Mutually Abelian-Bordered Pairs

• The paper develops a rigorous definition of mutually abelian-bordered pairs and outlines enumeration methods using lattice path techniques.
• Key results include explicit formulas for disjoint and overlapping cases that leverage abelian equivalence to generalize classical bordered word concepts.
• Implications extend to category theory and group theory, connecting combinatorics on words with broader algebraic and categorical frameworks.

Mutually abelian-bordered pairs are a combinatorial construct arising in the context of binary words and, more generally, finite alphabets. They generalize the classical notion of borders in words to the abelian setting and extend it from single words to pairs, requiring the existence of factors with abelian equivalence—i.e., equal Parikh vectors—at both ends of each word in the pair. This concept is rigorously developed and enumerated for binary words using lattice path methodologies and finds intersections with topics in combinatorics on words, homological algebra, category theory, and group theory.

1. Definitions and Formal Criteria

A finite word $w$ is traditionally called bordered if there exists a nonempty proper prefix that is also a suffix of $w$. For pairs of words $(u,v)$, the classical definition extends: $(u,v)$ is mutually bordered if there exists a nonempty proper prefix of $u$ which is a suffix of $v$, and a nonempty proper suffix of $u$ which is a prefix of $v$ (Gabric, 2020). In the abelian setting, this is relaxed to allow such borders to be abelian equivalent—matching not by literal string equality, but by letter multiset equality.

For mutually abelian-bordered (MAB) pairs of equal-length binary words, both of the following must hold:

• Internal abelian border: There exists a nonempty proper suffix of $u$ abelian equivalent to some proper prefix of $v$.
• External abelian border: There exists a nonempty proper prefix of $u$ abelian equivalent to some proper suffix of $v$.

Pairs that do not admit such abelian borders in either direction are called mutually abelian-unbordered (MAU) pairs (Maity et al., 25 Sep 2025).

2. Lattice Path Approach and Enumeration

A central method in enumerating MAB pairs utilizes the bijection between binary words and lattice paths in $\mathbb{Z}^2$. Each word of length $n$ is mapped to a path starting at the origin: every letter "a" translates to a unit step in one direction (e.g., east), "b" in another (e.g., north). Abelian borders then correspond to lattice path intersections between suitably aligned paths representing a word and its reverse or between two distinct words (Maity et al., 25 Sep 2025).

To compute the number $\mathcal{M}(n)$ of MAB pairs of length-$n$ binary words, the analysis decomposes into two cases:

• Disjoint case $\mathcal{M}^d(n)$: borders can be separated without overlap.

$$\mathcal{M}^d(n) = \sum_{i=1}^{n-1} \frac{(n-i)^2}{n-1}\binom{n-1}{i}\binom{n-1}{i-1}$$

• Overlapping case $\mathcal{M}^o(n)$: borders overlap and words are factored into internal segments, with enumeration relying on more intricate geometric conditions and multiple summations over combinatorial parameters.

Thus,

$$\mathcal{M}(n) = \mathcal{M}^d(n) + \mathcal{M}^o(n).$$

For MAU pairs with matching letter counts (external endpoints), one obtains

$$\overline{\mathcal{M}_{=}(n)} = 2 \sum_{r=1}^{n-1} \frac{1}{n-1}\binom{n-1}{r}\binom{n-1}{r-1}.$$

These enumeration results rely on properties of non-intersecting lattice paths and their interaction with boundary lines in the plane (e.g., $X+Y=n$).

3. Algebraic and Category-Theoretic Context

The concept of mutually abelian-bordered pairs admits generalization beyond combinatorics on words into categorical frameworks. In abelian categories, a pair of subcategories $(\mathcal{X}, \mathcal{Y})$ may be regarded as "mutually abelian-bordered" if each plays a symmetric role in bounding or controlling the homological properties of the ambient category (Li et al., 2015, Liu et al., 2023). The machinery of balanced pairs and cotorsion pairs formalizes this: relative (co)resolution dimensions and the vanishing of higher Ext groups imply the equivalence of right and left singularity categories,

$$D_{R\mathcal{X}\text{-sg}}(A) = D_{L\mathcal{Y}\text{-sg}}(A),$$

provided both dimensions are finite.

