Geodesic
A $q$-analog of the hook walk algorithm for random Young tableaux
Journal of Algebraic Combinatorics, Tome 2 (1993) no. 4, pp. 383-396.

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Summary: A probabilistic algorithm, called the q-hook walk, is defined. For a given Young diagram, it produces a new one by adding a random box with probabilities, depending on a positive parameter $q$. The corresponding Markov chain in the space of infinite Young tableaux is closely related to the knot invariant of Jones, constructed via traces of Hecke algebras. For $q = 1$, the algorithm is essentially the hook walk of Greene, Nijenhuis, and Wilf. The $q$-hook formula and a $q$-deformation of Young graph are also considered.
Keywords: Young diagram, random Young tableau, hook formula, q-analog, Hecke algebra 
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     author = {Kerov, S.},
     title = {A $q$-analog of the hook walk algorithm for random {Young} tableaux},
     journal = {Journal of Algebraic Combinatorics},
     pages = {383--396},
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Kerov, S. A $q$-analog of the hook walk algorithm for random Young tableaux. Journal of Algebraic Combinatorics, Tome 2 (1993) no. 4, pp. 383-396. http://geodesic.mathdoc.fr/item/JAC_1993__2_4_a2/