From Hammersley's lines to Hammersley's trees
Anne-Laure Basdevant (MODAL'X), Lucas Gerin (CMAP), Jean-Baptiste, Gouere, Arvind Singh (LM-Orsay)

TL;DR
This paper introduces Hammersley's tree process, a stationary random tree model extending Hammersley's line process, which counts heaps needed for random permutations and grows logarithmically with permutation size.
Contribution
It constructs a new stationary random tree model with prescribed offspring distribution, extending Hammersley's process and linking it to heap counting in permutations.
Findings
Number of heaps grows logarithmically with permutation size
Model extends Hammersley's line process to trees
Provides a combinatorial interpretation for the process
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 (2011) and Istrate and Bonchis (2015) in the case of regular trees. We show, in particular, that the number of heaps grows logarithmically with the size of the permutation.
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