Constructible Words in Combinatorics

• Constructible words are explicitly defined word families using finite procedures in combinatorics, coding theory, and automata theory.
• They are built through online extension, local motion, and morphic constructions that ensure square-avoidance and resilience under edits.
• Their study provides actionable frameworks for analyzing finite control over infinite behavior, aiding both algorithmic and enumerative research.

Searching arXiv for the cited papers and closely related work on constructible words, square-free words, and combinatorics on words. Constructible words, as a synthesized umbrella notion across combinatorics on words, coding theory, and automata on generalized words, are word classes specified by explicit generative mechanisms rather than by existence alone. In the literature this includes words extendable online while avoiding prescribed squares, words robust under deletions or insertions, words built from Motzkin-path or block-concatenation schemes, words generated by constrained walks, morphic words with twisted repetition-avoidance, universal Lyndon words recoverable from prefix-code data, finite words engineered for high distinct-square density, and generalized words over scattered linear orderings obtained by finitely many applications of concatenation and power operators (Sanderson, 2013, Dębski et al., 2022, Barcucci et al., 2014, Pratt-Hartmann, 2022, Rosenfeld, 2021, Birmajer et al., 2015, Abdullah et al., 16 May 2025, Carpi et al., 2014, Blanchet-Sadri et al., 2017, Braipson et al., 2 Jul 2026). This suggests that “constructibility” is best understood not as a single invariant, but as a family of explicit formation principles that make highly constrained words accessible to direct analysis.

1. Formal viewpoints and foundational notions

A first formal viewpoint treats constructibility as online extendability. For a finite alphabet $A$ with $|A|=\ell$, a square is a word $XX$ with $X\neq \varepsilon$, and for a finite set $I$ of half-lengths one defines $B(I)=\{vv\mid |v|\in I\}$. A word is $I$-square-free iff it contains no factor in $B(I)$. The sequential method starts from the empty word and appends one letter at a time, requiring every extension to remain $I$-square-free; the central question is whether this forward-only process can continue indefinitely or must reach a dead-end (Sanderson, 2013).

A second viewpoint treats constructibility as generation by local motion on an existing word. If $u=a_1\cdots a_n$ and $|A|=\ell$0 is a surjective map with $|A|=\ell$1, then $|A|=\ell$2 is generated by $|A|=\ell$3. A primitive generator of $|A|=\ell$4 is a generator not itself generated by any shorter word. Except for uniliteral words, every word has precisely two primitive generators, related by reversal (Pratt-Hartmann, 2022).

A third viewpoint makes constructibility an explicit closure property over generalized words. For words indexed by scattered linear orderings, the constructible words form the least set containing the empty word and one-letter words and closed under concatenation, $|A|=\ell$5, $|A|=\ell$6, $|A|=\ell$7, and $|A|=\ell$8. These words admit finite descriptions even when their domains are countable or uncountable scattered orderings (Braipson et al., 2 Jul 2026).

Taken together, these definitions show that the field uses constructibility to isolate words that can be described by finite procedures, finite operator terms, or finite-state-compatible local rules. The common feature is not a shared combinatorial shape but an explicit mechanism of formation.

2. Local extension, dead-ends, and avoidance of squares

The sequential extension problem for square avoidance is formalized through a de Bruijn-type graph $|A|=\ell$9 whose vertices are the $XX$0-square-free words of length $XX$1, where $XX$2 is the largest forbidden half-length. A directed edge corresponds to shifting and appending one letter. A vertex of outdegree $XX$3 is a dead-end, and the absence of dead-ends and dead-starts implies indefinite extendability by the sequential method (Sanderson, 2013).

This framework sharply separates existence from online constructibility. Over a binary alphabet there is no infinite square-free word. Over three or more letters, Thue’s existence theorem shows that infinite square-free words do exist, but this does not imply that a naïve online extension process can always continue. The paper isolates the finite-forbidden-set regime in which such local construction becomes analyzable (Sanderson, 2013).

For a forbidden set $XX$4, the number $XX$5 of forbidden half-lengths interacts decisively with $XX$6. If $XX$7, then the sequential method has no dead-ends. If $XX$8 and $XX$9, the paper gives an arithmetic criterion for the existence of dead-ends: they occur iff either $X\neq \varepsilon$0 is geometric,

$X\neq \varepsilon$1

or there exists $X\neq \varepsilon$2 with $X\neq \varepsilon$3 such that

$X\neq \varepsilon$4

For $X\neq \varepsilon$5, a sufficient condition for the absence of dead-ends is that every $X\neq \varepsilon$6-term subsequence is either geometric of ratio $X\neq \varepsilon$7 or satisfies the same “large tail” inequality; necessity is conjectured when $X\neq \varepsilon$8 (Sanderson, 2013).

