THE DIMENSION OF CENTRALISERS OF MATRICES OF ORDER

In this paper, we study the integer sequence $(E_{n})_{n\geq 1}$ , where $E_{n}$ counts the number of possible dimensions for centralisers of $n\times n$ matrices. We give an example to show another combinatorial interpretation of $E_{n}$ and present an implicit recurrence formula for $E_{n}$ , which may provide a fast algorithm for computing $E_{n}$ . Based on the recurrence, we obtain the asymptotic formula $E_{n}=\frac{1}{2}n^{2}-\frac{2}{3}\sqrt{2}n^{3/2}+O(n^{5/4})$ .