q} is the number of splittings into two proper subspaces, with V ⊕ W regarded as the same as W ⊕ V.

From §1.3 we have

\binom{n}{k}_{q} = \frac{γ_{n}}{γ_{k}γ_{n−k}q^{k(n−k)}} ,

showing the relationship between the number of subspaces of dimension k and the number of projections of rank k. From it we see that t there are q^{k(n−k)} complementary subspaces for a fixed subspace of dimension k.

1.11 Solutions of A^{k} = I

Let a_{n} be the number of n × n matrices A satisfying A^{2} = I. Such matrices correspond to group homomorphisms from the cyclic group of order 2 to GL_{n}(q). In characteristic other than two, the generating function for the a_{n} is

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = (∑_{m≥0} \frac{u_{m}}{γ_{m}} )^{2} ,

and so

a_{n} = ∑_{i=0}^{n} \frac{γ_{n}}{γ_{i}γ_{n−i}} .

These matrices are those which are diagonalizable and have only 1 and −1 for eigenvalues. If we fix two distinct elements λ_{1} and λ_{2} in the base field F_{q}, then a_{n} also gives the number of diagonalizable matrices having eigenvalues λ_{1} and λ_{2}. Taking the eigenvalues to be 0 and 1 gives the set of projections, and so we see that the number of projections is the same as the number of solutions to A^{2} = I, when q is not a power of 2. The proof in §2.4 for the number of projections is essentially the proof for the more general case. The sequence A053846 gives the number of solutions over F_{3}.

In characteristic two, A^{2} = I does not imply that A is diagonalizable and the previous formula does not hold. We have the formula

a_{n} = ∑_{0≤i≤n/2} \frac{γ_{n}}{q^{i(2n−3i)}γ_{i}γ_{n−2i}}

to be proved in §2.5. Because A^{2} = I is equivalent to (A + I)^{2} = 0, the formula for a_{n} also counts the number of n × n nilpotent matrices N such that N^{2} = 0. For q = 2 the sequence is A053722. Although there appears to be an error in the formula given in the OEIS entry, the initial terms of the sequence are correct. For n = 1, …, 8 they are

1, 4, 22, 316, 6976, 373024, 32252032, 6619979776.

For q = 4 the sequence is A053856 and the initial terms for n = 1, …, 7 are

1, 16, 316, 69616, 21999616, 74351051776, 374910580965376.

More generally, let k be a positive integer not divisible by p, where q is a power of p. We consider the solutions of A^{k} = I and let a_{n} be the number of n×n solutions with coefficients in F_{q}. Now z^{k} − 1 factors into a product of distinct irreducible polynomials

z^{k} − 1 = ϕ_{1}(z)ϕ_{2}(z) ⋯ ϕ_{r}(z).

Let d_{i} = deg ϕ_{i}. Then the generating function for the a_{n} is

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{i=1}^{r} ∑_{m≥0} \frac{q^{md_{i}}}{γ_{m}(q^{d_{i}})}.

Note that a_{n} counts the homormorphisms from the cyclic group of order k to GL_{n}(q). The generating function given here is a special case of the generating function given by Chigira, Takegahara, and Yoshida [3] for the sequence whose nth term is the number of homomorphisms from a finite group G to GL_{n}(q) under the assumption that the characteristic of F_{q} does not divide the order of G.

Now we consider some specific examples. Let k = 3 and let q be a power of 2. Then

z^{3} − 1 = z^{3} + 1 = (z + 1)(z^{2} + z + 1)

is the irreducible factorization. Thus, d_{1} = 1 and d_{2} = 2. The generating function is

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = (∑_{m≥0} \frac{u^{m}}{γ_{m}} ) (∑_{m≥0} \frac{u^{2m}}{γ_{m}(q^{2})}) .

With q = 2 we get the sequence A053725 with initial terms for n = 1, …, 8

1, 3, 57, 1233, 75393, 19109889, 6326835201, 6388287561729.

With q = 4 we get the sequence A053857 with initial terms for n = 1, …, 5

3, 63, 8739, 5790339, 25502129667.

For the next example, let k = 8 and let q be a power of 3. Then

z^{8} − 1 = (z − 1)(z + 1)ϕ_{3}(z)ϕ_{4}(z)ϕ_{5}(z),

where the last three factors all have degree 2. Thus, r = 5, d_{1} = d_{2} = 1, and d_{3} = d_{4} = d_{5} = 2. The generating function is given by

∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = (∑_{m≥0} \frac{u^{m}}{γ_{m}} )^{2} (∑_{m≥0} \frac{u^{2m}}{γ_{m}(q^{2})})^{3} .

For q = 3 this gives the sequence A053853 beginning for n = 1, …, 5

2, 32, 4448, 3816128, 26288771456.

These are the sequences in the OEIS for small values of k and q.

k q = 2  q = 3  q = 4
2 A053722 A053846 A053856
3 A053725 A053847 A053857
4 A053718 A053848 A053859
5 A053770 A053849 A053860
6 A053771 A053851 A053861
7 A053772 A053852 A053862
8 A053773 A053853 A053863
9 A053774 A053854
10 A053775 A053855
11 A053776
12 A053777

1.12 Nilpotent matrices

Fine and Herstein [5] proved that the number of nilpotent n × n matrices is q^{n(n−1)}, and Gerstenhaber [7] simplified the proof. Recently Crabb has given an even more accessible proof [4]. For q = 2 this is sequence A053763 with initial terms from n = 1 to n = 6

1, 1, 4, 64, 4096, 1048576, 1073741824.

