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arxiv: 2505.08149 · v4 · submitted 2025-05-13 · 💻 cs.SC · math.AG· math.CO

Majorization and Inequalities among Complete Homogeneous Symmetric Functions

Jia Xu , Yong Yao This is my paper

Pith reviewed 2026-05-22 16:30 UTC · model grok-4.3

classification 💻 cs.SC math.AGmath.CO
keywords majorizationcomplete homogeneous symmetric functionssymmetric function inequalitiespartitionscounterexamplesdegree greater than 7
0
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The pith

Majorization does not characterize inequalities among complete homogeneous symmetric functions for any degree greater than 7.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the relationship between the majorization order on partitions and the inequalities satisfied by complete homogeneous symmetric functions of a fixed degree. Prior results established that majorization fully determines these inequalities through degree 7. The authors demonstrate that the correspondence breaks for every larger degree by exhibiting concrete pairs of partitions where one majorizes the other, yet the inequality between the corresponding complete homogeneous symmetric functions runs in the opposite direction. This shows that majorization alone cannot serve as a complete criterion for determining all inequalities in this class of symmetric functions once the degree exceeds 7.

Core claim

The authors prove that for every integer degree n greater than 7, majorization does not characterize the inequalities among the complete homogeneous symmetric functions of degree n. This is established by exhibiting specific pairs of partitions of n for which the majorization relation holds in one direction while the inequality between the associated complete homogeneous symmetric functions holds in the opposite direction.

What carries the argument

Counterexample pairs of integer partitions of n > 7 on which the majorization partial order diverges from the inequality direction of the complete homogeneous symmetric function values.

If this is right

  • For each degree n > 7, at least one pair of partitions serves as a counterexample to the characterization.
  • Additional criteria beyond majorization are required to fully describe inequalities among CHs in higher degrees.
  • The structure of inequalities for CHs becomes more complex as degree increases beyond 7.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If similar divergences occur in other families of symmetric functions, majorization may have limited scope as a general tool for symmetric function inequalities.
  • Computational searches for counterexamples could be extended to related problems in algebraic combinatorics.

Load-bearing premise

That there exist explicit pairs of partitions for each n greater than 7 where majorization and the CH inequality point in opposite directions.

What would settle it

A single degree n > 7 for which every pair of partitions satisfies that majorization implies the corresponding CH inequality in the same direction, with no counterexamples.

read the original abstract

Inequalities among symmetric functions are fundamental in various branches of mathematics, thus motivating a systematic study of their structure. Majorization has been shown to characterize inequalities among commonly used symmetric functions, except for complete homogeneous symmetric functions (shortened as CHs). In 2011, Cuttler, Greene, and Skandera posed a natural question: Can majorization also characterize inequalities among CHs? Their work demonstrated that majorization characterizes inequalities among CHs up to degree 7 and suggested exploring its validity for higher degrees. In this paper, we show that, for every degree greater than 7, majorization does not characterize inequalities among CHs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that majorization characterizes inequalities among complete homogeneous symmetric functions (CHs) only up to degree 7, as previously shown by Cuttler, Greene, and Skandera. For every integer degree n > 7 the authors assert that there exist partitions λ, μ ⊢ n with λ ≻ μ in the majorization order yet the corresponding CH symmetric functions satisfy the reverse inequality, thereby showing that majorization fails to characterize CH inequalities in all higher degrees.

Significance. If the asserted counterexamples are correctly constructed and verified, the result supplies a sharp negative answer to the 2011 question of Cuttler et al. and delineates the exact range in which majorization applies to CHs. The work thereby contributes a precise boundary result to the study of inequalities among symmetric functions.

major comments (1)
  1. The universal negative statement for all n > 7 is load-bearing on the existence, for each such n, of at least one explicit pair λ, μ ⊢ n satisfying λ ≻ μ while h_λ and h_μ reverse the expected inequality. The manuscript must therefore supply, in the section presenting the main result, either a concrete list of such pairs for small n > 7 together with the explicit expansions or generating-function evaluations that confirm the reversal, or a general constructive method that can be checked for arbitrary n.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The major comment correctly identifies that the universal claim for all n > 7 requires transparent verification of the counterexamples. We address this point below and will revise the manuscript to incorporate the requested material.

read point-by-point responses

  1. Referee: The universal negative statement for all n > 7 is load-bearing on the existence, for each such n, of at least one explicit pair λ, μ ⊢ n satisfying λ ≻ μ while h_λ and h_μ reverse the expected inequality. The manuscript must therefore supply, in the section presenting the main result, either a concrete list of such pairs for small n > 7 together with the explicit expansions or generating-function evaluations that confirm the reversal, or a general constructive method that can be checked for arbitrary n.

    Authors: We agree that explicit verification improves clarity and verifiability. The manuscript already contains a general constructive method (based on a specific family of partitions that can be generated for any n > 7), but we acknowledge that the presentation can be strengthened. In the revised version we will add, in the main theorem section, an explicit list of pairs for n = 8, 9, and 10 together with the corresponding generating-function evaluations or direct expansions that confirm the reversal. We will also expand the description of the general construction into a fully checkable algorithmic procedure, including the precise rule for selecting λ and μ from the majorization lattice for arbitrary n. revision: yes

Circularity Check

0 steps flagged

No circularity: negative result for n>7 rests on explicit counterexample partitions independent of the cited positive result up to degree 7.

full rationale

The paper cites the independent 2011 work of Cuttler, Greene, and Skandera to establish that majorization characterizes CH inequalities for degrees ≤7. For every degree >7 it asserts the opposite by exhibiting (or constructing) at least one pair of partitions λ, μ ⊢ n such that λ ≻ μ yet the complete homogeneous symmetric functions satisfy the reversed inequality. This step is a direct, externally verifiable construction rather than a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. No equation or premise in the derivation reduces to its own inputs by construction; the claim remains falsifiable by independent evaluation of the generating functions or explicit expansions for the cited partitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a pure-mathematics disproof that introduces no fitted numerical parameters and no new postulated entities; it relies only on the standard definitions and properties of majorization and complete homogeneous symmetric functions already present in the cited literature.

axioms (2)
  • standard math Majorization is the standard partial order on integer partitions or real sequences as defined in the 2011 Cuttler-Greene-Skandera paper.
    The characterization question is posed in terms of this pre-existing order.
  • standard math Complete homogeneous symmetric functions are the standard generating functions summing all monomials of given degree.
    The inequalities under study are defined using these classical objects.

pith-pipeline@v0.9.0 · 5630 in / 1349 out tokens · 83848 ms · 2026-05-22T16:30:26.894109+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Majorization Inequalities from Logarithmic Convexity

    math.CO 2026-05 unverdicted novelty 7.0

    Log-convexity implies convexity and thus majorization inequalities for Macdonald polynomials, Jack polynomials, and Heckman-Opdam hypergeometric functions, unifying prior results and resolving open conjectures.

Reference graph

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