Abstract
We study fixed points in integer partitions viewed, respectively, as weakly increasing or weakly decreasing structures. A fixed point is a point with value i in position i. We also study matching points in weakly decreasing partitions. These are defined as positions where the partition and its reverse have the same size parts. From the generating functions, we also obtain asymptotic estimates as \(n\rightarrow \infty \) of some of the above statistics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G., Eriksson, K.: Integer Partitions. Cambridge University Press, Cambridge (2004)
Archibald, M., Blecher, A., Knopfmacher, A.: Fixed points in compositions and words. J. Integer. Sequences 23 (2020)
Archibald, M., Blecher, A., Elizalde, S., Knopfmacher, A.: Subdiagonal and superdiagonal partitions (in preparation)
Arratia, R., Tavaré, S.: The cycle structure of random permutations. Ann. Probab. 20, 1567–1591 (1992)
Auluck, F.C.: On some new types of partitions associated with generalised Ferrers graphs. Proc. Camb. Philos. Soc. 47, 679–686 (1951)
Blecher, A., Brennan, C., Mansour, T.: Compositions of n as alternating sequences of increasing and strictly decreasing partitions. Central Eur. J. Math. 10, 788–796 (2012)
Bóna, M.: On a balanced property of derangements. Electron. J. Comb. 13, R102 (2006)
Brualdi, R.A.: Introductory Combinatorics, 5th edn. Prentice-Hall, Hoboken (2010)
Cameron, P.J.: Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press, Cambridge (1994)
Deutsch, E., Elizalde, S.: The largest and the smallest fixed points of permutations. Eur. J. Comb. 31(5), 1404–1409 (2010)
Diaconis, P., Fulman, J., Guralnick, R.: On fixed points of permutations. J. Algebr. Comb. 28, 189–218 (2008)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2008)
Grabner, P., Knopfmacher, A., Wagner, S.: A general asymptotic scheme for moments of partition statistics. Combinatorics, Probability and Computing 23, 1057–1086 (2014). special issue dedicated to Philippe Flajolet
Han, G.-N., Xin, G.: Permutations with extremal number of fixed points. J. Comb. Theory Ser. A 116, 449–459 (2009)
Ingham, A.E.: A Tauberian theorem for partitions. Ann. Math. 42, 1075–1090 (1941)
Knopfmacher, A., Prodinger, H.: On Carlitz compositions. Eur. J. Comb. 19, 579–589 (1998)
Mansour, T., Rastegar, R.: Fixed points of a random restricted growth sequence. https://arxiv.org/pdf/2012.06891.pdf
Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge University Press, Cambridge (1986)
Wilf, H.S.: Generatingfunctionology. Peters (now CRC Press), Boca Raton (1990)
Yang, M.: Relaxed partitions. https://sites.math.rutgers.edu/ my237/RP.pdf
Acknowledgements
We would like to thank the referee for his/her very thorough reading of our manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Blecher, A., Knopfmacher, A. Fixed points and matching points in partitions. Ramanujan J 58, 23–41 (2022). https://doi.org/10.1007/s11139-022-00551-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-022-00551-x