1.13 Cyclic matrices

A matrix A is cyclic if there exists a vector v such that {A^{i}v|i = 0, 1, 2, …} spans the underlying vector space. (The term regular is also used. ) An equivalent description is that the minimal and characteristic polynomials of A are the same. Let a_{n} be the number of cyclic matrices over F_{q}. The generating function factors as

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{d≥1} (1 + \frac{1}{q^{d} − 1} \frac{u^{d}}{1 − (u/q)^{d}} )^{ν_{d}} ,

and can be put into the form

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = \frac{1}{1 − u} ∏_{d≥1} (1 + \frac{u^{d}}{q^{d}(q^{d} − 1)})^{ν_{d}} ,

where ν_{d} is the number of irreducible, monic polynomials of degree d over F_{q}. The generating function can be extracted from the proof of Theorem 1 in [6], which we present in section 2.

For q = 2 the sequence from a_{1} to a_{7} is

2, 14, 412, 50832, 25517184, 51759986688, 422000664182784.

Wall [13] proved that the probability that an n × n matrix is cyclic has a limit

lim_{n→∞} \frac{a_{n}}{q^{n^{2}}} = (1 − \frac{1}{q^{5}} ) ∏_{r≥3} (1 − \frac{1}{q^{r}} ) .

Fulman [6] also has a proof. For q = 2 this limit is 0.7403 . . ..

1.14 Semi-simple matrices

A matrix A is semi-simple if it diagonalizes over the algebraic closure of the base field. Let a_{n} be the number of semi-simple n × n matrices over F_{q}. Then the generating function has a factorization

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{d≥1} (1 + ∑_{j≥1} \frac{u^{jd}}{γ_{j}(q^{d})})^{ν_{d}} ,

where γ_{j}(q^{d}) = |GL_{j}(q^{d})|. For q = 2 the sequence from a_{1} to a_{7} is

2, 10, 218, 25426, 11979362, 24071588290, 195647202043778.

1.15 Separable matrices

A matrix is separable if it is both cyclic and semi-simple, which is equivalent to having a characteristic polynomial that is square-free. Let a_{n} be the number of separable n × n matrices over F_{q}. Then the generating function factors

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{d≥1} (1 + \frac{u^{d}}{q^{d} − 1})^{ν_{d}} .

This can also be factored as

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = \frac{1}{1 − u} ∏_{d≥1} (1 + \frac{u^{d}(u^{d} − 1)}{q^{d}(q^{d} − 1)})^{ν_{d}} .

For q = 2 the sequence from a_{1} to a_{7} is

2, 8, 160, 22272, 9744384, 20309999616, 165823024988160.

The number of conjugacy classes of separable n × n matrices is q^{n} − q^{n−1} for n ≥ 2 and q for n = 1. This is proved by Neumann and Praeger [11, Lemma 3.2] by showing that the number of square-free monic polynomials of degree n is q^{n}−q^{n−1}. There is a natural bijection between the set of conjugacy classes of separable matrices and the set of square-free monic polynomials obtained by associating to each conjugacy class the characteristic polynomial of the matrices in that class.

1.16 Conjugacy classes

Let a_{n} be the number of conjugacy classes of n×n matrices over F_{q}. The generating function for this sequence is

1 + ∑_{n≥1} a_{n}u^{n} = ∏_{r≥1} \frac{1}{1 − qu^{r}} .

The number of conjugacy classes grows like q^{n}. In fact,

lim_{n→∞} \frac{a_{n}}{q^{n}} = ∏_{r≥1} (1 − \frac{1}{q^{r}} )^{−1} ,

which is the reciprocal of the limiting probability that a matrix is invertible. See section 1.2. For q = 2 the sequence is A070933. The initial terms for n = 1, …, 10 are

2, 6, 14, 34, 74, 166, 350, 746, 1546, 3206.

For q = 3 the initial terms for n = 1, …, 10 are

3, 12, 39, 129, 399, 1245, 3783, 11514, 34734, 104754.

Let b_{n} be the number of conjugacy classes in the general linear group GL_{n}(q), the group of n × n invertible matrices over F_{q}. Then

1 + ∑_{n≥0} b_{n}u^{n} = ∏_{r≥1} \frac{1 − u^{r}}{1 − qu^{r}} .

For q = 2 the sequence is A006951, which starts

1, 3, 6, 14, 27, 60, 117, 246, 490, 1002.

For q = 3 the sequence is A006952, which starts

2, 8, 24, 78, 232, 720, 2152, 6528, 19578, 58944.

For q = 4, 5, 7 the sequences are A049314, A049315, and A049316.

The number of conjugacy classes is asymptotic to q^{n}:

lim_{n→∞} \frac{b_{n}}{q^{n}} = 1.

Hence, in the limit the ratio of the number of conjugacy classes of invertible matrices to the number of conjugacy classes of all matrices is the same as the limiting probability that a matrix is invertible. That is,

lim_{n→∞} \frac{b_{n}}{a_{n}} = ∏_{r≥1} (1 − \frac{1}{q^{r}} ) .

Sequence A070731 gives the size of the largest conjugacy class in GL_{n}(2). Starting with n = 1 the initial terms are

1, 3, 56, 3360, 833280, 959938560.

The minimal order of the centralizers of elements in GL_{n}(2) is given by the quotient of γ_{n} by the nth term in this sequence. The resulting sequence for n = 1, … 10 is

1, 2, 3, 6, 12, 21, 42, 84, 147, 294.

This sequence is A082877 in the OEIS.

