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$E_{n}=\frac{1}{2}n^{2}-\frac{2}{3}\sqrt{2}n^{3/2}+O(n^{5/4})$.