The structural analysis proceeds through orbit partitions, permutations $X\neq \varepsilon$9 enforcing candidate square constraints, and a difference graph whose chromatic number controls the minimal alphabet size needed to realize a dead-end pattern. Condition C requires each $I$0 to exceed the sum of all later $I$1’s, while Condition D captures an “almost C” regime in which the extremal pattern itself already contains a forbidden square. A notable binary example is $I$2, where $I$3 is a dead-end: every appended letter creates either a $I$4-square or a $I$5-square (Sanderson, 2013).

A common misconception is that the presence of arbitrarily long avoiding words guarantees a safe greedy construction. The local dead-end analysis shows otherwise: the obstruction is not global nonexistence, but the existence of prefixes from which every one-letter continuation is forbidden.

3. Robust square-free words and list-constrained construction

A different notion of constructibility studies square-free words that remain well behaved under elementary edits. A square-free word is steady if deletion of any single letter preserves square-freeness, and bifurcate if at every insertion position there exists at least one letter whose insertion preserves square-freeness. There exist arbitrarily long steady words over a $I$6-letter alphabet, and over a fixed alphabet with at least three letters every steady word is bifurcate (Dębski et al., 2022).

The quaternary existence result is derived from Dejean’s theorem. For $I$7, there exist arbitrarily long words with no factor of exponent greater than $I$8, and any such word with the separation property “every factor $I$9 satisfies $B(I)=\{vv\mid |v|\in I\}$0” is steady. The ternary obstruction is also explicit: any square-free word of length at least $B(I)=\{vv\mid |v|\in I\}$1 over a $B(I)=\{vv\mid |v|\in I\}$2-letter alphabet contains a factor $B(I)=\{vv\mid |v|\in I\}$3, forcing a deletion that creates a square. This is why $B(I)=\{vv\mid |v|\in I\}$4 is best possible for arbitrarily long steady words (Dębski et al., 2022).

The list version makes the constructive content stronger. If $B(I)=\{vv\mid |v|\in I\}$5 is a sequence of alphabets with $B(I)=\{vv\mid |v|\in I\}$6 for all $B(I)=\{vv\mid |v|\in I\}$7, then for every $B(I)=\{vv\mid |v|\in I\}$8 there exist at least $B(I)=\{vv\mid |v|\in I\}$9 steady words $I$0 with $I$1. The proof uses a weighted recurrence on the number $I$2 of steady words and the number $I$3 of one-step failures, with

$I$4

and yields $I$5. The conjectured optimal list size is $I$6 (Dębski et al., 2022).

Insertion-robustness admits a stronger global organization. There exists a complete bifurcate tree over an alphabet of size $I$7, and there exists a complete bifurcate tree of doubly infinite bifurcate words over an alphabet of size $I$8. By contrast, there is no complete bifurcate tree over a $I$9-letter alphabet whose words have length more than $B(I)$0. The $B(I)$1-letter construction is obtained from a square-free coloring of an infinite outerplanar graph using the Kündgen–Pelsmajer theorem (Dębski et al., 2022).

List-constrained square-free construction without edit-robustness has its own sharp results. For arbitrary $B(I)$2-list assignments, the number of square-free words of length $B(I)$3 is at least $B(I)$4. For $B(I)$5-list assignments with $B(I)$6, the number is at least $B(I)$7, and in particular an infinite square-free word respecting the lists exists. The proof uses weighted counting based on normalized prefixes of minimal squares of period at most $B(I)$8, together with coefficients $B(I)$9 satisfying an inequality with

$I$0

and the growth condition $I$1, instantiated with $I$2 (Rosenfeld, 2021).

These results show that constructibility under local constraints is not limited to pure existence. It can include deletion-resilience, insertion-resilience, and adversarial per-position choice constraints, with explicit exponential lower bounds on the number of admissible words.

4. Walks, paths, codes, and universal families

Constructibility also appears in models where words are recovered from combinatorial encodings. In the walk model, generation is controlled by left, right, or stationary moves on a shorter word. The central structural theorem is that the primitive generator of any word is unique up to reversal, and every uniliteral word has precisely one primitive generator while all other words have precisely two. A primitive word is perfect iff it contains no non-trivial palindrome as a factor, and equality of outputs from two walks is characterized by the defect equivalence relation generated by palindromic factors (Pratt-Hartmann, 2022).

