construct bijections

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 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.

Characterizing the implicit computational complexity of two-way bijections in reversible computing remains an open challenge. Existing models either require explicit reverse-computation management or introduce auxiliary garbage bits, violating zero-garbage constraints and obscuring intrinsic complexity. Method: We propose the first stack-based, zero-garbage reversible imperative model that inherently supports bidirectional bijections. Leveraging symmetric instruction design and stack discipline, it enforces strict reversibility and eliminates garbage generation without auxiliary bits or explicit inversion logic. Results: The model precisely captures the class of two-way bijections, preserves polynomial-time fidelity, and—crucially—achieves implicit, redundancy-free representation of bidirectional computation. It establishes the first implicit computational complexity framework for two-way bijections, providing both a theoretical foundation and a constructive paradigm for resource-sensitive semantics and efficient implementation of reversible programs.

This work addresses the absence of rigorous formalizations of abstract simplicial complexes and their stellar subdivisions in existing proof systems. It presents the first purely combinatorial formal framework for abstract simplicial complexes grounded in combinatorial topology, implemented in the Lean theorem prover. The framework encompasses fundamental operations such as morphisms, links, and joins, and systematically investigates their interaction with stellar subdivision. Key contributions include the first formalization of stellar subdivision in any proof assistant, the verification of several crucial identities—some previously undocumented in the literature—for the study of triangulated manifolds, and the proof of significant theorems such as the invariance of links under subdivision. This development establishes a reliable formal foundation for computational topology.

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 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 study addresses the problem of automatically discovering concise algebraic structures—such as Cayley graphs or lexicographic products—from graphs represented solely by their adjacency matrices. To this end, it introduces the first neurosymbolic collaborative framework that integrates a general-purpose large language model with the SageMath symbolic computation system via the Model Context Protocol (MCP), establishing a closed-loop pipeline of reasoning, construction, and verification without requiring fine-tuning of the language model. Evaluated on 100 highly symmetric two-orbit graphs, the method successfully derives verifiable algebraic representations in all cases, substantially outperforming template enumeration (+80%) and spectral lookup (+100%). Notably, it also yields the first explicit construction of the smallest known counterexample to the Bernhart–Kainen conjecture.

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.