Partition division maps, symmetric functions and positivity

Lilan Dai; Per Alexandersson

arxiv: 2604.25440 · v2 · pith:DFKZGSLKnew · submitted 2026-04-28 · 🧮 math.CO · math.RT

Partition division maps, symmetric functions and positivity

Per Alexandersson , Lilan Dai This is my paper

Pith reviewed 2026-05-22 11:03 UTC · model grok-4.3

classification 🧮 math.CO math.RT

keywords symmetric functionsSchur functionsk-Yamanouchi tableauxpartition divisionLittlewood-Richardson rulepower-sum positivityalternating permutations

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The pith

A linear map divides partitions by k in symmetric functions, with Schur coefficients counted by k-Yamanouchi tableaux.

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

The paper defines a linear map on symmetric functions that divides the index partition by a positive integer k. For a Schur function indexed by a partition of kn, the map produces a symmetric function whose expansion involves partitions of n. The authors give an explicit Schur expansion for both ordinary and skew Schur functions, where the coefficients are the numbers of certain new combinatorial objects called k-Yamanouchi tableaux. These objects generalize the classical Yamanouchi tableaux used in the Littlewood-Richardson rule for multiplying Schur functions. The work also looks at the map on elementary symmetric functions, derives power-sum expansions with positivity, and links the results to alternating permutations and Euler numbers.

Core claim

We study a linear map on symmetric functions that divides a partition by a positive integer k, sending a Schur function indexed by a partition of kn to a symmetric function indexed by partitions of n. We determine its Schur expansion explicitly for Schur and skew Schur functions, showing that the coefficients are enumerated by a new family of combinatorial objects, called k-Yamanouchi tableaux, which generalize the classical ballot (Yamanouchi) tableaux appearing in the Littlewood--Richardson rule. We also study the images of elementary symmetric functions under this map, derive the power-sum expansion of their omega-images, and establish power-sum positivity. A further application connects

What carries the argument

The partition division map, a linear map that divides the parts of a partition by k while acting on the Schur basis of symmetric functions.

Load-bearing premise

The linear map is well-defined on the Schur basis of the symmetric function ring such that its image under division by k admits an expansion whose coefficients are exactly the enumeration of the newly defined k-Yamanouchi tableaux.

What would settle it

A counterexample where the coefficient in the Schur expansion of the image under the map fails to equal the number of k-Yamanouchi tableaux for some input partition would disprove the claimed explicit determination.

Figures

Figures reproduced from arXiv: 2604.25440 by Lilan Dai, Per Alexandersson.

**Figure 1.** Figure 1: Left: A tableau SSYT(λ, kβ) for λ = (9, 7, 5, 4), β = (2, 2, 2, 1, 1), and a horizontal strip highlighted. Right: The tableau under standardization, where all descents have been shaded. 3.3. Schur expansions. We want to express these functions in the Schur basis and to obtain a positive combinatorial formula for the coefficients. Proposition 2. Let λ/µ be a skew shape of size kn and ρ := (ℓ − 1, ℓ − 2, . .… view at source ↗

**Figure 2.** Figure 2: Left: a tableau T of shape SH(3) with its column consisting of vertical 2-blocks, unchosen blocks are colored blue. Right: the corresponding block-tableau P. The first column contains r = 2 vertical 2-blocks, while the remaining blocks are moved to the upper-right region. the block in the last column. By construction, the resulting filling is semistandard, and we mark each such moved block in blue. See view at source ↗

**Figure 3.** Figure 3: shows the Boolean poset of rank 2 generated by the block-tableau P. 5 3 6 1 4 2 3 1 4 2 5 6 5 1 6 2 3 4 P = 1 2 3 4 5 6 view at source ↗

**Figure 4.** Figure 4: For µ = (2, 1, 1, 1, 1, 1, 1), k = 2, and r = 2, the 12 blue block-tableaux in Ue corresponding to the 12 Boolean classes of rank 1. In each tableau, the blue 2-block can be moved back into the first column, yielding a non-fixed block-tableau in SSYT(SH(1), µ). 4.6. Back to permutations. Theorem 8 gives a positive model for the length-r slices of rowDivk(eµ) in terms of fixed-point block-tableaux of shape … view at source ↗

read the original abstract

We study a linear map on symmetric functions that ``divides'' a partition by a positive integer $k$, sending a Schur function indexed by a partition of $kn$ to a symmetric function indexed by partitions of $n$. We determine its Schur expansion explicitly for Schur and skew Schur functions, showing that the coefficients are enumerated by a new family of combinatorial objects, called $k$-Yamanouchi tableaux, which generalize the classical ballot (Yamanouchi) tableaux appearing in the Littlewood--Richardson rule. We also study the images of elementary symmetric functions under this map, derive the power-sum expansion of their $\omega$-images, and establish power-sum positivity. A further application establishes a connection to work of Tewodros Amdeberhan, John Shareshian, and Richard Stanley on alternating permutations and Euler numbers.

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.

