Geodesic
Multiplicative distributions on Young graph
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Tome 240 (1997), pp. 245-256

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We describe all non-degenerate multiplicative distributions on the Young graph and on the graph of truncated Young diagrams. All such distributions are parametrized for the Young graph by two real numbers and by one real number for the graph of truncated Young diagrams. 
@article{ZNSL_1997_240_a14,
     author = {N. A. Rozhkovskaya},
     title = {Multiplicative distributions on {Young} graph},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {245--256},
     publisher = {mathdoc},
     volume = {240},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_240_a14/}
}
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N. A. Rozhkovskaya. Multiplicative distributions on Young graph. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Tome 240 (1997), pp. 245-256. http://geodesic.mathdoc.fr/item/ZNSL_1997_240_a14/