It is an old result that the number of nilpotent matrices with entries
in GF(q) is q^{n(n−1)}. Proofs have been given by
Philip Hall (1955), Fine & Herstein (1958), Gerstenhaber (1961),
Crabb (2006) and others.
In a more general setting
(of which this is the GL(n,q), that is, A_{n−1} case)
Steinberg (1968) showed that the number of nilpotent matrices equals
q^{|Φ|}, where Φ is the root system.
For example, the number of nilpotent skew-symmetric matrices
is q^{2m(m−1)} for n=2m, and q^{2m^2} for n=2m+1.

Clearly, for order 1 there is precisely 1. A little bit of work shows that the number of symmetric nilpotent matrices of order 3 equals q^3 + q^2 − q. Not so nice, but at least a polynomial in q. However, the number of symmetric nilpotent matrices of order 2 is 2q−1 for q=1 (mod 4), 1 for q=3 (mod 4), and q for q even. Not a polynomial in q. That means that we are doing something wrong.

If we choose a basis, then g is given by a matrix G, the Gram matrix
of the basis, via g(x,y) = x^{T} G y (with column vectors x,y).
Let us call the form g for which G is the identity matrix
the *standard form*.
It is given by g(x,y) = Σ x_{i}y_{i}.
A matrix is symmetric if and only if it is selfadjoint wth respect
to the standard form.

For even q, the symplectic and standard forms are easily distinguished: for a symplectic form G has zero diagonal, for a standard form it has not.

For odd q, and even n, the standard form is the orthogonal direct sum of n/2 2-dimensional spaces with standard form. This 2-dimensional space is hyperbolic for q=1 (mod 4), elliptic for q=3 (mod 4). This means that V (with standard form) is elliptic when q=3 (mod 4) and n/2 is odd, and is hyperbolic otherwise.

Let us call these counts e(2m), h(2m), p(2m+1) (for odd q), and z(2m), s(2m), s(2m+1) (for even q).

**Theorem.** All of e(2m), h(2m), p(2m+1), z(2m), s(2m), s(2m+1)
are polynomials in q.

**Theorem.** p(2m+1) = s(2m+1).

Put a(2m) = (h(2m)+e(2m))/2 and d(2m) = (h(2m)−e(2m))/2, so that h(2m) = a(2m)+d(2m) and e(2m) = a(2m)−d(2m).

**Theorem.** s(2m) = a(2m).

**Theorem.** z(2m) = q^{2m^2}.

**Theorem.** p(2m+1) = q^{2m}a(2m) + q^{m}d(2m).

**Theorem.** p(2m+1) = (q^{2m}−1)a(2m) + z(2m).

**Theorem.** a(2m) = q^{2m−1} p(2m−1).

This proves the corollary, conjectured by Sheekey:

**Corollary.**
p(2m+1) = q^{2m−1} (q^{2m}−1) p(2m−1) + q^{2m^2}.

**Conjecture.**

(i) p(2m+1,2s+1) = [2m−2s] p(2m+1,2s).

(ii) a(2m,2s+1) = [2m−2s−1] a(2m,2s).

(iii) d(2m,2s) = [2m−2s] d(2m,2s−1).

(iv) [2m]a(2m,r) = [2m−r]p(2m+1,r).

(v) p(2m+1,2s) = q^{s(s+1)} Π_{0≤i≤s−1} [2m−2i] .
Σ_{0≤i≤s}
q^{(s−i)(2m−2s−1)} Qnom(m−s−1+i,i,q^{2}).

(vi) d(2m,2s+1) = (q−1)q^{m+s(s+1)−1} Π_{1≤i≤s} [2m−2i] .
Σ_{0≤i≤s}
q^{(s−i)(2m−2s−3)} Qnom(m−s−1+i,i,q^{2}).

Once more, as a picture:

so that

We can give an explicit recurrence for these functions.
Let p_{0}, h_{0}, e_{0} count the
selfadjoint nilpotent N such that Nx=0 for a fixed nonzero isotropic vector x.
Define a_{0}, d_{0} in the obvious way.

**Proposition.**
These functions are recursively given by the following:

(i) [2m+1−r] p(2m+1,r) = [2m] p_{0}(2m+1,r) +
q^{2m}(q−1) a(2m,r) + q^{m}(q−1) d(2m,r).

(ii) [2m−r] a(2m,r) = [2m−1] a_{0}(2m,r) +
q^{m−1}(q−1) d_{0}(2m,r) + q^{2m−1}(q−1) p(2m−1,r).

(iii) [2m−r] d(2m,r) = [2m−1] d_{0}(2m,r) +
q^{m−1}(q−1) a_{0}(2m,r) − q^{m−1}(q−1) p(2m−1,r).

And for f any of p,h,e,a,d:

(iv) f_{0}(n,r) = q^{r} f(n−2,r) +
(q−1)q^{r−1} f(n−2,r−1) +
[n−r]q^{r−1} f(n−2,r−2).

Here f(n,r) = f_{0}(n,r) = 0 for r < 0 or r > n or r = n > 0.
As start of the induction only h(0,0) = 1 is needed.

Let A_{n}(X) = Σ_{r} a(n,r) X^{r}.
Conjecture: A_{2m}(−1/q) = q^{2m(m−2)+1}.

Let D_{n}(X) = Σ_{r} d(n,r) X^{r}.
Conjecture: D_{2m}(−1/q) = −(q−1)q^{m(2m−3)}.

Suppose n > 0. Then the exponent is n iff the rank is n−1. The number of symmetric nilpotent matrices of order n and rank n−1 equals

p(2m+1,2m) = qfor n = 2m+1. If q is odd and n=2m is even, then^{m^2}[2] [4] ... [2m]

a(2m,2m−1) = d(2m,2m−1) = qSo, twice this if the form is hyperbolic, 0 if the form is elliptic. For even q, the count is a(2m,2m−1) if the form is standard, and q^{m(m−1)}[m] [2] [4] ... [2m−2].

Recursion via f_{0}, as was done by Lusztig.

Recursion from all matrices to nilpotent matrices via Fitting decomposition, as done by Gow & Sheekey.

Send corrections, additions, theorems, proofs to aeb@cwi.nl.