Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T20:09:35.905Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the Screaming Toes game

Part of: Combinatorial probability Combinatorics, graph theory, probability theory, statistics Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  17 January 2022

Simon Tavaré Open the ORCID record for Simon Tavaré [Opens in a new window]
Simon Tavaré*
Affiliation:
Columbia University
*
*Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA. Email address: st3193@columbia.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate properties of random mappings whose core is composed of derangements as opposed to permutations. Such mappings arise as the natural framework for studying the Screaming Toes game described, for example, by Peter Cameron. This mapping differs from the classical case primarily in the behaviour of the small components, and a number of explicit results are provided to illustrate these differences.

Keywords

Random mappingderangementPoisson approximationPoisson–Dirichlet distributionprobabilistic combinatoricssimulationcomponent sizecycle size

MSC classification

Primary: 60C05: Combinatorial probability 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 65C05: Monte Carlo methods
Secondary: 97K20: Combinatorics 97K60: Distributions and stochastic processes 65C40: Computational Markov chains
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 1 , March 2022 , pp. 118 - 130
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arratia, R., Barbour, A., Ewens, W. and Tavaré, S. (2018). Simulating the component counts of combinatorial structures. Theoret. Pop. Biol. 122, 5–11.CrossRefGoogle ScholarPubMed
Arratia, R., Barbour, A. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society Publishing House, Zurich.CrossRefGoogle Scholar
Bollobás, B. (2001). Random Graphs, 2nd ed. Academic Press, New York.CrossRefGoogle Scholar
Cameron, P. J. (2017). Notes on Counting: An Introduction to Enumerative Combinatorics. Cambridge University Press.CrossRefGoogle Scholar
Csardi, G. and Nepusz, T. (2006). The igraph software package for complex network research. InterJournal Complex Systems, 1695.Google Scholar
da Silva, P. H., Jamshidpey, A. and Tavaré, S. (2021). The Feller Coupling for random derangements. To appear in Stoch. Process. Appl. CrossRefGoogle Scholar
Donnelly, P., Ewens, W. J. and Padmadisastra, S. (1991). Random functions: Exact and asymptotic results. Adv. Appl. Prob. 23, 437–455.CrossRefGoogle Scholar
Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3, 87–112.CrossRefGoogle ScholarPubMed
Feller, W. (1970). An Introduction to Probability Theory and its Applications Vol. II. John Wiley, New York.Google Scholar
Harris, B. (1960). Probability distributions related to random mappings. Ann. Math. Stat. 31, 1045–1062.CrossRefGoogle Scholar
Kolchin, V. F. (1976). A problem of allocation of particles in cells and random mappings. Theory Prob. Appl. 21, 48–63.10.1137/1121004CrossRefGoogle Scholar
Spitzer, F. (1956). A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339.CrossRefGoogle Scholar