apply counting arguments

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 addresses permutation pattern avoidance and containment via a declarative constraint programming (CP) approach. Methodologically, it introduces a composable and extensible library of permutation constraints, enabling unified modeling of six pattern avoidance/containment relations, eight classical permutation properties, and five combinatorial statistics; it further achieves the first dynamic composition and incremental solving of arbitrary pattern constraints. As a key application, the framework enumerates inversions in 1324-avoiding permutations—extending the tractable instance size to length 16 for the first time—and discovers that the resulting inversion distribution corresponds to a novel integer sequence not yet cataloged in OEIS. Experiments demonstrate substantial improvements in both efficiency and flexibility for generating and analyzing permutations under complex combinatorial constraints. The proposed framework establishes a reusable, declarative paradigm for enumerative combinatorics, bridging CP methodology with structural enumeration problems.

This work proposes a novel architecture based on adaptive feature fusion and dynamic reasoning to address the limited generalization of existing methods in complex scenarios. By incorporating a multi-scale context-aware module and a learnable reasoning path selection strategy, the approach significantly enhances model robustness under distribution shifts and noisy interference. Extensive experiments demonstrate that the proposed method consistently outperforms state-of-the-art models across multiple benchmark datasets, achieving an average accuracy improvement of 2.3% while maintaining low computational overhead. The primary contribution lies in the development of a general-purpose reasoning framework that effectively balances accuracy and efficiency, offering a promising direction for deploying intelligent systems in open-world environments.

This work addresses the problem of verbose and semantically opaque partial assignments induced by CNF conversion in SAT/SMT enumeration. We systematically evaluate the suitability of Tseitin versus Plaisted–Greenbaum (PG) encodings in enumeration contexts. Theoretically and empirically, we show that Tseitin encoding inherently impedes generation of short partial assignments, whereas PG encoding—when combined with negation normal form (NNF) preprocessing—guarantees that each enumerated solution corresponds to a minimal, semantically transparent partial assignment. This synergistic approach is the first to provably eliminate encoding-induced assignment redundancy in enumeration. Evaluated on SMT-LIB benchmarks, it reduces both the number of partial solutions and total runtime by 1–3 orders of magnitude, demonstrating strong theoretical soundness and practical efficacy.

This work addresses the challenge of verifying equivalence between target representations—such as decision-DNNF—and their original CNF encodings in knowledge compilation. Methodologically, it introduces (1) Partitioned-Operation Graphs (POGs) as a unified intermediate representation; (2) the Certified POG (CPOG) proof framework, enabling structured, correctness-preserving compilation from CNF to POG; and (3) full formal verification in Lean 4 of the compiler, proof generator, and model counter. Contributions include: the first end-to-end, machine-checked correctness guarantee for the entire knowledge compilation pipeline; automated verification of D4-generated POGs; empirical evaluation on standard model counting benchmarks; and the first mathematically verified toolchain supporting both weighted and unweighted model counting. The framework ensures semantic equivalence at every compilation step, thereby bridging the gap between practical knowledge compilation tools and formal correctness guarantees.

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 systematically evaluates the rigorous proof reasoning and explicit construction capabilities of large language models on Olympiad-level combinatorics problems. To this end, we introduce a benchmark comprising 100 expert-annotated competition problems, categorizing tasks into analytical (proof-oriented) and constructive (implementation-oriented) types. We propose a unified evaluation protocol that integrates rubric-guided proof assessment with deterministic verification of constructions, enhanced by a Best@4 multi-solution sampling strategy. Experimental results show that the strongest model achieves an average score of 65.4% overall (75.3% under Best@4), with markedly divergent performance across the two task types, revealing current limitations in creative mathematical reasoning—particularly on existence and construction problems. This work presents the first fine-grained distinction and joint evaluation of these capabilities, offering a new benchmark and diagnostic framework for mathematical reasoning research.

This work addresses the explicit vertex enumeration problem for the cut polytope CUT(n), a fundamental object in combinatorial optimization. Despite longstanding interest, no closed-form expression for its vertex count has been available. We introduce a novel framework based on consistency probability modeling of symmetric Bernoulli variables, establishing an exact bijection between cut vectors (binary encodings) and alternating cyclic functions. This yields the first explicit analytic formula for the number of vertices of CUT(n), reducing computational complexity from exponential enumeration to constant-time evaluation. Furthermore, we uncover structural properties of the scaled-encoding vertex sequence—including near-linear growth, palindromicity, and recursive self-similarity. Integrating probabilistic methods, combinatorial coding, convex polyhedral theory, and integer sequence analysis, our results provide a new analytically tractable and computationally efficient theoretical tool for NP-hard problems such as Max-Cut.

This work investigates the parameterized counting complexity of $k$-vertex induced subgraphs satisfying a fixed graph property $\Phi$, with a focus on symmetry conditions dictated by the structure of their automorphism groups. By initiating from the $k$-clique problem and employing a refined parameterized reduction based on a “clique gadget” construction, the study establishes—for the first time—that counting $k$-vertex induced subgraphs whose automorphism group is exactly a given finite group $Q$ is $\#\mathbf{W}[1]$-hard for any finite group $Q$. This result not only confirms the $\#\mathbf{W}[1]$-hardness in the case of trivial automorphism groups but also generalizes it to arbitrary finite groups, thereby overcoming limitations inherent in existing Fourier-analytic approaches and resolving a long-standing open problem in this direction.

When does a linear combination of induced subgraph counts—i.e., a motif parameter of the form ∑_H c_H · #IndSub(H, G)—admit a combinatorial interpretation? Specifically, which integer coefficient vectors preserve “counting semantics”? Method: We introduce a categorical framework generalizing the problem to colored graphs and finite vector spaces; integrate relativized #P-closure, the Ikenmeyer–Pak algebraic method, and Ramsey-theoretic arguments. Contribution/Results: We establish the first dichotomy theorem for combinatorial interpretability of motif parameters: such a linear combination admits a combinatorial interpretation if and only if all motifs H are isolated-vertex-free and all coefficients c_H are positive integers; parameters with negative coefficients are uncomputable even relative to a #P oracle. This yields a complete classification of combinatorially interpretable motif parameters over graphs and relational structures, resolving a fundamental question in parameterized counting complexity.