computational enumeration

This work addresses the efficient counting of permutation patterns. We introduce a unified modeling and algorithmic framework based on double posets. Our key innovation is the formalization of corner trees as a special class of double posets—termed *twin-tree double posets*—and their generalization to a broader family of tree-structured double posets. Building on this structural characterization, we extend the Even–Zohar–Leng algorithm to this new setting, yielding a subquadratic counting algorithm with time complexity $O(n^{5/3})$. This approach substantially transcends the expressive limitations of the original corner tree model, significantly expanding the class of permutation patterns amenable to efficient enumeration. The framework provides a novel structural tool and algorithmic paradigm for permutation pattern theory and combinatorial enumeration.

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.

This work addresses four #P-complete graph counting problems in computational chemistry—Kekulé structure enumeration, Hosoya index, Merrifield–Simmons index, and matching/independent set entropy—by introducing the first fixed-parameter tractable (FPT) framework parameterized by treewidth (tw) and pathwidth (pw). We design a unified dynamic programming algorithm for these chemical topological indices, achieving time complexity $O^*(2^{O( ext{tw})})$, substantially improving upon existing exponential-time baselines. Theoretical analysis and empirical evaluation on the full PubChem dataset (>100 million compounds) demonstrate that >99.9% of real chemical graphs satisfy $ ext{tw} leq 6$, validating the small-treewidth hypothesis. Our implementation scales efficiently to large-scale chemical graphs, delivering speedups of several orders of magnitude over naive enumeration. This work bridges parameterized algorithms and cheminformatics, establishing a new paradigm for exact computation of #P-hard chemical graph metrics.

This work addresses the computationally challenging problem of efficiently enumerating small-scale pseudoline arrangements and abstract order types. We propose a customized algorithmic framework integrating symbolic computation, backtracking search, and combinatorial constraint pruning, augmented by canonical-form normalization and isomorphism testing to eliminate structural redundancies. Our approach achieves, for the first time, the complete and provably correct enumeration of all abstract order types on 12 points and all pseudoline arrangements of 11 pseudolines—thereby filling a critical gap in existing combinatorial geometry databases. The resulting experimental toolkit substantially enhances both the feasibility and efficiency of systematic exploration of discrete geometric configurations. It provides essential infrastructure for empirical research in computational geometry, discrete geometry, and formal verification, enabling rigorous experimentation with complex combinatorial structures previously beyond reach.

This study addresses the problem of maximizing the number of triangular faces in simple pseudoline arrangements formed by an odd number of pseudolines. By employing a depth-first search that branches only on generators at even positions, combined with geometric pruning and reduced word enumeration, the approach systematically handles structural biases arising when $n \equiv 1 \pmod{6}$. Arrangements are hierarchically classified according to swap, Euclidean, and projective equivalence relations. The work achieves, for the first time, a complete enumeration of maximal-triangle arrangements together with a multi-level classification into equivalence classes, fully recovering the symmetry groups of each projective class and the orbit–stabilizer structures of their Euclidean subclasses. The enumeration yields 85,562,064 wiring diagrams for $n=27$, grouped into 56,646 projective classes, and provides initial results extending up to $n=93$.

This study addresses the efficient enumeration of connected graphs whose automorphism groups act with exactly two orbits. To this end, the authors propose a novel methodology that integrates Goursat’s lemma to construct candidate groups, performs graph enumeration under automorphism group constraints, and incorporates group-theoretic pruning to enhance computational efficiency. This approach yields the first complete enumeration of all connected two-orbit graphs on up to 27 vertices, resulting in a total of 10,094,721 such graphs. The method substantially surpasses the limitations of traditional brute-force enumeration techniques, dramatically expanding the scale of instances that can be feasibly solved within this class of symmetry-constrained graph enumeration problems.

This work addresses the need for efficient higher-order structural analysis in complex networks by studying the counting of hypertriangles—patterns formed by three pairwise-intersecting hyperedges—in hypergraphs. Inspired by graph orientation and degeneracy-based algorithms, the authors generalize the concepts of graph orientation and degeneracy ordering to hypergraphs for the first time, proposing DITCH, a provably efficient counting algorithm. DITCH integrates hypergraph orientation, degeneracy ordering, and combinatorial enumeration to effectively handle the diverse intersection structures inherent in hypertriangles. Experimental results demonstrate that DITCH achieves speedups of 10–100× over state-of-the-art methods while substantially reducing memory consumption.

This work addresses the enumeration and listing of projected tree patterns in graphs—a generalization of subgraph isomorphism central to database querying. We present the first efficient enumeration algorithm featuring polynomial preprocessing time and polylogarithmic delay. Under natural conditions, we establish a general equivalence between enumeration and listing for this problem. Our approach integrates fast (rectangular and output-sensitive) matrix multiplication, parameterized analysis via submodular width, and fine-grained complexity lower bounds. For a projected tree with $k$ nodes, the algorithm achieves $\tilde{O}(n^{17.42})$ preprocessing time and polylogarithmic delay, and extends to hypergraphs with preprocessing time $\tilde{O}(m^{17.42 \cdot \text{subw}(H)})$, where $\text{subw}(H)$ denotes the submodular width of the hypergraph $H$.