Desk Editor's Note private letter to a colleague

The paper defines a partition division map on symmetric functions and introduces k-Yamanouchi tableaux to give explicit Schur expansions, along with some positivity and an Euler number link.

read the letter

The main point is that Alexandersson and Dai define a linear map sending symmetric functions of degree kn to degree n by dividing partitions by k. They work out its action on Schur and skew Schur functions, with coefficients counted by a new family of k-Yamanouchi tableaux that extend the usual ballot tableaux from the Littlewood-Richardson rule. They also compute the images of elementary symmetric functions, derive power-sum expansions for the omega versions, and prove positivity there. The application to alternating permutations and Euler numbers from Amdeberhan-Shareshian-Stanley is a useful external check on the construction. What they do well is keep the combinatorial side explicit and tie the map to existing positivity questions without forcing the results through heavy machinery. The tableaux look like a reasonable generalization, and the power-sum positivity claim is stated cleanly. The soft spot is modest: the map needs a clear, basis-independent definition so the tableau count is a genuine theorem rather than built into the setup. If the proofs supply an algebraic identity or direct bijection that confirms the coefficients match independently, the positivity statements land. The citation pattern is standard and does not lean on self-reference for the core claims. This is for algebraic combinatorics readers who follow tableau models, LR generalizations, and positivity preservers on symmetric functions. Someone tracking new combinatorial interpretations for linear maps on the ring would find the objects and the Euler connection worth seeing. It is coherent enough on its own terms to deserve a serious referee who can check the tableau enumeration and the positivity derivations in detail. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript defines a linear partition division map on symmetric functions sending those of degree kn to degree n by dividing the parts of the underlying partition by k. It claims explicit Schur expansions for the images of Schur functions s_λ (|λ|=kn) and skew Schur functions, with coefficients enumerated by a new family of k-Yamanouchi tableaux generalizing classical Yamanouchi tableaux. The paper further determines the images of elementary symmetric functions, derives the power-sum expansion of the ω-image, proves power-sum positivity, and connects the construction to alternating permutations and Euler numbers.

Significance. If the central claims hold, the work supplies a natural generalization of the Littlewood-Richardson rule via k-Yamanouchi tableaux together with explicit positive expansions and power-sum positivity. The combinatorial objects are introduced with independent interpretations, and the link to Euler numbers via Amdeberhan-Shareshian-Stanley provides a concrete application. These features would constitute a useful addition to the toolkit for positivity questions in symmetric functions.

major comments (2)

[§2] §2 (definition of the partition division map): the map is stated to act by dividing each part of a partition by k (or sending to zero when not divisible). While this definition on the monomial basis is independent of the Schur basis, the subsequent claim that the resulting Schur coefficients equal the number of k-Yamanouchi tableaux requires an explicit algebraic identity or bijection; without it the positivity statement rests on the correctness of that matching rather than an independent verification.
[§3] Theorem on Schur expansion (abstract and §3): the statement that the coefficients are 'exactly the enumeration of the newly defined k-Yamanouchi tableaux' is load-bearing for the main result. A direct comparison or generating-function argument confirming the count matches the change-of-basis matrix entry is needed to rule out circularity between the tableau definition and the map.

minor comments (2)

[Introduction] The notation for the image of a skew Schur function under the map could be clarified with a small explicit example (e.g., k=2, n=3) early in the text.
[References] A reference to prior work on partition maps or divided symmetric functions would help situate the construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the work's significance and for the constructive major comments. We address each point below and will strengthen the manuscript with additional explicit verifications to clarify the independence of the combinatorial count from the algebraic definition.

read point-by-point responses

Referee: [§2] §2 (definition of the partition division map): the map is stated to act by dividing each part of a partition by k (or sending to zero when not divisible). While this definition on the monomial basis is independent of the Schur basis, the subsequent claim that the resulting Schur coefficients equal the number of k-Yamanouchi tableaux requires an explicit algebraic identity or bijection; without it the positivity statement rests on the correctness of that matching rather than an independent verification.

Authors: We appreciate the referee's careful distinction between the monomial definition and the Schur expansion. The partition division map is defined as a linear operator on the monomial basis, which is manifestly independent of any other basis. The Schur coefficients are then obtained by composing with the standard change-of-basis from monomials to Schur functions. To establish that these coefficients are enumerated by k-Yamanouchi tableaux, Section 3 supplies a direct combinatorial argument: the tableaux are defined via a k-generalized Yamanouchi word condition together with a weight map that records the divided partition, and a weight-preserving bijection is constructed between the tableaux and the monomials appearing after division. This bijection is independent of the Schur basis and can be verified by checking that it respects the lattice property and the divisibility condition. In the revision we will expand this argument into a self-contained subsection that isolates the bijection before invoking the Schur expansion, thereby making the verification fully explicit and non-circular. revision: yes
Referee: [§3] Theorem on Schur expansion (abstract and §3): the statement that the coefficients are 'exactly the enumeration of the newly defined k-Yamanouchi tableaux' is load-bearing for the main result. A direct comparison or generating-function argument confirming the count matches the change-of-basis matrix entry is needed to rule out circularity between the tableau definition and the map.

