The Young lattice of integer partitions is well known for its role in representation theory and the study of symmetric functions. It is simultaneously the Bratelli diagram of the symmetric groups, expressing how irreducible representations of restrict to , and also responsible for how Schur functions multiply --- as distilled in the celebrated Littlewood-Richardson rule.
In 1988 R. Stanley introduced a similar lattice called the Young-Fibonacci (YF) lattice, which consists of Fibonacci words: i.e. binary words made of the digits 1 and 2 and ranked according to the sum of their digits. In 1994 Okada showed that the YF-lattice is the Bratelli diagram of a tower of semi-simple algebras and also responsible for
the YF-analogue of a Littlewood-Richardson rule governing multiplication of "clone" Schur functions.
In joint work with L. Petrov, we introduce a system of coherent measures on the YF-lattice using certain positive biserial specializations of Okada's clone Schur functions. We characterize these specializations using the theory of totally positive tridiagonal matrices and describe the asymptotic behaviour of random Fibonacci words sampled with respect to the associated coherent measures. Our results have connections with Stieltjes moment sequences, orthogonal polynomials from the (q-)Askey scheme, and residual allocation (stick-breaking) models.