Journal of Integer Sequences, Vol. 23 (2020), Article 20.11.1

Fixed Points in Compositions and Words

M. Archibald, A. Blecher, and A. Knopfmacher
The John Knopfmacher Centre for
Applicable Analysis and Number Theory
University of the Witwatersrand
South Africa


We study fixed points in compositions (ordered partitions) of integers and words. A fixed point is a point with value i in position i. Using generating functions and probabilistic arguments, we enumerate the compositions and words with no fixed points and p fixed points and also how many fixed points occur on average. We briefly discuss the average maximum (respectively minimum) fixed point and the sum of sizes of fixed points. Moreover we provide asymptotic results for the above parameters.

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(Concerned with sequences A026097 A099036 A238349 A238351 A240736 A240737 A335712 A335713 A335714.)

Received March 23 2020; revised version received July 15 2020; October 28 2020. Published in Journal of Integer Sequences, November 7 2020.

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