Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unimodality and certain bivariate formal Laurent series

Nian Hong Zhou

TL;DR

The paper develops a unifying semiring framework for formal bivariate Laurent series with nonnegative coefficients, showing the sets $\mathscr{U}_{x,y}^\nu$ and $\mathscr{T}_{x,y}^\nu$ are closed under addition and multiplication. This structure yields broad unimodality results across diverse combinatorial and geometric objects, including Andrews’s generalized Gauss polynomials, refined plane partitions, refined color partitions, various rank statistics (Garvan $k$-ranks, concave/convex compositions, and related sequences), and Hilbert-scheme invariants such as Betti numbers and genus-0 Gromov–Witten invariants. Key results include the unimodality of the refined Gauss-type polynomials $\left[ {m+n \choose m} \right]_q$ (and many variants), parity/unimodality phenomena for Betti numbers, and unimodality statements for several rank distributions, with several open directions notably for certain $k$-ranks. The methods hinge on embedding generating functions into the semiring classes, deriving coefficient-inequality criteria, and applying them to obtain monotonicity and strict unimodality, offering a cohesive perspective bridging partition theory, cohomological Hall algebras, and Hilbert schemes. Overall, the work provides a powerful algebraic mechanism to certify unimodality across a range of combinatorial and geometric settings, and it resolves notable open problems in generalized Gauss polynomials while exposing a rich landscape of unimodal phenomena in geometric invariants.

Abstract

In this paper, we examine the unimodality and strict unimodality of certain formal bivariate Laurent series with non-negative coefficients. We show that the sets of these formal bivariate Laurent series form commutative semirings under the operations of addition and multiplication of formal Laurent series. This result is used to establish the unimodality of sequences involving Gauss polynomials and certain refined color partitions. In particular, we solve an open problem posed by Andrews on the unimodality of generalized Gauss polynomials and establish an unimodal result for a statistic of plane partitions. We also establish many unimodal results for rank statistics in partition theory, including the rank statistics of concave and convex compositions studied by Andrews, as well as certain unimodal sequences studied by Kim-Lim-Lovejoy. Additionally, we establish the unimodality of the Betti numbers and Gromov-Witten invariants of certain Hilbert schemes of points.

Unimodality and certain bivariate formal Laurent series

TL;DR

The paper develops a unifying semiring framework for formal bivariate Laurent series with nonnegative coefficients, showing the sets and are closed under addition and multiplication. This structure yields broad unimodality results across diverse combinatorial and geometric objects, including Andrews’s generalized Gauss polynomials, refined plane partitions, refined color partitions, various rank statistics (Garvan -ranks, concave/convex compositions, and related sequences), and Hilbert-scheme invariants such as Betti numbers and genus-0 Gromov–Witten invariants. Key results include the unimodality of the refined Gauss-type polynomials (and many variants), parity/unimodality phenomena for Betti numbers, and unimodality statements for several rank distributions, with several open directions notably for certain -ranks. The methods hinge on embedding generating functions into the semiring classes, deriving coefficient-inequality criteria, and applying them to obtain monotonicity and strict unimodality, offering a cohesive perspective bridging partition theory, cohomological Hall algebras, and Hilbert schemes. Overall, the work provides a powerful algebraic mechanism to certify unimodality across a range of combinatorial and geometric settings, and it resolves notable open problems in generalized Gauss polynomials while exposing a rich landscape of unimodal phenomena in geometric invariants.

Abstract

In this paper, we examine the unimodality and strict unimodality of certain formal bivariate Laurent series with non-negative coefficients. We show that the sets of these formal bivariate Laurent series form commutative semirings under the operations of addition and multiplication of formal Laurent series. This result is used to establish the unimodality of sequences involving Gauss polynomials and certain refined color partitions. In particular, we solve an open problem posed by Andrews on the unimodality of generalized Gauss polynomials and establish an unimodal result for a statistic of plane partitions. We also establish many unimodal results for rank statistics in partition theory, including the rank statistics of concave and convex compositions studied by Andrews, as well as certain unimodal sequences studied by Kim-Lim-Lovejoy. Additionally, we establish the unimodality of the Betti numbers and Gromov-Witten invariants of certain Hilbert schemes of points.
Paper Structure (14 sections, 24 theorems, 128 equations)

This paper contains 14 sections, 24 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Unimodality and Gauss polynomials
  3. On Andrews's generalized Gauss polynomials
  4. On a refined plane partition
  5. Unimodality and certain refined color partitions
  6. Unimodality and strict unimodality
  7. On certain refined color partitions
  8. Unimodality and ranks in partition theory
  9. Garvan $k$-ranks
  10. Ranks of concave and convex compositions
  11. Ranks of certain unimodal sequences
  12. Unimodality and certain Hilbert scheme
  13. Betti numbers of certain Hilbert scheme
  14. Gromov-Witten invariants of certain Hilbert scheme

Key Result

Proposition 1

For each $\nu \in \{ 1, 2 \}$, the sets $\mathscr{U}_x^\nu$ and $\mathscr{T}_x^\nu$ both form commutative semirings under the operations of addition and multiplication of Laurent polynomials.

Theorems & Definitions (46)

  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Remark 1.1
  • Theorem 1
  • proof
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • ...and 36 more