Hearts of twin cotorsion pairs further illustrate these ideas: the intersection of two abelian subcategories yields a heart that is abelian if and only if additional extension conditions are satisfied, effectively requiring mutual abelian border control over morphisms and objects (Liu et al., 2023).

4. Asymptotic and Complexity Considerations

Enumeration analyses show that the number of mutually abelian-bordered pairs behaves asymptotically as $c\cdot k^{2n}$ for a $k$-letter alphabet (constant $c$ is specified in related work for non-abelian case) (Gabric, 2020). The expected length of the shortest abelian overlap between two binary words is bounded above by a constant, independent of word length—a reflection of strong combinatorial regularity.

Complexity results for single infinite words show that if only finitely many abelian unbordered factors exist, the number of abelian equivalence classes for each factor length is bounded (Charlier et al., 2015):

$$\forall n\geq 1, \quad |\{[v]_{ab} : v \in \text{Fac}(w), |v| = n\}| \leq N.$$

This property extends to the paired context, suggesting bounded factorizational structure when the abelian mutual border condition is imposed.

5. Connections to Group Theory and Broader Mathematical Structures

The notion of border equivalence and its abelian generalization has analogies in the paper of Gelfand pairs and wreath products. The pair $(G\wr S_n, G\wr S_{n-1})$ is a Gelfand pair if and only if $G$ is abelian, highlighting the centrality of abelian structure for mutual properties in group-theoretic constructs (Tout, 2020). While direct analogues in combinatorics on words require further development, this indicates that mutual abelian constraints are a recurring structural motif across diverse areas.

6. Implications, Applications, and Open Problems

The rigorous enumeration and combinatorial classification of mutually abelian-bordered pairs have implications for pattern matching, digital communications, and structural analysis in theoretical computer science. The methodology utilizing lattice paths connects border problems in word combinatorics to well-studied areas in enumerative combinatorics. Theoretical developments in abelian categories and cotorsion theory suggest further abstraction and cross-disciplinary applications.

Open questions include determining explicit asymptotic ratios ($\mathcal{M}(n)/2^{2n}$), extending enumeration to arbitrary finite alphabets, characterizing abelian mutual border properties for unequal word lengths, and investigating connections with bounded complexity results in infinite words.

7. Key Formulas and Illustrative Example

Fundamental formulas governing MAB pair enumeration include:

• Disjoint case:

$$\mathcal{M}^{\textup{d}}(n)=\sum_{i=1}^{n-1} \frac{(n-i)^2}{n-1}\binom{n-1}{i}\binom{n-1}{i-1}$$

• External/internal abelian-bordered pairs:

$$\text{Number} = \frac{2^{2n} - \mathcal{M}(n) - \overline{\mathcal{M}(n)}}{2}$$

• Abelian equivalence:

$$u \sim_{ab} v \;\iff\; \forall a\in\Sigma,\ |u|_a=|v|_a$$

A diagrammatic illustration for the disjoint case depicts two lattice paths corresponding to $u$ and $v^R$ meeting transversally on $X+Y=n$:

\begin{tikzpicture}[scale=0.8] \draw[step=0.5cm,gray,very thin] (0,0) grid (5,5); \draw[line width=0.5mm,->] (0,0) -- (5,0); \draw[line width=0.5mm,->] (0,0) -- (0,5); \draw[line width=0.5mm] (0,0) -- (2,3) node[midway,above] {%%%%37%%%%}; \draw[line width=0.5mm,dotted] (0,0) -- (2,3) node[midway,below] {%%%%38%%%%}; \draw[dotted, line width=0.5mm] (0,0) -- (5,5) node[midway, right] {%%%%39%%%%}; \fill[red] (2,3) circle (2pt) node[right] {%%%%40%%%%}; \end{tikzpicture}

This encapsulates the intersection structure underlying the abelian border condition.

---

Mutually abelian-bordered pairs thus constitute a mathematically robust generalization of borderedness in words, characterized by combinatorial, geometric, and homological symmetry. The exact enumeration formulas and geometric methods developed provide tools for further paper of abelian equivalence in the combinatorics of words and its categorical extensions.