2 Selected Proofs

2.1 The cycle index and generating functions

In sections 1.6-1.13 we make heavy use of generating functions of the form

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n},

where the sequence a_{n} counts some class of n × n matrices. These generating functions come from the cycle index for matrices that was first defined by Kung [9] and later extended by Stong [12]. We follow Fulman’s notation in [6]. The cycle index for conjugation action of GL_{n}(q) on M_{n}(q) is a polynomial in the indeterminates x_{ϕ,λ}, where ϕ ranges over the set Φ of monic irreducible polynomials over F_{q} and λ ranges over the partitions of the positive integers. First we recall that the conjugacy class of a matrix A is determined by the isomorphism type of the associated F_{q}[z]-module on the vector space F_{q}^{n} in which the action of z is that of A. This module is isomorphic to a direct sum

⨁_{i=1}^{k} ⨁_{j=1}^{l_{i}} F_{q}[z]/(ϕ_{i}^{λ_{i,j}})

where ϕ_{1}, …, ϕ_{k} are distinct monic irreducible polynomials; for each i, λ_{i,1} ≥ λ_{i,2} ≥ ⋯ ≥ λ_{i,l_{i}} is a partition of n_{i} = ∑_{j} λ_{i,j}. Since A is n × n, then n = ∑_{i} n_{i} deg ϕ_{i} = ∑_{i,j} λ_{i,j} deg ϕ_{i}.Let λ_{i} denote the partition of n_{i} given by the λ_{i,j} and define |λ_{i}| = n_{i}. The conjugacy class of A in M_{n}(q) is determined by the data consisting of the finite list of distinct monic irreducible polynomials ϕ_{1}, …, ϕ_{k} and the partitions λ_{1}, …, λ_{k}.

The cycle index is defined to be

\frac{1}{γ_{n}} ∑_{A∈M_{n}(q)} ∏_{ϕ∈Φ} x_{ϕ,λ_{ϕ}(A)},

where λ_{ϕ}(A) is the partition associated to ϕ in the conjugacy class data for A. If ϕ does not occur in the polynomials associated to A, then λ_{ϕ}(A) is the empty partition, and we define x_{ϕ,λ_{ϕ}(A)} = 1.

We construct the generating function for the cycle index

1 + ∑_{n≥1} \frac{u^{n}}{γ_{n}} ∑_{A∈M_{n}(q)} ∏_{ϕ∈Φ} x_{ϕ,λ_{ϕ}(A)}.

This generating function has a formal factorization over the monic irreducible polynomials. For the proof we refer to the paper of Stong [12, p. 170] .

Lemma 2.1.

1 + ∑_{n=1}^{∞} \frac{u^{n}}{γ_{n}} ∑_{A∈M_{n}(q)} ∏_{ϕ} x_{ϕ,λ_{ϕ}(A)} = ∏_{ϕ} ∑_{λ} \frac{x_{ϕ,λ}u^{|λ| deg ϕ}}{c_{ϕ}(λ)}

where c_{ϕ}(λ) is the order of the group of module automorphisms of the F_{q}[z]-module ⨁_{j} F_{q}[z]/(ϕ^{λ_{j}}).

Lemma 2.2. Fix a monic irreducible ϕ. Then

∑_{λ} \frac{u^{|λ| deg ϕ}}{c_{ϕ}(λ)} = ∏_{r≥1} (1 − \frac{u^{deg ϕ}}{q^{r deg ϕ}} )^{−1} .

Proof. We use the formula for c_{ϕ}(λ) proved by Kung [9, Lemma 2, p. 146]. Let λ be a partition of n and let b_{i} be the number of parts of size i. Let d_{i} = b_{1} + 2b_{2} + ⋯ + ib_{i} + i(b_{i+1} + ⋯ + b_{n}). Then

c_{ϕ}(λ) = ∏_{i} ∏_{k=1}^{b_{i}}(q^{d_{i} deg ϕ} − q^{(d_{i}−k) deg ϕ}).

We see from this formula that c_{ϕ}(λ) is a function of λ and q^{deg ϕ}. Therefore it is sufficient to prove the lemma for a single polynomial of degree 1, say ϕ(z) = z, and then to replace u by u^{deg ϕ} and q by q^{deg ϕ}. So, now we will prove that for ϕ(z) = z,

∑_{λ} \frac{u^{|λ|}}{c_{ϕ}(λ)} = ∏_{r≥1} (1 − \frac{u}{q^{r}} )^{−1} .

We split the left side into an outer sum over n and an inner sum over the partitions of n

∑_{λ} \frac{u^{|λ|}}{c_{ϕ}(λ)} = 1 + ∑_{n≥1} ∑_{|λ|=n} \frac{u^{n}}{c_{ϕ}(λ)}.

To evaluate the inner sum we note that γ_{n} ∑_{|λ|=n} \frac{1}{c_{ϕ}(λ)} is the number of n × n nilpotent matrices, which is q^{n(n−1)} from the theorem of Fine and Herstein [5]. Therefore, the u^{n} coefficient of the left side is

∑_{|λ|=n} \frac{1}{c_{ϕ}(λ)} = \frac{1}{q^{n}(1 − \frac{1}{q}) … (1 − \frac{1}{q^{n}})}.

For the right side we use a formula of Euler, which is a special case of Cauchy’s identity and a limiting case of the q-binomial theorem. A convenient reference is the book of Hardy and Wright [8, Theorem 349, p. 280]. For |a| < 1, |y| < 1, the coefficient of a^{n} in the infinite product

∏_{r≥1} \frac{1}{(1 − ay^{r})}

is

\frac{y^{n}}{(1 − y)(1 − y^{2}) … (1 − y^{n})}.

