generating functions

This paper addresses the fundamental challenge of extending asymptotic analysis of combinatorial systems from basic constructions (Cartesian product, disjoint union) to general symbolic constructions (e.g., sets, cycles). Methodologically, it integrates analytic combinatorics, complex analysis, and symbolic computation, employing singularity analysis of generating functions—systematically handling algebraic-logarithmic singularities—and invoking the Schanuel conjecture to resolve transcendental singularities arising from set and cycle constructions. The main contribution is a near-complete algorithmic pipeline that automatically derives asymptotic expansions directly from combinatorial specifications, covering a broad class of constructions including sets and cycles for the first time. Asymptotics for Cartesian products and disjoint unions are rigorously established without number-theoretic assumptions; results for other constructions hold conditionally under the Schanuel conjecture. This work significantly extends both the scope and automation level of combinatorial asymptotics.

This study addresses the construction of functions in algebraic combinatorics subject to stringent distributional constraints and the discovery of previously unknown combinatorial symmetries. To this end, we propose the SLURP framework, which integrates MapSeek-Functional and MapSeek-Symbolic approaches through alternating pseudo-label supervised learning, symbolic regression, and formal verification in Lean 4. The framework yields the first combinatorial interpretation of $q,t$-Narayana polynomials based on non-crossing partitions and provides a combinatorial proof of symmetry in previously unresolved cases by leveraging newly discovered statistics. All code and formalized results are publicly released to ensure reproducibility and rigorous verification.

This work studies efficient approximation of two natural counting problems under the Lovász Local Lemma (LLL) framework: the probability of the intersection of bad events and the dimension of the intersection of subspaces. Specifically, it addresses counting satisfying assignments for classical CNF formulas and counting the dimension of the satisfying subspace in quantum SAT. We propose a unified approximation framework based on cluster expansion. Our contributions include: (i) the first fully polynomial-time approximation scheme (FPTAS) for commuting projection operators; (ii) for general (non-commuting) projections, FPTAS under either inclusion–exclusion stability or spectral gap conditions, along with a novel affine approximation paradigm. The approach integrates cluster expansion, inclusion–exclusion principles, spectral analysis, and quantum satisfiability modeling—breaking reliance on commutativity or stringent constraint assumptions. This significantly extends the applicability of the LLL to counting problems beyond traditional limitations.

This study addresses the enumeration of admissible sequences and capacity characterization for discrete noiseless channels under cost constraints. Methodologically, it models the channel structure as a weighted labeled directed graph and establishes an exact correspondence between this graph and the singularities of a bivariate generating function; asymptotic analysis then leverages multivariate analytic combinatorics, spectral theory of directed graphs, and singularity analysis to derive precise asymptotics for the number of admissible sequences. The key contributions are threefold: (i) it identifies, for the first time, that channel capacity under cost constraints is determined by cost-dependent singularities of the associated generating function; (ii) it provides a novel proof of the equivalence between the combinatorial and probabilistic definitions of cost-constrained capacity; and (iii) it unifies information-theoretic coding theory with enumerative combinatorics, thereby generalizing and extending Shannon’s classical capacity theorem.

This work addresses parameterized approximation algorithms for Vertex Cover and 3-Hitting Set. Methodologically, it introduces a novel randomized branching paradigm grounded in an equivalence between the algorithm’s recursive structure and a binary stochastic process. Leveraging a type-theoretic adaptation of Sanov’s theorem, the framework performs large-deviation analysis on bivariate recurrence relations, yielding an analytically tractable master theorem for asymptotic running time. Contribution-wise, this is the first unified theoretical framework providing rigorous approximation-ratio–dependent guarantees across multiple approximation factors. It substantially improves worst-case time complexity over prior deterministic branching approaches and overcomes fundamental analytical limitations inherent in traditional branching analysis. The framework establishes a general methodology for characterizing the asymptotic performance of parameterized approximation algorithms, bridging stochastic analysis and combinatorial optimization.

This work addresses the challenges in combinatorial counting arising from intricate structural and arithmetic constraints, which hinder manual derivation and cause existing methods to break problem symmetries. To overcome these limitations, the paper introduces Cofola, a typed declarative language that unifies combinatorial counting as a weighted first-order model counting (WFOMC) problem with coefficient extraction constraints—the first such formulation. Cofola naturally expresses common combinatorial structures including sets, multisets, permutations, and partitions. Its three-stage compilation pipeline integrates preprocessing, symmetry-preserving decomposition, and ordering axiom encoding—such as lexicographic symmetry breaking and sequence/cycle axioms—to enable efficient solving while preserving inherent symmetries. Experimental results demonstrate that Cofola substantially outperforms existing frameworks in both expressiveness and computational efficiency across a diverse benchmark suite, ranging from textbook examples to complex multi-object scenarios.

This work investigates which combinatorial and number-theoretic counting functions are computable in logarithmic space, i.e., belong to the complexity class #L. By developing a framework that counts accepting paths of nondeterministic logspace Turing machines and integrating tools from combinatorial encoding, discrete geometry, and representation theory, the study systematically establishes the #L-computability of numerous classical functions. Key contributions include the first unified inclusion of Catalan numbers, Stirling numbers, and the number of standard Young tableaux within #L; proofs that multinomial coefficients, linear extensions of trees, and GL₂-plethysm coefficients under bounded outer partitions lie in #L or are verifiable in log² space; and a novel conditional approach to refuting their #P-completeness, thereby substantially expanding the theoretical frontier of low-complexity counting problems.

This study addresses the efficient enumeration of hole-free row-convex polyominoes. By interpreting the area of a polyomino as an integer partition of its row lengths and employing generating functions to encode horizontal alignments between adjacent rows, the work establishes—for the first time—a direct connection between integer partitions and the enumeration of row-convex polyominoes. This approach yields an exact generating function and leads to the asymptotic formula \( S(N) \approx A \cdot 2^N \cdot \cos(N\theta + \phi) \), where \( \theta = \arctan(\sqrt{7}/3) \). Numerical validation confirms high accuracy for small areas, offering a concise and effective framework for both exact and asymptotic analysis of this class of combinatorial structures.

This work establishes, for the first time, a bijection between the OEIS “Genesis sequence,” the number of records in rooted trees, and the girth of connected endofunctions. By constructing this combinatorial bijection and integrating it with Cayley tree functions and generating function techniques, the authors derive exact generating functions for the distribution of records in both trees and forests. Building upon this unified framework, they provide a concise and novel proof of the classical enumeration formula for Cayley forests. The approach offers fresh insight and a coherent interpretation of these fundamental combinatorial structures, revealing deeper connections among seemingly disparate objects in enumerative combinatorics.