Geodesic
The boundary of the refined Kingman graph
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 58-74

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We introduce the refined Kingman graph $\mathbb D$ whose vertices are indexed by the set of compositions of positive integers and multiplicity function reflects the Pieri rule for quasisymmetric monomial functions. We show that the Martin boundary of $\mathbb D$ can be identified with the space $\Omega$ of all sets of disjoint open subintervals of $[0,1]$ and coincides with the minimal boundary of $\mathbb D$. 
@article{ZNSL_2018_468_a5,
     author = {M. V. Karev and P. P. Nikitin},
     title = {The boundary of the refined {Kingman} graph},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {58--74},
     publisher = {mathdoc},
     volume = {468},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a5/}
}
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M. V. Karev; P. P. Nikitin. The boundary of the refined Kingman graph. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 58-74. http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a5/