Let a = u and y = \frac{1}{q} to see that the u^{n} coefficient of

∏_{r≥1} (1 − \frac{u}{q^{r}} )^{−1}

is

\frac{1}{q^{n}(1 − \frac{1}{q}) … (1 − \frac{1}{q^{n}})}.

Therefore, the coefficients of u^{n} of the left and right sides are equal and so we have proved that for ϕ(z) = z,

∑_{λ} \frac{u^{|λ|}}{c_{ϕ}(λ)} = ∏_{r≥1} (1 − \frac{u}{q^{r}} )^{−1} .

Lemma 2.3. Let Φ′ be a subset of the irreducible monic polynomials. Let a_{n} be the number of n × n matrices whose conjugacy class data involves only polynomials ϕ ∈ Φ′. Then

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{ϕ∈Φ′} ∑_{λ} \frac{u^{|λ| deg ϕ}}{c_{ϕ}(λ)}

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{ϕ∈Φ′} ∏_{r≥1} (1 − \frac{u^{deg ϕ}}{q^{r deg ϕ}} )^{−1} .

Proof. Using Lemma 1 we set x_{ϕ,λ} = 1 or 0 according to whether ϕ ∈ Φ′ or not. Then on the left side the inner sum is over the matrices A that do not have factors of ϕ ∈ Φ′ in their characteristic polynomials, and so the coefficient of u^{n} is simply the number of such n × n matrices. This gives the first equality in the statement of the lemma

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{ϕ∈Φ′} ∑_{λ} \frac{u^{|λ| deg ϕ}}{c_{ϕ}(λ)} .

Then from Lemma 2 we get the second equality.

Lemma 2.4.

\frac{1}{1 − u} = ∏_{ϕ≠z} ∑_{λ} \frac{u^{|λ| deg ϕ}}{c_{ϕ}(λ)}

\frac{1}{1 − u} = ∏_{ϕ≠z} ∏_{r≥1} (1 − \frac{u^{deg ϕ}}{q^{r deg ϕ}} )^{−1} .

Proof. Using Lemma 3 we let Φ′ be the complement of the polynomial ϕ(z) = z. The matrices not having factors of z in their characteristic polynomials are the invertible matraices. Thus, a_{n} = γ_{n}, and so the left side is 1 + ∑_{n≥1} u^{n}.

We can generalize the first part of Lemma 3 to allow for conjugacy class data in which the allowed partitions vary with the polynomials. The proof follows immediately from Lemma 1.

Lemma 2.5. For each monic, irreducible polynomial ϕ let L_{ϕ} be a subset of all partitions of the positive integers. Let a_{n} be the number of n × n matrices A such that λ_{ϕ}(A) ∈ L_{ϕ} for all ϕ. Then

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{ϕ} ∑_{λ∈L_{ϕ}} \frac{u^{|λ| deg ϕ}}{c_{ϕ}(λ)} .

Lemma 2.6.

1 − u = ∏_{ϕ} (1 − \frac{u^{deg ϕ}}{q^{deg ϕ}} ) = ∏_{d≥1} (1 − \frac{u^{d}}{q^{d}} )^{ν_{d}}

Proof. Unique factorization in the ring of polynomials F_{q}[z] means that each of the q^{n} monic polynomials of degree n has a unique factorization as a product of monic irreducible polynomials. This implies the factorization of the generating function

∑_{n≥0} q^{n}u^{n} = ∏_{ϕ} ∑_{k≥0} u^{k deg ϕ}.

The left side and the inner sum on the right are geometric series, and so

\frac{1}{1 − qu} = ∏_{ϕ} \frac{1}{1 − u^{deg ϕ}} .

Grouping the factors on the right according to degree we get

\frac{1}{1 − qu} = ∏_{d≥1} (\frac{1}{1 − u^{d}} )^{ν_{d}} .

Substituing u/q for u gives

\frac{1}{1 − u} = ∏_{d≥1} (\frac{1}{1 − (u/q)^{d}} )^{ν_{d}} .

The lemma follows by taking reciprocals.

The final lemma below is used to evaluate limiting probabilities such as lim_{n→∞} a_{n}/γ_{n}. The proof is straightforward and omitted.

Lemma 2.7. If

∑_{n≥0} α_{n}u^{n} = \frac{1}{1 − u}F (u)

where F (u) is analytic and the series for F (1) is convergent, then

lim_{n→∞} α_{n} = F (1).

Now with these lemmas we are ready to prove several results stated in §1.

2.2 Linear and Projective Derangements

In this section we prove the results given in §1.8. Linear derangements are matrices with no eigenvalues of 0 or 1, which means that their characteristic polynomials do not have factors of z or z − 1.

Theorem 2.1. Let e_{n} be the number of n × n linear derangements. Then

1 + ∑_{n≥1} \frac{e_{n}}{γ_{n}} u^{n} = \frac{1}{1 − u} ∏_{r≥1} (1 − \frac{u}{q^{r}} ) .

Proof. In Lemma 3 let Φ′ = Φ \ {z, z − 1} to see that

1 + ∑_{n≥1} \frac{e_{n}}{γ_{n}} u^{n} = ∏_{ϕ∈Φ′} ∏_{r≥1} (1 − \frac{u^{deg ϕ}}{q^{r deg ϕ}} )^{−1} .

On the right side multiply and divide by the product corresponding to ϕ(z) = z, which is

∏_{r≥1} (1 − \frac{u}{q^{r}} )^{−1} ,

to see that

1 + ∑_{n≥1} \frac{e_{n}}{γ_{n}} u^{n} = ∏_{ϕ≠z} ∏_{r≥1} (1 − \frac{u^{deg ϕ}}{q^{r deg ϕ}} )^{−1} ∏_{r≥1} (1 − \frac{u}{q^{r}} ) .

