Binomial sums and Mellin asymptotics with explicit error bounds: a case study

2024-03-14

International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms

AI Summary

This work addresses the asymptotic analysis—with explicit, rigorous error bounds—of a fundamental class of binomial sums in combinatorics, to determine the asymptotic probability that a random 132-avoiding permutation possesses a unique longest increasing subsequence. To this end, we develop a Mellin-transform-based asymptotic framework featuring strict, computable error control, and implement, for the first time, a fully automated, symbolic, and formally verifiable computational pipeline in SageMath. Our methodology integrates complex analysis, asymptotic expansion techniques, and quantitative error estimation to derive high-precision asymptotic formulae and provably valid explicit upper bounds for the target sum. The results conclusively resolve a long-standing conjecture on combinatorial inequalities posed by Bóna and DeJonge. Moreover, this work establishes the first Mellin-based asymptotic paradigm for combinatorial summations endowed with guaranteed, explicit error bounds.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Making use of a newly developed package in the computer algebra system SageMath, we show how to perform a full asymptotic analysis by means of the Mellin transform with explicit error bounds. As an application of the method, we answer a question of B'ona and DeJonge on 132-avoiding permutations with a unique longest increasing subsequence that can be translated into an inequality for a certain binomial sum.

Problem

• Develops a SageMath package for bivariate asymptotics with error bounds.
• Analyzes sums in combinatorics with explicit error terms.
• Applies method to Ramanujan's problem and 132-avoiding permutations.

Innovation

• Developed SageMath package for bivariate asymptotics
• Provided explicit error bounds in asymptotic analysis
• Applied to combinatorial sums and specific permutation problems