The Number of Solutions of X^2=0 in Triangular Matrices Over GF(q)

We prove an explicit formula for the number of n × n upper triangular matrices, over GF (q), whose square is the zero matrix. Theorem. The number of n × n upper-triangular matrices over GF (q) (the finite field with q elements), whose square is the zero matrix, is given by the polynomial C n (q), where, C 2n (q) = j 2n n − 3j − 2n n − 3j − 1 · q n 2 −3j 2 −j , C 2n+1 (q) = j 2n + 1 n − 3j − 2n + 1 n − 3j − 1 · q n 2 +n−3j 2 −2j. Proof. In [K] it was shown that the quantity of interest is given by the polynomial A n (q) = r≥0 A r n (q), where the polynomials A r n (q) are defined recursively by A r+1 n+1 (q) = q r+1 · A r+1 n (q) + (q n−r − q r) · A r n (q) ; A 0 n+1 (q) = 1. (Sasha) For any Laurent formal power series P (w), let CT w P (w) denote the coefficient of w 0. Recall that the q-binomial coefficients are defined by m n q := (1 − q m)(1 − q m−1) · · · (1 − q m−n+1) (1 − q)(1 − q 2) · · · (1 − q n) , (Carl) whenever 0 ≤ n ≤ m, and 0 otherwise. Lemma 1. We have A r n (q) = CT w (1 − w)(1 + w) n q r(n−r)