Then use Lemma 4 to give

1 + ∑_{n≥1} \frac{e_{n}}{γ_{n}} u^{n} = \frac{1}{1 − u} ∏_{r≥1} (1 − \frac{u}{q^{r}} ) .

From the generating function we can derive a recursive formula for e_{n}. Corollary 2.1.

e_{n} = e_{n−1}(q^{n} − 1)q^{n−1} + (−1)^{n}q^{n(n−1)/2}, e_{0} = 1.

Proof. From Theorem 8 it follows that e_{n}/γ_{n} is the sum of the u^{i} coefficients of

∏_{r≥1}(1 − u/q^{r})

for i = 0, 1, …, n. Now the u^{i} coefficient is

(−1)^{i} ∑_{1≤r_{1}<r_{2}<···<r_{i}} \frac{1}{q^{r_{1}+r_{2}+···+r_{i}}} .

By induction one can easily show that this coefficient is

\frac{(−1)^{i}}{(q^{i} − 1)(q^{i−1} − 1) ⋯ (q − 1)}.

Therefore

\frac{e_{n}}{γ_{n}} = 1 + ∑_{1≤i≤n} \frac{(−1)^{i}}{(q^{i} − 1)(q^{i−1} − 1) ⋯ (q − 1)}.

Next,

\frac{e_{n}}{γ_{n}} = \frac{e_{n−1}}{γ_{n−1}} + \frac{(−1)^{n}}{(q^{n} − 1) ⋯ (q − 1)}.

Making use of the formula for γ_{n} and γ_{n−1} and canceling where possible we see that e_{n} = e_{n−1}(q^{n} − 1)q^{n−1} + (−1)^{n}q^{n(n−1)/2}.

We present the next result because the sequence e_{n} for q = 2 is given in the OEIS as 2^{n(n−1)/2}a_{n} where a_{n} satisfies a recursive formula. Corollary 2.2. Let

a_{n} = \frac{e_{n}}{q^{n(n−1)/2}} .

Then a_{n} satisfies the recursion

a_{n} = a_{n−1}(q^{n} − 1) + (−1)^{n}, a_{0} = 1.

Proof. The proof follows immediately from the recursive formula for e_{n}.

Corollary 2.3. The asymptotic probability that an invertible matrix is a linear derangement is

lim_{n→∞} \frac{e_{n}}{γ_{n}} = ∏_{r≥1} (1 − \frac{1}{q^{r}} ) .

The asymptotic that any matrix is a linear derangement is

lim_{n→∞} \frac{e_{n}}{q^{n^{2}}} = ∏_{r≥1} (1 − \frac{1}{q^{r}} )^{2} .

Proof. We use Lemma 7 for the first statement. Then

lim_{n→∞} \frac{e_{n}}{q^{n^{2}}} = lim_{n→∞} \frac{e_{n}}{γ_{n}} \frac{γ_{n}}{q^{n^{2}}}

= ( lim_{n→∞} \frac{e_{n}}{γ_{n}} ) ( lim_{n→∞} \frac{γ_{n}}{q^{n^{2}}} ) .

We have just computed the first limit on the right and the second limit (from §1.2) is the same.

The proof of Theorem 8 easily adapts to give the generating function for the number of projective derangements.

Theorem 2.2. Let d_{n} be the number of n × n projective derangements. Then

1 + ∑_{n≥1} \frac{d_{n}}{γ_{n}} u^{n} = \frac{1}{1 − u} ∏_{r≥1} (1 − \frac{u}{q^{r}} )^{q−1} .

Proof. In this case Φ′ = Φ \ {z − a|a ∈ F_{q}}. As in the proof of Theorem 8 this time we multiply and divide by the products corresponding to all the linear polynomials except ϕ(z) = z − 1. There are q − 1 of these and so we get

1 + ∑_{n≥1} \frac{e_{n}}{γ_{n}} u^{n} = ∏_{ϕ≠z} ∏_{r≥1} (1 − \frac{u^{deg ϕ}}{q^{r deg ϕ}} )^{−1} ∏_{r≥1} (1 − \frac{u}{q^{r}} )^{q−1} .

Use Lemma 4 to finish the proof.

It would be interesting to find a recursive formula for the d_{n} analagous to that given in Corollary 9 for the e_{n}.

2.3 Diagonalizable matrices

Theorem 2.3. Let d_{n} be the number of diagonalizable n × n matrices. Then

1 + ∑_{n≥0} \frac{d_{n}}{γ_{n}} u^{n} = (∑_{m≥0} \frac{u^{m}}{γ_{m}} )^{q}

and

d_{n} = ∑_{n_{1}+···+n_{q}=n} \frac{γ_{n}}{γ_{n_{1}} ⋯ γ_{n_{q}}} .

Proof. We use Lemma 5. Diagonalizable matrices have conjugacy class data that only involves the linear polynomials ϕ(z) = z−a, a ∈ F_{q} and partitions λ_{ϕ}(A) that are either empty or have the form 1 ≥ 1 ≥ … ≥ 1. These partitions are indexed by non-negative integer. For ϕ(z) = z − a and for λ the partition consisting of m 1’s, the corresponding m × m matrix in the canonical form is the diagonal matrix with a on the main diagonal. The centralizer subgroup of this matrix is the full general linear group GL_{m}(q) and so c_{ϕ}(λ) = γ_{m}. Then from Lemma 5 we see that

1 + ∑_{n≥0} \frac{d_{n}}{γ_{n}} u^{n} = ∏_{a∈F_{q}} ∑_{m≥0} \frac{u^{m}}{γ_{m}}

= (∑_{m≥0} \frac{u^{m}}{γ_{m}} )^{q} .

