p-Dyck Paths: Enumerative and Structural Insights

• p-Dyck paths are generalized lattice paths defined by constraints such as pattern avoidance and specific step sets, extending classical Dyck paths.
• The framework employs operator calculus and transfer matrix methods to derive recurrence relations and explicit polynomial enumeration formulas.
• These methodologies yield structural insights with applications to combinatorial analysis, pattern avoidance in lattice paths, and asymptotic enumeration.

A p-Dyck path is a generalization of the classical Dyck path in which the defining constraints, step sets, or avoidance patterns are extended to incorporate a parameter p, yielding a spectrum of combinatorial objects subsuming classical, rational, and pattern-avoiding Dyck paths, as well as various restricted or decorated families. The p-Dyck framework serves as a unifying concept, permitting systematic paper of structural, enumerative, and bijective properties of lattice paths under diverse constraints. The development and analysis of p-Dyck paths leverage operator calculus, transfer matrix methods, pattern avoidance techniques, and connections to other combinatorial structures such as polyominoes and non-crossing partitions. This article surveys the principal definitions, methodologies, structural properties, analytic techniques, and implications regarding p-Dyck paths, emphasizing rigorous details appropriate for the combinatorics research community.

1. Definitions: Generalized and Pattern-Avoiding Dyck Paths

A standard Dyck path is a lattice path from (0,0) to (n, n) using steps u = (1, 0) (up/right) and r = (0, 1) (right/up), remaining weakly above the line y = x. A ballot path is a more general lattice path from (0, 0) to (m, n) (with possibly m ≠ n) using steps u and r, such that at no point does the path fall below y = x (i.e., at each step, the number of r's is at least the number of u's). Every Dyck path is a ballot path, but not vice versa.

A p-Dyck path can be instantiated in multiple, but intimately related, fashions:

• Slope generalization: Paths from (0,0) to (m, n) with steps u and r, staying weakly above y = (n/m)x (e.g., rational or “(a, b)-Dyck” paths, classical case is m = n). This encompasses so-called Fuss–Catalan or rational Dyck paths (Fukukawa, 2013, Gotti, 2017, Ceballos et al., 2015).
• Pattern avoidance: Dyck or ballot paths that avoid a specified forbidden pattern p in the set of allowed steps (e.g., avoiding a particular sequence of u’s and r’s, subwords such as "rur") (Niederhausen et al., 2010, Bernini et al., 2013). For depth-zero patterns (whose reverse is also a ballot path), the enumeration exhibits strong polynomial and algebraic structure.
• Weight constraints or operator restrictions: Paths constructed to avoid specific "depth" or "bifix" patterns, sometimes studied via operator calculus or pattern posets (Niederhausen et al., 2010, Bernini et al., 2013).
• Other variant definitions: p-Dyck may refer to distinct step sets (e.g., up steps of type (1, 1), down steps of type (1, –p)), or combinations of the above (see (Bacher, 2013) for step sets, (Eremin, 2019) for lattice geometry).

For general pattern-avoidance in ballot paths, a “p-Dyck path” is often implicit in the class of Dyck paths (i.e., ballot paths terminating at y = x) that avoid the pattern p. In all cases:

• The diagonal y = x (or more generally y = (n/m)x) acts as the critical lower bound for the path's altitude.
• For p-Dyck paths defined via pattern avoidance, the forbidden subpath p is a contiguous sequence of allowed steps; the path strictly avoids forming p at any position.

2. Recurrence Relations, Operator Methods, and Polynomial Enumeration

Enumeration of p-Dyck paths (pattern-avoiding ballot/Dyck paths) is achieved via recursive relations, operator calculus, and algebraic generating functions.

Given a forbidden pattern p, the number Sₙ(m) of ballot paths from (0,0) to (n, m) avoiding p (for m ≥ n) satisfies recurrences that, for bifix-free patterns (patterns with no internal overlap), simplify to: $$S_n(m; p) = S_{n-1}(m; p) + S_n(m-1; p) - S_{n-a}(m-c; p)$$ where a, c are the number of r's and u's in p, respectively. Initial conditions include S₀(m) = 1 and Sₙ(n-1) = 0, enforcing the non-falling constraint at the diagonal.

For more complex patterns (multiple bifixes), the recurrence accumulates correction terms corresponding to overlapping forbidden subpatterns.

These recurrences are most naturally analyzed by finite operator calculus. Let Sₙ(x) denote the sequence in the variable x (number of right/r steps). Translating the recursion to delta operator notation (e.g., V = 1 – E^-1, where E^a is the shift operator), polynomiality is established: $V = B - B E^{-1} + B E^{-2}$ with B an unknown basic operator, and Sₙ(x) is the Sheffer sequence for the operator algebra generated. Using the transfer formula (Theorem 5), the basic sequence for B can be derived and Sₙ(x) admits an explicit Abel-type formula: $S_n(x) = (x - n + 1) \, b_n(x + 1)$ where b_n(x) encodes the operator structure. This method proves that for depth-zero patterns, Sₙ(m) is a polynomial in m.

For Dyck paths (ending at (n,n)), the enumeration becomes Sₙ(n), yielding explicit polynomial formulas for the number of p-Dyck paths of semilength n avoiding pattern p. In special cases, these reduce to known results: for the pattern "rur", the number of Dyck paths can be expressed in terms of Catalan numbers.

