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A computer algebra package for bivariate asymptotics with explicit error terms

Benjamin Hackl, Stephan Wagner

TL;DR

The paper presents a SageMath-based toolkit, dependent_bterms, for obtaining rigorous multivariate asymptotics with explicit error terms for binomial-sum expressions by extending SageMath's $B$-term framework to a dependent variable. It demonstrates the approach on two concrete problems: Ramanujan's Q-function and a Bóna–DeJonge inequality for 132-avoiding permutations, delivering fully quantified expansions and computable error bounds. The method combines a Mellin-transform framework, region-splitting, residue calculus, and automated error tracking to produce precise asymptotics and certify results for large $n$ while providing verified checks for small $n$. This yields both theoretical guarantees and practical computational tools for rigorous asymptotic analysis in combinatorics and related fields, streamlining what used to be highly manual, case-by-case labor.

Abstract

Making use of a newly developed package in the computer mathematics system SageMath, we show how to perform a full asymptotic analysis of certain types of sums that occur frequently in combinatorics, including explicit error bounds. We present two applications of the general approach to illustrate its use: the first concerns a classical problem due to Ramanujan, while the second one concerns a question of Bóna and DeJonge on 132-avoiding permutations with a unique longest increasing subsequence that can be translated into an inequality for a certain binomial sum.

A computer algebra package for bivariate asymptotics with explicit error terms

TL;DR

The paper presents a SageMath-based toolkit, dependent_bterms, for obtaining rigorous multivariate asymptotics with explicit error terms for binomial-sum expressions by extending SageMath's -term framework to a dependent variable. It demonstrates the approach on two concrete problems: Ramanujan's Q-function and a Bóna–DeJonge inequality for 132-avoiding permutations, delivering fully quantified expansions and computable error bounds. The method combines a Mellin-transform framework, region-splitting, residue calculus, and automated error tracking to produce precise asymptotics and certify results for large while providing verified checks for small . This yields both theoretical guarantees and practical computational tools for rigorous asymptotic analysis in combinatorics and related fields, streamlining what used to be highly manual, case-by-case labor.

Abstract

Making use of a newly developed package in the computer mathematics system SageMath, we show how to perform a full asymptotic analysis of certain types of sums that occur frequently in combinatorics, including explicit error bounds. We present two applications of the general approach to illustrate its use: the first concerns a classical problem due to Ramanujan, while the second one concerns a question of Bóna and DeJonge on 132-avoiding permutations with a unique longest increasing subsequence that can be translated into an inequality for a certain binomial sum.
Paper Structure (12 sections, 1 theorem, 99 equations, 1 figure)

This paper contains 12 sections, 1 theorem, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. B-terms and asymptotics with explicit error bounds
  3. Asymptotic Expansions in SageMath
  4. The dependent_bterms package
  5. A detailed example: the Ramanujan Q-function
  6. Contribution of the large range
  7. Contribution of the middle range
  8. Contribution of the small range
  9. On a Question of Bóna and DeJonge
  10. Asymptotic analysis
  11. Collection of all contributions
  12. Sketch of an alternative approach

Key Result

Theorem 1

For $n\geq 10000$, the binomial sum $F(n)$ satisfies the asymptotic formula

Figures (1)

  • Figure 1: Comparison of the absolute value of the asymptotic main term $-n^2/8 + n/24$ (red) against the accumulated total error (blue) on the left. The right plot depicts the ratio of the error bound to the main term.

Theorems & Definitions (1)

  • Theorem 1