From this the formula for d_{n} follows immediately.

It should be noted that the formula for d_{n} follows directly from the knowledge of the centralizer subgroup of a diagonalizable matrix and does not require the full machinery of the cycle index generating function. Also, there is an alternative approach to this problem given in [10].

2.4 Projections

Theorem 2.4. Let p_{n} be the number of n × n projections. Then

1 + ∑_{n≥0} \frac{p_{n}}{γ_{n}} u^{n} = (∑_{m≥0} \frac{u^{m}}{γ_{m}} )^{2}

and

p_{n} = ∑_{0≤i≤n} \frac{γ_{n}}{γ_{i}γ_{n−i}} .

Proof. Since projections are diagonalizable matrices with eigenvalues restricted to be in the set {0, 1}, it follows that the product in the generating function for the cycle index is over the two polynomials z and z −1 and the partitions are restricted as described in the previous theorem. Thus, we see that

1 + ∑_{n≥0} \frac{p_{n}}{γ_{n}} u^{n} = ∏_{z,z−1} ∑_{m≥0} \frac{u^{m}}{γ_{m}}

= (∑_{m≥0} \frac{u^{m}}{γ_{m}} )^{2} .

The formula for p_{n} follows immediately.

2.5 Solutions of A^{k} = I

Theorem 2.5. Let a_{n} be the number of n × n matrices A satisfying A^{2} = I, and assume that the base field F_{q} has characteristic two. Then

a_{n} = ∑_{0≤i≤n/2} \frac{γ_{n}}{q^{i(2n−3i)}γ_{i}γ_{n−2i}} .

Proof. The rational canonical form for A is a direct sum of companion matrices for z −1 and (z − 1)^{2}. No other polynomials occur. Thus, the partition λ_{ϕ}(A) for ϕ(z) = z − 1 consists of b_{1} repetitions of 1 and b_{2} repetitions of 2, where b_{1} + 2b_{2} = n. From Lemma 5 we see that

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∑_{b_{1}, b_{2}} \frac{u^{b_{1}+2b_{2}}}{c_{z−1}(b)},

where b = (b_{1}, b_{2}) denotes the partition. To compute c_{z−1}(b) we refer to the formula stated in the proof of Lemma 2. In that formula we have d_{1} = b_{1} + b_{2} and d_{2} = b_{1} + 2b_{2}, and then

c_{z−1}(b) = ∏_{i=1}^{2} ∏_{k=1}^{b_{i}}(q^{d_{i}} − q^{d_{i}−k}).

This becomes

c_{z−1}(b) = ∏_{k=1}^{b_{1}}(q^{d_{1}} − q^{d_{1}−k}) ∏_{k=1}^{b_{2}}(q^{d_{2}} − q^{d_{2}−k}).

Now let i = d_{1} − k in the first product and let i = d_{2} − k in the second product, so that

c_{z−1}(b) = ∏_{i=b_{2}}^{d_{1}−1}(q^{d_{1}} − q^{i}) ∏_{i=d_{2}−b_{2}}^{d_{2}−1}(q^{d_{2}} − q^{i}).

From each factor of the first product we pull out a factor of q^{b_{2}} and from each factor of the second product we remove a factor of q^{d_{2}−b_{2}}. This gives

c_{z−1}(b) = q^{b_{1}b_{2}}γ_{b_{1}}q^{(d_{2}−b_{2})b_{2}}γ_{b_{2}}.

Then

\frac{a_{n}}{γ_{n}} = ∑_{b_{1}+2b_{2}=n} \frac{1}{c_{z−1}(b)},

where we note that d_{2} = n and b_{1} = n − 2b_{2}. Therefore,

\frac{a_{n}}{γ_{n}} = ∑_{b_{1}+2b_{2}=n} \frac{1}{q^{b_{2}(3n−2b_{2})}γ_{b_{1}}γ_{b_{2}}} ,

which is equivalent to the formula to be proved.

Theorem 2.6. Assume that k is a positive integer relatively prime to q. Let

z^{k} − 1 = ϕ_{1}(z)ϕ_{2}(z) ⋯ ϕ_{r}(z)

be the factorization of z^{k} − 1 over F_{q} into distinct irreducible polynomials with d_{i} = deg ϕ_{i}, and let a_{n} be the number of n × n matrices over F_{q} that are solutions of A^{k} = I. Then

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{i=1}^{r} ∑_{m≥0} \frac{q^{md_{i}}}{γ_{m}(q^{d_{i}})}.

Proof. The rational canonical form of a matrix A satisfying A^{k} = I is a direct sum of any number of copies of the companion matrices of the ϕ_{i}. Thus, with Lemma 5 the product is taken over the ϕ_{i} for i = 1, …, r and the subset L_{ϕ_{i}} is the same for all i and consists of the partitions in which all parts are 1. Hence, L_{ϕ_{i}} can be identified with the non-negative integers m = 0, 1, 2, . . .. For the partition λ given by m 1’s, the value of c_{ϕ_{i}}(λ) is the order of the group of automorphisms of the F_{q}[z]-module F_{q}[z]/(ϕ_{i}^{m}), but this module is the direct sum of m copies of the extension field of degree d_{i} over F_{q}. Therefore, the group of automorphisms is the general linear group GL_{m}(q^{d_{i}}), and so c_{ϕ_{i}}(λ) = γ_{m}(q^{d_{i}}).