3. Combinatorial Consequences and Structural Results

For depth-zero patterns (whose reverse is also a ballot path), the main combinatorial outcome is:

• The enumeration of ballot paths avoiding pattern p is always a polynomial in the number of steps of each type (e.g., m). Evaluated at the diagonal (m = n), these give the explicit count of p-Dyck paths for each n.

More generally, for patterns with bifixes, recurrences and enumeration carry over by adapting the delta operator and basic sequence analysis as above. Furthermore, by explicit combinatorial decompositions, these results yield structural insights:

• For p-Dyck paths associated with pattern avoidance, the family is closed under natural operations (e.g., concatenation and first return).
• For certain forbidden patterns, Dyck paths avoiding p correspond bijectively to known families (such as Motzkin paths, or subsets of Young tableaux) (Bernini et al., 2013).

Moreover, the methods delineated here extend beyond bifix-free or single-bifix cases to all depth-zero patterns. Polynomial enumeration may fail for patterns of nonzero depth or when the avoidance condition is relaxed, an open direction (Niederhausen et al., 2010).

4. Extension to Generalized and Rational Dyck Paths

The restriction and enumeration methodologies for p-Dyck paths smoothly generalize to rational Dyck paths (paths from (0,0) to (m, n), always staying below y = (n/m)x), as well as to other generalized paths.

In the general case, such “rational” Dyck or p-Dyck paths may be prescribed as:

• Paths from (0,0) to (m, n) via steps in {u, r}, weakly above y = (n/m)x.
• Enumeration via formulas:

$$C(m, n) = \frac{1}{m + n} \binom{m+n}{n} \quad \text{if } \gcd(m, n) = 1,$$

which generalizes the Catalan number. For m = kn, this yields the Fuss–Catalan number:

$C(kn, n) = \frac{1}{kn + 1} \binom{(k+1)n}{n}.$

• For arbitrary m, n, explicit formulas involve sum-over-sequence expansions and cyclic equivalence classes (see (Fukukawa, 2013)).

Pattern avoidance remains compatible in this setting, as patterns are defined in terms of local step sequences, and the delta operator/recurrence techniques generalize, provided the pattern’s depth or “bifix” structure allows.

5. Algebraic, Analytic, and Asymptotic Insights

The operator and recurrence framework for counting p-Dyck and p–ballot paths have deep analytic implications:

• For every depth-zero pattern p, the enumeration Sₙ(m) is a polynomial in m, with explicit recurrence and generating function determined by the operator algebra.
• The degrees and coefficients of these polynomials reflect the structural complexity of pattern p (length, bifix structure).
• When specializing to Dyck paths (setting m = n), one obtains strong asymptotic statements: for fixed pattern p, the avoidance sequence is polynomial in n, with degree determined by the shape and depth of p. For instance, the number of Dyck paths of length n avoiding certain patterns is always O(n^k) for an explicit k (Bernini et al., 2013).

Open problems include the extension to non-depth-zero patterns, for which no general polynomiality is proved, and the refinement of the operator calculus for more intricate recursive structures.

6. Broader Implications and Applications

The enumeration of p-Dyck paths via pattern avoidance, recurrence relations, and operator calculus is foundational for multiple areas:

• Pattern avoidance in lattice paths: The approach provides a systematic technique for enumerating constrained paths in ballot and Dyck classes, generalizing Catalan, Motzkin, and Schröder families.
• Algebraic combinatorics: The generated Sheffer sequences and their associated operator algebras connect to representation theory and symmetric function theory.
• Enumerative and asymptotic analysis: The tools yield both finite and asymptotic formulas for constrained paths, supporting analysis in algorithmics and statistical physics.
• Further combinatorial objects: Similar techniques apply to pattern-avoiding permutations, polyominoes, and pattern-avoiding non-crossing partitions via structural bijections (Flórez et al., 2023, Bernini et al., 2013).

The extensibility of the operator calculus, transfer formula, and recurrence framework to other constrained path families underscores a broad research program in the enumeration of structured lattice paths.

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Table: Key Concepts in p-Dyck Path Enumeration

Concept	Definition/Method	Key Formula or Description
Ballot path	Lattice path staying weakly above y = x	Steps u, r; m ≥ n at every point
Dyck path	Ballot path ending at (n, n)	Standard Catalan model
p-Dyck path	Ballot path (or Dyck path) avoiding pattern p	Enumerated via Sₙ(m; p), recurrences, operator calculus
Recursion for bifix-free p	Sₙ(m; p) = Sₙ₋₁(m; p) + Sₙ(m–1; p) – Sₙ₋ₐ(m–c; p)	a = #r in p, c = #u in p
Operator calculus	Expresses recursions as delta operator equations	Sₙ(x) polynomial in x; uses Transfer Formula
Dyck path enumeration	Sₙ(n; p) = count of Dyck paths of length n avoiding p	Explicit polynomials for depth-zero p
Generalization to (m,n)	Lattice paths from (0,0) to (m, n) staying below y = (n/m)x	C(m, n) = 1/(m+n)·binom{m+n}{n} for coprime m, n

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This article synthesizes the combinatorial, algebraic, and analytic aspects of p-Dyck paths, detailing the recurrence, operator, and generating function techniques for their enumeration. The transferability of these methods to broader classes of constrained paths underpins ongoing research at the intersection of lattice path enumeration, pattern avoidance, and algebraic combinatorics.