arXiv subjects

Shaoshi Chen

Publications and source records attributed to Shaoshi Chen.

At least 19 recordsLinked to original sources

arXiv2026

Automated Formal Proofs of Combinatorial Identities via Wilf-Zeilberger Guidance and LLMs

Automating formal proofs of combinatorial identities is challenging for LLM-based provers, as long-horizon proof planning is required and unconstrained search quickly explodes. Symbolic methods such as the Wilf-Zeilberger (WZ) method can achieve a mechanized proof of combinatorial identities by constructing special auxiliary functions and demonstrating that they satisfy specific recurrence relations. We propose WZ-LLM, a neuro-symbolic framework that turns WZ proof plans into executable proof sketches in Lean 4 and uses an LLM-based prover to discharge the resulting machine-checkable subgoals. We also train a dedicated WZ-Prover via a Lean-kernel-verified bootstrapping loop with expert-verified iteration, followed by DAPO-based refinement. Experiments show that WZ-LLM achieves a 34% proof success rate on LCI-Test (100 classic combinatorial identities), outperforming strong baselines such as DeepSeek-V3 and Goedel-Prover-V2, and delivering consistent gains on CombiBench and PutnamBench-Comb. These results indicate that our framework provides two complementary strengths: improved direct proving for identities beyond the scope of WZ, and substantially higher end-to-end success when WZ sketches guide a specialized prover.

cs.LG↗

arXiv2026

Stable patterns on permutations of multisets

In this paper, we study patterns on permutations of multisets whose multivariate distribution generating functions are symmetric. We interpret this phenomenon through the lens of group actions and define such a pattern as stable. Although various stability results are already implicit in existing enumerative work, we explicitly summarize them here and provide bijective proofs. These bijections offer new combinatorial insight into the symmetry of the generating functions. We also establish instability results. In particular, we provide a complete characterization of stable classical patterns, showing that the only such patterns are those of length one or two. For consecutive patterns, we reprove the stability of all monotone patterns and also identify a large class of unstable patterns. We conjecture that monotone patterns are the only stable consecutive patterns. As an application, we use stability to derive recurrence relations for the ascent distribution over permutations of restricted multisets, yielding a generalization of Eulerian numbers.

math.CO↗

arXiv2026

Symbolic Integration in Weierstrass-like Extensions

This paper studies the integration problem in differential fields that may involve quantities reminiscent of the Weierstrass $\wp$ function, which are defined by a first-order nonlinear differential equation. We extend the classical notion of special polynomials to elements of Weierstrass-like extensions and present algorithms for reduction in such extensions. As an application of these results, we derive some new formulae for integrals of powers of $\wp$.

cs.SC↗

arXiv2026

On the Summability Problem of Multivariate Rational Functions in the Mixed Case

Continuing previous work, this paper focuses on the summability problem of multivariate rational functions in the mixed case in which both shift and $q$-shift operators can appear. Our summability criteria rely on three ingredients including orbital decompositions, Sato's isotropy groups, and difference transformations. This work settles the rational case of the long-term project aimed at developing algorithms for symbolic summation of multivariate functions.

cs.SC↗

arXiv2026

Complete Reduction for Derivatives in a Transcendental Liouvillian Extension

Transcendental Liouvillian extensions are differential fields, in which one can model poly-logarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension, we construct, over the subfield of constants, a complement of the subspace of derivatives, and develop an algorithm that decomposes any element of the field into the sum of a derivative and a component lying in the complement. Consequently, an element is a derivative if and only if its complementary component vanishes. Moreover, the algorithm enables us to determine elementary integrability over the extension by computing parametric logarithmic parts, and leads to a reduction-based approach to constructing telescopers for elements in the extension, provided that an a priori order bound is given.

cs.SC↗

arXiv2026

A zero-test for D-algebraic transseries

Consider formal power series $f_1,\ldots, f_k\in\mathbb{Q}[[z]]$ that are defined as the solutions of a system of polynomial differential equations together with a sufficient number of initial conditions. Given $P\in \mathbb{Q}[F_1,\ldots,F_k]$, several algorithms have been proposed in order to test whether $P(f_1,\ldots,f_k)=0$. In this paper, we present such an algorithm for the case where $f_1,\ldots,f_k$ are so-called transseries instead of power series.

cs.SC↗

arXiv2026

Symbolic Integration of Differential Forms: From Abel to Zeilberger

This paper focuses on symbolic integration of differential forms, with a particular emphasis on historical and modern developments, from Abel's addition theorems for Abelian integrals to Zeilberger's creative telescoping for parameterized integrals. It explores closed rational $p$-forms and provides algorithmic approaches for their integration, extending classical results like Hermite reduction and Liouville's theorem. The integration of closed differential forms with parameters is further examined through telescopers, offering a unified framework for handling both algebraic and transcendental cases.

math.CA↗

arXiv2025

Single-exponential bounds for diagonals of D-finite power series

D-finite power series appear ubiquitously in combinatorics, number theory, and mathematical physics. They satisfy systems of linear partial differential equations whose solution spaces are finite-dimensional, which makes them enjoy a lot of nice properties. After attempts by others in the 1980s, Lipshitz was the first to prove that the class they form in the multivariate case is closed under the operation of diagonal. In particular, an earlier work by Gessel had addressed the D-finiteness of the diagonals of multivariate rational power series. In this paper, we give another proof of Gessel's result that fixes a gap in his original proof, while extending it to the full class of D-finite power series. We also provide a single exponential bound on the degree and order of the defining differential equation satisfied by the diagonal of a D-finite power series in terms of the degree and order of the input differential system.

