Young-Fibonacci Lattices

• Young-Fibonacci Lattices are graded, self-dual differential posets where elements are Fibonacci words over {1,2} with rank given by digit sum.
• They are characterized by recursive covering relations and explicit generating functions, linking enumeration to the Fibonacci sequence.
• Their structure supports applications in game theory, spectral analysis, and representation theory, highlighting deep combinatorial and computational properties.

The Young-Fibonacci lattice (often written YF or 𝕐𝔉) is a graded, self-dual, infinite differential poset whose elements are words over the alphabet {1,2} with fixed total weight, corresponding bijectively to the combinatorial structures underlying Fibonacci numbers. As the unique infinite, connected differential poset (other than the Young lattice) in Stanley’s theory, YF serves as a central object interlinking enumerative combinatorics, algebraic combinatorics, and probabilistic path models. Its Hasse diagram, join/meet structure, spectral and probabilistic boundary, and application to combinatorial game theory and representation theory are topics of active research.

1. Combinatorial Structure and Order Relations

The elements of the Young-Fibonacci lattice YF at rank $r$ are words $v = v_1v_2...v_k$ in $\{1,2\}$ with $v_1 + \cdots + v_k = r$; such a word is called a Fibonacci word of weight $r$. The partial order is defined by the following covering relations: $u$ covers $v$ if $u$ can be obtained from $v$ by a sequence of “down-steps,” each of which either:

• deletes a single $1$ from a leftmost block of $1$’s, or
• replaces the leftmost $1$ with a $2$.

Alternatively, the covering relation $v' \lessdot v$ holds if, for some $i$ such that $u_1,...,u_i$ contains no $1$’s, $v = u_1...u_i 1 u_{i+1}...u_{|u|}$ or, for $v = u_1...u_{i-1} 2 u_{i+1}...u_{|u|}$, with $u_i$ the leftmost $1$ in $u$ (Choi et al., 16 Jun 2024).

The rank function $\rho(v)$ is the sum of the digits of $v$, i.e., $\rho(v) = \sum_j v_j$. The minimal element is the empty word $\epsilon$ with rank 0. Hasse diagrams of YF up to rank five illustrate its combinatorial depth: for example, at rank three there are three words $\{111,12,21\}$, at rank four five words, and so on, matching the initial Fibonacci sequence (Choi et al., 16 Jun 2024, Evtushevsky, 2020).

2. Enumeration and Generating Functions

The cardinality of words at rank $r$ is the $r$th Fibonacci number $F_r$, where $F_0=1$, $F_1=1$, and $F_r=F_{r-1}+F_{r-2}$. The generating function is

$$G(x) = \sum_{r \geq 0} F_r x^r = \frac{1}{1 - x - x^2}.$$

Cumulatively, the number of vertices up to rank $n$ is $F_{n+2}-1$ (Choi et al., 16 Jun 2024, Evtushevsky, 2020). The underlying path enumeration also possesses closed forms; the number of downward paths $d(x,y)$ from $y$ to $x$ is given by

$$d(x, y) = \sum_{i=0}^{|x|} f(x,i,h(x,y)) \prod_{j=1}^{d(y)} (g(y,j)-i),$$

where $h(x, y)$ is the length of the maximal common suffix, $g(y,j)$ are “hook-length” statistics tied to block decompositions, and $f(x,i,z)$ are explicitly defined auxiliary functions (Evtushevsky, 2020, Bochkov et al., 2020).

3. Lattice Structure and Self-Duality

YF is a (modular, distributive) lattice and a 1-differential poset. The meet $u \wedge v$ of two elements is calculated positionwise: align $u$ and $v$ at the left, pad the shorter word with leading zeros, and at each position take $\min \{ \text{entries} \}$, discarding leading zeros to recover a word in $\{1,2\}$.

The join operation is induced by the self-duality involution: map $v$ to its reversed complement (swap $1 \leftrightarrow 2$ and reverse), take the meet, then invert back (Choi et al., 16 Jun 2024). This involution gives the poset a natural order-reversing symmetry, and truncated YF is rank-symmetric and unimodal per differential poset theory.

Certain distributive lattices built from YF encode topological types of plane branches and admit a unique order-reversing involution; the enumeration of such self-dual elements again produces Fibonacci numbers (Pereira et al., 2013). In addition, the lattice structure underlies connections to crystal bases and representation theory for $\mathfrak{sl}_3$, with explicit rank generating functions and symmetry properties (Donnelly et al., 2020).