2.6 Cyclic matrices

Theorem 2.7. Let a_{n} be the number of cyclic n × n matrices. Then

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{d≥1} (1 + \frac{1}{q^{d} − 1} \frac{u^{d}}{1 − (u/q)^{d}} )^{ν_{d}}

and

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = \frac{1}{1 − u} ∏_{d≥1} (1 + \frac{u^{d}}{q^{d}(q^{d} − 1)})^{ν_{d}} .

Proof. In terms of the conjugacy class data a matrix is cyclic if λ_{ϕ} has at most one part for each ϕ. Thus, L_{ϕ} = {∅, 1, 2, …}, where m means the partition of m having just one part. For these partitions we have c_{ϕ}(∅) = 1 and c_{ϕ}(m) = q^{m deg ϕ} − q^{(m−1) deg ϕ}. Let a_{n} be the number of cyclic matrices of size n × n. Beginning with Lemma 5 we find the generating function.

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{ϕ} ∑_{L_{ϕ}} \frac{u^{|λ| deg ϕ}}{c_{ϕ}(λ)}

= ∏_{ϕ} (1 + ∑_{m≥1} \frac{u^{m deg ϕ}}{q^{m deg ϕ} − q^{(m−1) deg ϕ}} )

= ∏_{d≥1} (1 + ∑_{m≥1} \frac{u^{md}}{q^{md} − q^{(m−1)d}} )^{ν_{d}}

= ∏_{d≥1} (1 + \frac{1}{q^{d} − 1} ∑_{m≥1} \frac{u^{md}}{q^{(m−1)d}} )^{ν_{d}}

= ∏_{d≥1} (1 + \frac{1}{q^{d} − 1} \frac{u^{d}}{1 − (u/q)^{d}} )^{ν_{d}} .

For the second part, we use Lemma 6 to get

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = \frac{1}{1 − u} ∏_{d≥1} (1 − \frac{u^{d}}{q^{d}} )^{ν_{d}} ∏_{d≥1} (1 + \frac{1}{q^{d} − 1} \frac{u^{d}}{1 − (u/q)^{d}} )^{ν_{d}} .

Combine the products and simplify to finish the proof.

2.7 Semi-simple matrices

Theorem 2.8. Let a_{n} be the number of n × n semi-simple matrices over F_{q}. Then

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{d≥1} (1 + ∑_{j≥1} \frac{u^{jd}}{γ_{j}(q^{d})})^{ν_{d}} .

Proof. In order for a matrix to be semi-simple the partition associated to a polynomial ϕ has no parts greater than 1. Thus, for all ϕ, L_{ϕ} = {∅, 1, 1^{2}, …, 1^{j}, …}, where 1^{j} means the partition of j consisting of j copies of 1. Then c_{ϕ}(1^{j}) is the order of the automorphism group of the F_{q}[z]-module which is the direct sum of j copies of F_{q}[z]/(ϕ), which can be identified with (F_{q^{d}})^{j}, where d = deg ϕ. Furthermore, the automorphisms of the module (F_{q}[z]/(ϕ))^{⊕j} can be identified with the F_{q^{d}}-linear automorphisms of (F_{q^{d}})^{j}. This group of automorphisms is GL_{j}(q^{d}), and so c_{ϕ}(1^{j}) = γ_{j}(q^{d}).

From Lemma 5 we find

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{ϕ} ∑_{L_{ϕ}} \frac{u^{|λ| deg ϕ}}{c_{ϕ}(λ)}

= ∏_{ϕ} (1 + ∑_{j≥1} \frac{u^{j deg ϕ}}{γ_{j}(q^{d})} )

= ∏_{d≥1} (1 + ∑_{j≥1} \frac{u^{jd}}{γ_{j}(q^{d})})^{ν_{d}} .

2.8 Separable matrices

Theorem 2.9. Let a_{n} be the number of separable n × n matrices over F_{q}. Then

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{d≥1} (1 + \frac{u^{d}}{q^{d} − 1})^{ν_{d}}

and

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = \frac{1}{1 − u} ∏_{d≥1} (1 + \frac{u^{d}(u^{d} − 1)}{q^{d}(q^{d} − 1)})^{ν_{d}} .

Proof. For a matrix to be separable the allowed partitions are either empty or the unique partition of 1. So the sum over L_{ϕ} in Lemma 5 is a sum of two terms. For λ = ∅,

\frac{u^{|λ| deg ϕ}}{c_{ϕ}(λ)} = 1,

and for λ = 1,

\frac{u^{|λ| deg ϕ}}{c_{ϕ}(λ)} = \frac{u^{deg ϕ}}{q^{deg ϕ} − 1}.

Therefore,

1 + ∑_{n≥1} \frac{a_{n}}{γ_{n}} u^{n} = ∏_{ϕ} ∑_{L_{ϕ}} \frac{u^{|λ| deg ϕ}}{c_{ϕ}(λ)}

= ∏_{ϕ} (1 + \frac{u^{deg ϕ}}{q^{deg ϕ} − 1})

= ∏_{d≥1} (1 + \frac{u^{d}}{q^{d} − 1})^{ν_{d}} .

For the second statement of the theorem multiply the right side in the line above by

\frac{1}{1 − u} ∏_{d≥1} (1 − \frac{u^{d}}{q^{d}} )^{ν_{d}} ,

which is equal to 1 by Lemma 6. Then combine the products.

2.9 Conjugacy classes

Theorem 2.10. Let a_{n} be the number of conjugacy classes of n × n matrices. Then the ordinary generating function for the sequence {a_{n}} is given by

1 + ∑_{n≥1} a_{n}u^{n} = ∏_{r≥1} \frac{1}{1 − qu^{r}} .

