Random Fibonacci Words via Clone Schur Functions

Abstract We investigate positivity and probabilistic properties arising from the Young–Fibonacci lattice YF, a 1-differential poset on words composed of 1’s and 2’s (Fibonacci words) and graded by the sum of the digits. Building on Okada’s theory of clone Schur functions, we introduce clone coherent measures on YF which give rise to random Fibonacci words of increasing length. Unlike coherent systems associated to classical Schur functions on the Young lattice of integer partitions, clone coherent measures are generally not extremal on YF. Our first main result is a complete characterization of Fibonacci positive specializations – parameter sequences which yield positive clone Schur functions on YF. Second, we establish a broad array of correspondences that connect Fibonacci positivity with: (i) the total positivity of tridiagonal matrices; (ii) Stieltjes moment sequences; (iii) the combinatorics of set partitions; and (iv) families of univariate orthogonal polynomials from the (q -) Askey scheme. We further link the moment sequences of broad classes of orthogonal polynomials to combinatorial structures on Fibonacci words, a connection that may be of independent interest. Our third family of results concerns the asymptotic behavior of random Fibonacci words derived from various Fibonacci positive specializations. We analyze several limiting regimes for specific examples, revealing stick-breaking-like processes (connected to GEM distributions), dependent stick-breaking processes of a new type, or limits supported on the discrete component of the Martin boundary of the Young–Fibonacci lattice. Our stick-breaking-like scaling limits significantly extend the result of Gnedin–Kerov on asymptotics of the Plancherel measure on YF. We also establish Cauchy-like identities for clone Schur functions whose right-hand side is presented as a quadridiagonal determinant rather than a product, as in the case of classical Schur functions. We construct and analyze models of random permutations and involutions based on Fibonacci positive specializations along with a version of the Robinson–Schensted correspondence for YF.