Cross-bifix-free sets provide a code-theoretic version of explicit construction. For $I$3 and fixed length $I$4, words over $I$5 are encoded as lattice paths in which $I$6 is a fall step, $I$7 is a rise step, and $I$8 are colored horizontal steps. Using $I$9-colored Motzkin paths, the set

$u=a_1\cdots a_n$0

is constructed explicitly, and it is both cross-bifix-free and non-expandable on $u=a_1\cdots a_n$1. The construction is algorithmic: generate Motzkin words, identify elevated blocks $u=a_1\cdots a_n$2, form the three families by explicit concatenation rules, and then use balance and elevated-block arguments to prove non-expandability (Barcucci et al., 2014).

Universal Lyndon words furnish another explicit universal construction. A universal Lyndon word of degree $u=a_1\cdots a_n$3 is a word over $u=a_1\cdots a_n$4 of length $u=a_1\cdots a_n$5 such that all conjugates are Lyndon words, each for a unique total order. Such words exist for every $u=a_1\cdots a_n$6. One construction uses Eulerian cycles in the Jackson graph, whose vertices are permutations of length $u=a_1\cdots a_n$7 and whose edges encode overlap by $u=a_1\cdots a_n$8 letters. A more intrinsic characterization states that $u=a_1\cdots a_n$9 is a universal Lyndon word iff every cyclic factor $|A|=\ell$00 satisfies

$|A|=\ell$01

The shortest unrepeated prefixes of the conjugates form $|A|=\ell$02, a Hamiltonian lex-code, and every Hamiltonian lex-code yields a universal Lyndon word. This gives an algorithmic route to constructing all universal Lyndon words by generating lex-codes and finding Hamiltonian cycles in the associated digraph $|A|=\ell$03 (Carpi et al., 2014).

These constructions clarify that “constructible” may mean recoverable from a path system, a code, or a walk relation. The underlying word is then a canonical output of a finite combinatorial object rather than an arbitrary member of an avoidance language.

5. Block constructions, morphisms, and extremal finite words

A systematic enumerative theory of constructible restricted words is developed through building blocks. For fixed $|A|=\ell$04, the set $|A|=\ell$05 consists of length-$|A|=\ell$06 words over $|A|=\ell$07 that start with $|A|=\ell$08 and whose remaining letters lie in $|A|=\ell$09. Choosing subsets $|A|=\ell$10 with $|A|=\ell$11 defines a family $|A|=\ell$12 of words constructible by concatenation of allowed blocks. The count is given by the invert transform and partial Bell polynomials: $|A|=\ell$13 which enumerates words in the enlarged alphabet model $|A|=\ell$14. This unifies constructions for words avoiding runs of zeros of specified lengths, words with spacing constraints after nonzero symbols, and words avoiding patterns such as $|A|=\ell$15 (Birmajer et al., 2015).

A morphic form of constructibility is developed for strongly $|A|=\ell$16-free words. If $|A|=\ell$17 is a permutation of the alphabet, a strongly $|A|=\ell$18-repetition is a factor $|A|=\ell$19 with equal block lengths and $|A|=\ell$20. With a cyclic permutation $|A|=\ell$21 on an alphabet $|A|=\ell$22 of size $|A|=\ell$23, the cyclic shift morphism is

$|A|=\ell$24

Its fixed point $|A|=\ell$25 is strongly $|A|=\ell$26-free whenever $|A|=\ell$27, $|A|=\ell$28, $|A|=\ell$29, and $|A|=\ell$30. For $|A|=\ell$31, the morphism reduces to the Thue–Morse morphism. The paper further conjectures linear factor complexity of the form

$|A|=\ell$32

for sufficiently large $|A|=\ell$33 (Abdullah et al., 16 May 2025).

At the opposite end of the spectrum, constructibility can be directed toward extremal density objectives. Finite words can be engineered to maximize or nearly maximize the density of distinct square factors. For each positive integer $|A|=\ell$34, the word

$|A|=\ell$35

has length $|A|=\ell$36 and begins with $|A|=\ell$37 FS-double-square positions, each with $|A|=\ell$38 and $|A|=\ell$39. Its distinct-square-sequence is

$|A|=\ell$40

and its distinct-square-density tends to $|A|=\ell$41. A second family,

$|A|=\ell$42

has distinct-square-density approaching $|A|=\ell$43 as $|A|=\ell$44. These words are explicitly built to control FS-double-squares and last occurrences of distinct squares (Blanchet-Sadri et al., 2017).

