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 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.

Explicitly constructing combinatorial bijections remains a long-standing challenge in algebraic combinatorics, especially given the intractability of manually analyzing massive combinatorial datasets. Method: We propose the first machine learning–driven, interpretable discovery framework for combinatorial bijections. Leveraging the attention mechanism of Transformer models, our approach analyzes paired combinatorial structures—such as Dyck paths—to uncover latent bijection patterns. We then introduce the Scaffolding Map algorithm, which systematically translates opaque attention patterns into verifiable, generalizable combinatorial mapping rules. Contribution/Results: Our framework automatically derives a novel explicit construction of the zeta map directly from model attention—marking the first data-driven, mathematically rigorous derivation of this fundamental bijection. It overcomes the traditional reliance on human insight while preserving formal correctness, significantly enhancing both the efficiency and interpretability of discovering complex combinatorial bijections.

This paper addresses the optimization of $n$-ary functions over integer boxes. We propose a reducibility-based compression method grounded in *equivalent small-domain substitution*: leveraging integer lattice theory and domain reduction, we construct an equivalent function with a significantly smaller domain, thereby upgrading originally weakly polynomial-time algorithms to strongly polynomial-time guarantees. This work establishes the first unified framework for systematically strengthening a broad class of weakly polynomial algorithms to strong polynomial complexity, while simultaneously simplifying both the analysis and construction of reducibility bounds. By decoupling computational efficiency from input size—particularly the bit-length of coefficients—the method overcomes a fundamental scalability bottleneck. It yields a qualitative leap in computational efficiency for integer programming and discrete optimization, offering a novel paradigm for large-scale combinatorial optimization.

This work addresses #P-hard counting problems—such as counting independent sets in general graphs and #2-SAT—that are inapproximable in polynomial time and prohibitively expensive to solve exactly. The authors propose a novel framework based on bounded, unweighted self-reducibility, which recursively decomposes problem instances and aggregates upper bounds from subproblems at a square-root recursion depth. By integrating enumeration with a hybrid sampling estimator, the approach substantially reduces the base of the exponential time complexity. The method achieves improved runtimes of O*(1.1869ⁿ) for independent set counting and O*(1.2373ⁿ) for #2-SAT approximation, outperforming the best known exact algorithms. It further extends to counting maximum cliques, minimal separators, and perfect matchings in subcubic graphs, and admits black-box quantum speedup.

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.

For #P problems where decision is tractable but exact counting is intractable, this paper introduces spanALP—a novel counting complexity class defined via alternating Turing machines with transducer-style output mechanisms—and establishes a general sufficient condition for admitting a fully polynomial-time randomized approximation scheme (FPRAS). We strictly locate spanALP between #L and TotP, revealing an intrinsic connection between alternation-based computation and approximate solvability. Theoretically, we prove that every problem in spanALP admits an FPRAS. Practically, we design the first FPRAS for graph path queries under Dyck language constraints. This work bridges a fundamental gap between alternating computation models and randomized approximate counting, providing a systematic complexity-theoretic framework for approximation algorithms in counting problems.

This work addresses the quadratic-time bottleneck in counting k-order subhypergraphs (hypergraphlets) in hypergraphs by proposing the first subquadratic-time algorithm. Leveraging a novel structural assumption termed (α,β)-niceness, the input hypergraph is decomposed into low-rank and low-degree components. The algorithm combines color coding, rank- and degree-based divide-and-conquer strategies, and randomized sampling to process these components separately before merging the results. This approach enables both efficient exact counting and uniform sampling of hypergraphlets. Empirical evaluation on real-world hypergraph datasets demonstrates over an order-of-magnitude speedup compared to the naive quadratic algorithm, substantially breaking through the prevailing complexity barrier.

This work investigates the fine-grained complexity of approximately counting occurrences of a length-$k$ permutation pattern within a length-$n$ permutation. Under the Exponential Time Hypothesis (ETH), we establish via fine-grained reductions that no algorithm running in time $f(k) \cdot n^{o(k/\log k)}$ can approximate the count within a multiplicative error of $n^{(1/2 - \varepsilon)k}$ for any $\varepsilon > 0$. This result refutes the conjecture that approximate counting of small patterns is significantly easier than exact counting, and it provides the first conditional equivalence between approximate and exact counting in this setting. Moreover, our lower bound yields an almost tight trade-off between approximation error and running time, nearly matching the known upper bound of $n^{k/2}$.

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.