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From Hammersley’s lines to Hammersley’s trees

  • A.-L. Basdevant
  • L. Gerin
  • J.-B. Gouéré
  • A. SinghEmail author
Article

Abstract

We construct a stationary random tree, embedded in the upper half plane, with prescribed offspring distribution and whose vertices are the atoms of a unit Poisson point process. This process which we call Hammersley’s tree process extends the usual Hammersley’s line process. Just as Hammersley’s process is related to the problem of the longest increasing subsequence, this model also has a combinatorial interpretation: it counts the number of heaps (i.e. increasing trees) required to store a random permutation. This problem was initially considered by Byers et al. (ANALCO11, workshop on analytic algorithmics and combinatorics, pp 33–44, 2011) and Istrate and Bonchis (Partition into Heapable sequences, heap tableaux and a multiset extension of Hammersley’s process. Lecture notes in computer science combinatorial pattern matching, pp 261–271, 2015) in the case of regular trees. We show, in particular, that the number of heaps grows logarithmically with the size of the permutation.

Keywords

Hammersley’s process Heap sorting Patience sorting Longest increasing subsequences Interacting particles systems 

Mathematics Subject Classification

60K35 60G55 

Notes

Acknowledgements

The authors warmly thank Nathanaël Enriquez for stimulating discussions on the topic.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Laboratoire Modal’XUniversité Paris NanterreNanterreFrance
  2. 2.CMAP, Ecole PolytechniquePalaiseauFrance
  3. 3.Laboratoire de MathématiquesUniversité de ToursToursFrance
  4. 4.Laboratoire de Mathématiques d’Orsay, CNRSUniv. Paris-SudVillejuifFrance

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