The shared methodological point is that a word family can be constructed to satisfy a target specification—enumerative, avoidance-theoretic, or extremal—by choosing blocks, substitutions, or overlap patterns whose local combinatorics are fully analyzable.

6. Constructible generalized words and rational languages on scattered orderings

The most abstract notion of constructible words arises for words indexed by scattered linear orderings. A generalized word is a function $|A|=\ell$45 where $|A|=\ell$46 is a linear ordering, and automata on such words assign states to cuts of $|A|=\ell$47, using successor transitions and left- and right-limit transitions. The constructible words are the least class containing $|A|=\ell$48 and singleton-letter words and closed under concatenation and the four powers $|A|=\ell$49, $|A|=\ell$50, $|A|=\ell$51, and $|A|=\ell$52 (Braipson et al., 2 Jul 2026).

These words do not exhaust all scattered words. Nevertheless, for an automaton $|A|=\ell$53 one may quotient words by the equivalence relation $|A|=\ell$54 meaning that $|A|=\ell$55 and $|A|=\ell$56 admit exactly the same paths $|A|=\ell$57. The quotient is finite and carries a semigroup structure compatible with concatenation. The proof that every $|A|=\ell$58-class contains a constructible word combines this finite semigroup with Colcombet’s theorem on Ramseyan splits of multiplicative labelings (Braipson et al., 2 Jul 2026).

A crucial technical fact is that automata on scattered orderings cannot distinguish powers $|A|=\ell$59 and $|A|=\ell$60 when $|A|=\ell$61 and $|A|=\ell$62 are regular uncountable ordinals; similarly for the reverse powers. This is why $|A|=\ell$63 and $|A|=\ell$64 suffice as canonical uncountable representatives in the definition of constructibility. Inductively decomposing a general word through a Ramseyan split then yields an equivalent term built from concatenation and the four power operations (Braipson et al., 2 Jul 2026).

The main characterization theorem states that if two automata on scattered linear orderings accept the same constructible words, then they accept the same language. Equivalently, a rational language of words indexed by scattered linear orderings is determined by its intersection with the constructible words. This generalizes the role of ultimately periodic words in $|A|=\ell$65-regular language theory to the scattered-ordering setting, including both countable and uncountable domains (Braipson et al., 2 Jul 2026).

A frequent misunderstanding is to conflate “constructible” with “all scattered.” The theorem does not say that every scattered word is constructible; it says that constructible words form a behaviorally complete test class for rational languages on scattered linear orderings.

7. Conceptual synthesis and open directions

Across these literatures, constructible words are characterized by finite control over infinite or highly constrained behavior. In the sequential square-avoidance setting, the issue is whether a local extension rule can avoid dead-ends. In edit-robust square-freeness, the issue is stability under deletion or insertion. In Motzkin, lex-code, and walk models, the word is reconstructed from a finite combinatorial witness. In block concatenation and morphic systems, finite parameters generate infinite or large structured families. In the scattered-ordering setting, finite operator expressions suffice to represent one word from every automaton-equivalence class (Sanderson, 2013, Dębski et al., 2022, Barcucci et al., 2014, Pratt-Hartmann, 2022, Birmajer et al., 2015, Abdullah et al., 16 May 2025, Braipson et al., 2 Jul 2026).

Several open directions recur. The $|A|=\ell$66-list steady conjecture remains open, as do the existence of an infinite sequence of quaternary bifurcate words under single-letter extensions and a complete bifurcate tree over $|A|=\ell$67 letters (Dębski et al., 2022). Grytczuk’s conjecture is proved only for $|A|=\ell$68-lists taken from a fixed $|A|=\ell$69-letter alphabet, not for arbitrary $|A|=\ell$70-lists over larger alphabets (Rosenfeld, 2021). For strongly $|A|=\ell$71-free morphic words, the case $|A|=\ell$72 and the factor-complexity conjecture remain open (Abdullah et al., 16 May 2025). No general counting formula is known for universal Lyndon words of degree $|A|=\ell$73 (Carpi et al., 2014). In the extremal distinct-square problem, the constructions approaching density $|A|=\ell$74 leave open the true maximal asymptotic density (Blanchet-Sadri et al., 2017). For automata on scattered orderings, the characterization by constructible words is expected to support further theoretical development rather than to close the subject (Braipson et al., 2 Jul 2026).

The unifying lesson is that constructibility is a methodological principle. It identifies word families whose combinatorial, algebraic, or automata-theoretic behavior can be encoded by explicit local rules, finite decompositions, or canonical finite witnesses. In that sense, constructible words occupy a central position between abstract existence theorems and algorithmically manageable structure.