4. Spectral Measures, Martin Boundary, and Asymptotics

Central (harmonic) measures on path spaces of YF are parametrized by infinite words $w \in \{1,2\}^\infty$ and a parameter $\beta \in [0, 1]$, corresponding to Martin boundary extremal measures. The Plancherel measure ($\beta=0$) gives explicit weights via hook-formulas, while for each $w$ the $\beta=1$ measure is supported on paths converging to $w$.

A principal result is that, for any $w$ and $\beta=1$, the measure $\mu_{w,1}$ is ergodic, i.e., almost every infinite path converges to $w$. This underscores a dichotomy with the classical Young lattice, which has a higher-dimensional Thoma parameterization. For intermediate $\beta\in(0,1)$, ergodicity remains an open subject but is being addressed in ongoing research (Bochkov et al., 2020).

In the context of “clone Schur functions,” a hierarchy of random measures (Fibonacci-positive specializations) yields diverse probabilistic asymptotics: stick-breaking (GEM-type) processes emerge in the Plancherel and Charlier cases, while other specializations produce discrete or “freezing” limits. The structure of asymptotic random Fibonacci words thus reflects the full richness of the YF Martin boundary and extends classical results for Schur measures (Petrov et al., 30 Dec 2024).

5. Connections with Game Theory and Algebraic Combinatorics

YF supports rich combinatorial game-theoretic dynamics. The Ungar game, recently studied combinatorially, is played by two players alternately executing “Ungar moves”: replacing a word $v$ by the meet of any nonempty collection of elements covered by $v$. A precise classification of losing positions (P-positions, or Eeta-wins) in YF is known and determined by explicit parity and prefix rules on the words’ structure. Specifically, the count of such positions in rank $r\geq 2$ is $F_{r-2} + (-1)^r$, and the set of Grundy values is explicitly computable (Choi et al., 16 Jun 2024).

Further, the f-statistic (number of saturated chains from the bottom to a given element) satisfies a hook-length-type formula, and induces a binary tree—the “Macdonald tree”—on elements of odd f-statistic. The distribution of odd residues of f-statistics is equidistributed modulo powers of two for large enough ranks, connecting YF to number-theoretic phenomena (Bi et al., 2017).

The combinatorics of YF underpins representation-theoretic constructions. In symmetric Fibonaccian distributive lattices, YF arises for $n=3$, and the associated vector spaces with natural edge-colorings support actions of the Lie algebra $\mathfrak{sl}_3$, with the weight-basis given by lattice elements and explicit formulas for weight multiplicities and character polynomials (Donnelly et al., 2020).

6. Logical and Computational Properties

The first-order theory of the YF lattice (as a poset), including the order relation and word constants, is undecidable and not finitely axiomatizable. Arithmetic can be interpreted in YF: the words $1^n$ represent the natural number $n$, and addition and multiplication are first-order definable. Moreover, every word is first-order definable, and the maximal definability property holds; any invariant relation under the unique non-trivial automorphism is first-order definable (Evtushevsky, 24 Nov 2024). This sharply contrasts with many naturally occurring posets and lattices, highlighting the subtle computational complexity of YF as a combinatorial object.

7. Further Research Directions and Applications

Research on YF is positioned at the crossroads of algebraic combinatorics, probability, and algebraic geometry. Open avenues include refining the Martin boundary classification for arbitrary $\beta$, developing $q$-analogues, relating path enumeration to cluster algebras, and constructing explicit bijections between YF structures and tilings, tableaux, or representation-theoretic crystals.

Significant advances link YF’s combinatorics to random permutation models, structure constants of the Okada algebra, and probabilistic laws for runs and hikes in random Fibonacci words derived from measure-theoretic and symmetric function-theoretic frameworks (Petrov et al., 30 Dec 2024). Applications to singularity theory and the theory of plane branches emerge via encoding topological types as elements of self-dual distributive lattices with Fibonacci enumeration features (Pereira et al., 2013).

In summary, the Young-Fibonacci lattice stands as a central combinatorial object exhibiting deep connections to enumerative, algebraic, probabilistic, and computational themes, with ongoing research further extending its reach and unifying capacity within discrete mathematics and mathematical physics.