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THE DIMENSION OF CENTRALISERS OF MATRICES OF ORDER $n$

Part of: Combinatorics

Published online by Cambridge University Press:  26 September 2016

DONG ZHANG  and
HANCONG ZHAO
DONG ZHANG
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China email [email protected], [email protected]
HANCONG ZHAO*
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China email [email protected]
*
✉[email protected]
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Abstract

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

Keywords

centraliserasymptotic enumerationpartitioncommutator

MSC classification

Primary: 05A05: Permutations, words, matrices
Secondary: 05A16: Asymptotic enumeration 05A17: Partitions of integers
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 3 , December 2016 , pp. 353 - 361
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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