math.CO↗

arXiv2025

Telescoping Algorithms for $\Sigma^*$-Extensions via Complete Reductions

A complete reduction on a difference field is a linear operator that enables one to decompose an element of the field as the sum of a summable part and a remainder such that the given element is summable if and only if the remainder is equal to zero. In this paper, we present a complete reduction in a tower of $\Sigma^*$-extensions that turns to a new efficient framework for the parameterized telescoping problem. Special instances of such $\Sigma^*$-extensions cover iterative sums such as the harmonic numbers and generalized versions that arise, e.g., in combinatorics, computer science or particle physics. Moreover, we illustrate how these new ideas can be used to reduce the depth of the given sum and provide structural theorems that connect complete reductions to Karr's Fundamental Theorem of symbolic summation.

cs.SC↗

arXiv2025

Creative Telescoping

These notes on creative telescoping are based on a series of lectures at the Institut Henri Poincare in November and December 2023.

cs.SC↗

arXiv2025

Flip Graphs for Polynomial Multiplication

Flip graphs were recently introduced in order to discover new matrix multiplication methods for matrix sizes. The technique applies to other tensors as well. In this paper, we explore how it performs for polynomial multiplication.

cs.SC↗

arXiv2025

Non-minimality of minimal telescopers explained by residues

Elaborating on an approach recently proposed by Mark van Hoeij, we continue to investigate why creative telescoping occasionally fails to find the minimal-order annihilating operator of a given definite sum or integral. We offer an explanation based on the consideration of residues.

cs.SC↗

arXiv2025

A Unified Reduction for Hypergeometric and q-Hypergeometric Creative Telescoping

We adapt the theory of normal and special polynomials from symbolic integration to the summation setting, and then built up a general framework embracing both the usual shift case and the $q$-shift case. In the context of this general framework, we develop a unified reduction algorithm, and subsequently a creative telescoping algorithm, applicable to both hypergeometric terms and their $q$-analogues. Our algorithms allow to split up the usual shift case and the $q$-shift case only when it is really necessary, and thus instantly reveal the intrinsic differences between these two cases. Computational experiments are also provided.

cs.SC↗

arXiv2024

Bijections around Springer numbers

Arnol'd proved in 1992 that Springer numbers enumerate the Snakes, which are type $B$ analogs of alternating permutations. Chen, Fan and Jia in 2011 introduced the labeled ballot paths and established a ``hard'' bijection with snakes. Callan conjectured in 2012 and Han--Kitaev--Zhang proved recently that rc-invariant alternating permutations are counted by Springer numbers. Very recently, Chen--Fang--Kitaev--Zhang investigated multi-dimensional permutations and proved that weakly increasing $3$-dimensional permutations are also counted by Springer numbers. In this work, we construct a sequence of ``natural'' bijections linking the above four combinatorial objects.

math.CO↗

arXiv2024

Patterns in Multi-dimensional Permutations

In this paper, we propose a general framework that extends the theory of permutation patterns to higher dimensions and unifies several combinatorial objects studied in the literature. Our approach involves introducing the concept of a "level" for an element in a multi-dimensional permutation, which can be defined in multiple ways. We consider two natural definitions of a level, each establishing connections to other combinatorial sequences found in the Online Encyclopedia of Integer Sequences (OEIS). Our framework allows us to offer combinatorial interpretations for various sequences found in the OEIS, many of which previously lacked such interpretations. As a notable example, we introduce an elegant combinatorial interpretation for the Springer numbers: they count weakly increasing 3-dimensional permutations under the definition of levels determined by maximal entries.

math.CO↗

arXiv2024

How to generate all possible rational Wilf-Zeilberger forms?

Wilf-Zeilberger pairs are fundamental in the algorithmic theory of Wilf and Zeilberger for computer-generated proofs of combinatorial identities. Wilf-Zeilberger forms are their high-dimensional generalizations, which can be used for proving and discovering convergence acceleration formulas. This paper presents a structural description of all possible rational such forms, which can be viewed as an additive analog of the classical Ore-Sato theorem. Based on this analog, we show a structural decomposition of so-called multivariate hyperarithmetic terms, which extend multivariate hypergeometric terms to the additive setting.

cs.SC↗

arXiv2024

Parallel Summation in P-Recursive Extensions

We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive sequence and its shifts. There is a distinction between so-called normal and so-called special polynomials. Under the assumption that the corresponding difference field has no unnatural constants, we are able to predict the normal polynomials appearing in the denominator of a potential closed form. We can also handle the numerator. Our method is incomplete so far as we cannot predict the special polynomials appearing in the denominator. However, we do have some structural results about special polynomials for the setting under consideration.

math.CO↗

arXiv2023

Stability Problems on D-finite Functions

This paper continues the studies of symbolic integration by focusing on the stability problems on D-finite functions. We introduce the notion of stability index in order to investigate the order growth of the differential operators satisfied by iterated integrals of D-finite functions and determine bounds and exact formula for stability indices of several special classes of differential operators. With the basic properties of stability index, we completely solve the stability problem on general hyperexponential functions.

cs.SC↗