A derangement is a permutation that has no fixed points. In this paper, we are interested in the proportion of derangements of the finite affine general linear groups. We prove a remarkably simple and explicit formula for this proportion. We also give a formula for the proportion of derangements of prime power order. Both formulae rely on a result of independent interest on partitions: we determine the generating function for the partitions with m parts and with the kth largest part not k, for every k∈ N.
Spiga, P. (2017). On the number of derangements and derangements of prime power order of the affine general linear groups. JOURNAL OF ALGEBRAIC COMBINATORICS, 45(2), 345-362 [10.1007/s10801-016-0709-3].
On the number of derangements and derangements of prime power order of the affine general linear groups
Spiga, P
2017
Abstract
A derangement is a permutation that has no fixed points. In this paper, we are interested in the proportion of derangements of the finite affine general linear groups. We prove a remarkably simple and explicit formula for this proportion. We also give a formula for the proportion of derangements of prime power order. Both formulae rely on a result of independent interest on partitions: we determine the generating function for the partitions with m parts and with the kth largest part not k, for every k∈ N.
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