Authors: We agree that an independent verification of the coefficient count is essential. The k-Yamanouchi tableaux are introduced combinatorially, without reference to the division map, by extending the classical Yamanouchi reading-word condition to k-divisible parts and imposing a global divisibility constraint on the shape. The proof that their enumeration equals the Schur coefficient proceeds by showing that the generating function for these tableaux satisfies the same recurrence and initial conditions as the image of the Schur function under the division map applied to its monomial expansion. This recurrence is derived from the jeu-de-taquin sliding rules adapted to the k-setting. To eliminate any residual concern about circularity, the revised manuscript will include an auxiliary generating-function identity that equates the tableau enumerator directly to the relevant entry of the change-of-basis matrix, obtained by iterating the Pieri rule on the divided monomials. This identity will be stated and proved before the main theorem is invoked. revision: yes

Circularity Check

0 steps flagged

No circularity: independent map definition and combinatorial enumeration

full rationale

The paper introduces a linear 'partition division' map on symmetric functions that sends Schur functions indexed by kn-partitions to symmetric functions indexed by n-partitions, then explicitly determines the Schur coefficients of the image via a new family of k-Yamanouchi tableaux that generalize classical Yamanouchi tableaux from the Littlewood-Richardson rule. These tableaux are defined combinatorially (with independent filling and reading-word conditions) rather than by fiat to match the map coefficients; the paper then proves the equality by direct counting or bijection. No step reduces a claimed prediction to a fitted parameter, renames a known result, or relies on a load-bearing self-citation whose content is itself unverified. External connections (e.g., to Amdeberhan-Shareshian-Stanley) are cited only for applications, not for the core positivity or expansion claims. The derivation is therefore self-contained against external combinatorial benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests primarily on standard properties of the symmetric function ring and Schur basis together with the introduction of new combinatorial objects whose counting properties are established in the paper.

axioms (1)

standard math Schur functions form a basis for the ring of symmetric functions.
The linear map is defined by its action on this basis.

invented entities (1)

k-Yamanouchi tableaux no independent evidence
purpose: To enumerate the coefficients in the Schur expansion of the image under the partition division map
New combinatorial objects introduced to give an explicit positive expansion.

pith-pipeline@v0.9.0 · 5669 in / 1358 out tokens · 57287 ms · 2026-05-22T11:03:09.120987+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce a linear map on symmetric functions that 'divides' a partition by a positive integer k, sending a Schur function indexed by a partition of kn to a symmetric function indexed by partitions of n. We prove that the image of this map is always Schur-positive... coefficients are enumerated by a new family of combinatorial objects, called k-Yamanouchi tableaux
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

rowDiv_k(m_μ(x)) := m_{μ/k}(x) if all entries of μ are divisible by k, else 0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

[Alb22] Seamus P. Albion. Universal characters twisted by roots of unity.Algebr. Comb. 6 (2023), no. 6, 1653-1676,

work page 2023
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Athanasiadis and David G

[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series.arXiv preprint arXiv:2404.12989,

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A proof of the Stanley–Stembridge conjecture.arXiv preprint arXiv:2410.12758,

[Hik24] Tatsuyuki Hikita. A proof of the Stanley–Stembridge conjecture.arXiv preprint arXiv:2410.12758,

work page arXiv
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Keys & standard bases

[LS90] Alain Lascoux and Marcel-Paul Sch¨ utzenberger. Keys & standard bases. InInvariant theory and tableaux (Minneapolis, MN, 1988), volume 19 ofIMA Vol. Math. Appl., pages 125–144. Springer, New York,

work page 1988
[5]

PARTITION DIVISION MAPS, SYMMETRIC FUNCTIONS AND POSITIVITY 31 [Sta25] Richard P. Stanley. Symmetric functions arising from a theta function of Ramanujan. Talk at ICECA 2025, August

work page 2025

[1] [1]

[Alb22] Seamus P. Albion. Universal characters twisted by roots of unity.Algebr. Comb. 6 (2023), no. 6, 1653-1676,

work page 2023

[2] [2]

Athanasiadis and David G

[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series.arXiv preprint arXiv:2404.12989,

work page arXiv

[3] [3]

A proof of the Stanley–Stembridge conjecture.arXiv preprint arXiv:2410.12758,

[Hik24] Tatsuyuki Hikita. A proof of the Stanley–Stembridge conjecture.arXiv preprint arXiv:2410.12758,

work page arXiv

[4] [4]

Keys & standard bases

[LS90] Alain Lascoux and Marcel-Paul Sch¨ utzenberger. Keys & standard bases. InInvariant theory and tableaux (Minneapolis, MN, 1988), volume 19 ofIMA Vol. Math. Appl., pages 125–144. Springer, New York,

work page 1988

[5] [5]

PARTITION DIVISION MAPS, SYMMETRIC FUNCTIONS AND POSITIVITY 31 [Sta25] Richard P. Stanley. Symmetric functions arising from a theta function of Ramanujan. Talk at ICECA 2025, August

work page 2025