Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The Number of Solutions of X 2 = 0 in Triangular Matrices Over GF . | The Number of Solutions of X2 0 in Triangular Matrices Over GF q Shalosh B. EKHAD1 and Doron ZEILBERGER1 Abstract We prove an explicit formula for the number of n X n upper triangular matrices over GF q whose square is the zero matrix. This formula was recently conjectured by Sasha Kirillov and Anna Melnikov KM . Theorem. The number of n X n upper-triangular matrices over GF q the Snite held with q elements whose square is the zero matrix is given by the polynomial Cn q where C2n q j 2n _ n - 3j 2n n 3j 1 qn2-3j2-j C2n 1 q j 2n A _ i 2n 1 n 3 j n 3 j 1 qn2 n-3j2-2j Proof. In K it was shown that the quantity of interest is given by the polynomial An q A r 0 An q where the polynomials Arn q are defined recursively by An 1 q qr 1 An 1 q qn-r qr Affiq A q 1. Sasha For any Laurent formal power series P w let CTw P w denote the coefficient of w . Recall that the q-binomial coefficients are defined by m _ 1 qm 1 qm-1 1 qm-n 1 n q 1 q 1 q2 1 qn Carl whenever 0 n m and 0 otherwise. Lemma 1. we have Arn q CTw 1 w 1 w nqr n-r wr lb 1 V i 0 i 1 i 2-i n-2r i n 2r wi Anna Proof. Call the right side of Eq. Anna sn q . Since sn 1 q 1 the lemma would follow by induction if we could show that sn 1 q qr 1 sn 1 q qn-r qr sn q 0. Sasha 1 Department of Mathematics Temple University Philadelphia PA 19122 USA. ekhad zeilberg @ http fekhad zeilberg ftp pub ekhad zeilberg . Supported in part by the NSF. Nov. 28 1995. 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 3 1996 R2 2 Using the linearity of CTw manipulating the series using the dehnition Carl of the q binomial coefficients and simplifying brings the left side of Sasha to be CTw n q w where n is zero except when n is odd and r n 1 2 in which case it is a monomial in q times 1-w N I1w and applying CTw kills it all the same thanks to the symmetry of the Chu-Pascal triangle. Summing the expression proved for An q yields that An q CTw 1 w 1 w n T T 1 iqr n-r - i 1 i 2-i n-2r ii n 2 i wi-r