Proof. Recall the ordinary power series generating function for partitions factors as the infinite product

∑_{n≥0} p_{n}u^{n} = ∏_{r≥1} \frac{1}{1 − u^{r}} ,

where p_{n} is the number of partitions of the integer n. A conjugacy class of n × n matrices is uniquely specified by the choice of a partition λ_{ϕ} for each monic irreducible polynomial ϕ such that ∑_{ϕ} |λ_{ϕ}| deg ϕ = n. Consider the infinite product over ϕ

∏_{ϕ} ∑_{n≥0} p_{n}u^{n deg ϕ}.

The coefficient of u^{n} is a sum of terms of the form p_{n_{1}}u^{n_{1} deg ϕ_{1}} ⋯ p_{n_{k}}u^{n_{k} deg ϕ_{k}} where n = ∑ n_{k} deg ϕ_{k}. Therefore, the u^{n}-coefficient is the number of conjugacy classes of n × n matrices. Then

1 + ∑_{n≥1} a_{n}u^{n} = ∏_{ϕ} ∑_{n≥0} p_{n}u^{n deg ϕ}

= ∏_{ϕ} ∏_{r≥1} \frac{1}{1 − u^{r deg ϕ}}

= ∏_{r≥1} ∏_{ϕ} \frac{1}{1 − u^{r deg ϕ}}

= ∏_{r≥1} ∏_{d≥1} (\frac{1}{1 − u^{rd}} )^{ν_{d}}

Then by substituting qu^{r} for u in Lemma 6 and inverting we see that

∏_{d≥1} (\frac{1}{1 − u^{rd}} )^{ν_{d}} = \frac{1}{1 − qu^{r}} .

Therefore

1 + ∑_{n≥1} a_{n}u^{n} = ∏_{r≥1} \frac{1}{1 − qu^{r}} .

References

[1] Edward A. Bender and Jay R. Goldman, Enumerative uses of generating functions, Indiana Univ. Math. J. 20 (1970/1971), 753–765.

[2] Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, Cambridge, 1994.

[3] Naoki Chigira, Yugen Takegahara, and Tomoyuki Yoshida, On the number of homomorphisms from a finite group to a general linear group, J. Algebra 232 (2000), 236–254.

[4] M. C. Crabb, Counting nilpotent endomorphisms, Finite Fields Appl. 12 (2006), 151– 154.

[5] N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math. 2 (1958), 499–504.

[6] Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 51–85 (electronic).

[7] Murray Gerstenhaber, On the number of nilpotent matrices with coefficients in a finite field, Illinois J. Math. 5 (1961), 330–333.

[8] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Ed., The Clarendon Press Oxford University Press, New York, 1979.

[9] Joseph P. S. Kung, The cycle structure of a linear transformation over a finite field, Linear Algebra Appl. 36 (1981), 141–155.

[10] Kent E. Morrison, q-exponential families, Electron. J. Combin. 11 (2004), Research Paper 36, 11 pp. (electronic).

[11] Peter M. Neumann and Cheryl E. Praeger, Cyclic matrices over finite fields, J. London Math. Soc. 52 (1995), 263–284.

[12] Richard Stong, Some asymptotic results on finite vector spaces, Adv. in Appl. Math. 9 (1988), 167–199.

[13] G. E. Wall, Counting cyclic and separable matrices over a finite field, Bull. Austral. Math. Soc. 60 (1999), 253–284.

2000 Mathematics Subject Classification: Primary 05A15, 15A33; Secondary 05A10, 05A30, 11B65, 11B73. Keywords: Integer sequences, matrices over finite fields, cycle index for matrices.

(Concerned with sequences \underline{A002416}, \underline{A002820}, \underline{A002884}, \underline{A005329}, \underline{A006116}, \underline{A006117}, \underline{A006118}, \underline{A006119}, \underline{A006120}, \underline{A006121}, \underline{A006122}, \underline{A015195}, \underline{A015196}, \underline{A015197}, \underline{A015198}, \underline{A015199}, \underline{A015200}, \underline{A015201}, \underline{A015202}, \underline{A015203}, \underline{A015204}, \underline{A015205}, \underline{A015206}, \underline{A015207}, \underline{A015208}, \underline{A015209}, \underline{A015210}, \underline{A015211}, \underline{A015212}, \underline{A015213}, \underline{A015214}, \underline{A015215}, \underline{A015216}, \underline{A015217}, \underline{A022166}, \underline{A022167}, \underline{A022168}, \underline{A022169}, \underline{A022170}, \underline{A022171}, \underline{A022172}, \underline{A022173}, \underline{A022174}, \underline{A022175}, \underline{A022176}, \underline{A022177}, \underline{A022178}, \underline{A022179}, \underline{A022180}, \underline{A022181}, \underline{A022182}, \underline{A022183}, \underline{A022184}, \underline{A022185}, \underline{A022186}, \underline{A022187}, \underline{A022188}, \underline{A053718}, \underline{A053722}, \underline{A053725}, \underline{A053763}, \underline{A053770}, \underline{A053771}, \underline{A053772}, \underline{A053773}, \underline{A053774}, \underline{A053775}, \underline{A053776}, \underline{A053777}, \underline{A053846}, \underline{A053847}, \underline{A053848}, \underline{A053849}, \underline{A053851}, \underline{A053852}, \underline{A053853}, \underline{A053854}, \underline{A053855}, \underline{A053856}, \underline{A053857}, \underline{A053859}, \underline{A053860}, \underline{A053861}, \underline{A053862}, \underline{A053863}, \underline{A069777}, \underline{A070933}, \underline{A076831}, and \underline{A082877}.

Received May 13 2005; revised version received February 10 2006. Published in Journal of Integer Sequences, April 17 2006.

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