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Hall--Littlewood expansions of chromatic quasisymmetric polynomials using linked rook placements

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Pith's one-line read Linked rook placements give a Hall–Littlewood expansion of chromatic quasisymmetric functions.

desk verdict The main Hall-Littlewood expansion is false for γ=N^2E^2; the linked rook model is promising but the paper's central theorems fail in the simplest case. read the letter →

arxiv 2506.23082 v1 pith:FJNHJPKY submitted 2025-06-29 math.CO

Jang Soo Kim , Seung Jin Lee , Meesue Yoo This is my paper · ORCID
classification math.CO MSC 05A1505A30
keywords chromaticquasisymmetricfunctionsHall–LittlewoodpolynomialslinkedrookplacementsDyckpathsunicellularLLTmodularlawfree-cellstatisticq-rook
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to prove a new expansion: for a Dyck path γ, the chromatic quasisymmetric function $X_\gamma(x;q)$ of the associated natural unit interval order is a sum over partitions μ of Hall–Littlewood polynomials $P_\mu(x;q)$, with coefficients built from linked rook placements of type μ and a free-cell statistic. This is a $q$-analogue of the classical expansion of $X_\gamma$ into monomial symmetric functions via ordinary rook placements, and it refines $q$-rook polynomials by recording the type of each placement. The proof shows that the proposed right-hand side obeys the modular law and a multiplicativity property, so a known uniqueness criterion for functions on Dyck paths forces it to equal $X_\gamma$. A corollary applies the plethystic relation between $X_\gamma$ and unicellular LLT polynomials to write unicellular LLT polynomials in modified transformed Hall–Littlewood bases; that second statement is more delicate and should be checked path by path.

What carries the argument

The central object is the linked rook placement: a set of chains of non-attacking rooks on the Ferrers board under a Dyck path, where consecutive rooks $(i,j)$ and $(j,k)$ in a chain share the middle coordinate. The extended linked rook placement adds one diagonal cell per chain and an extra row, and the rank of an extended rook in its column or row decides whether a cell above the path is free; a free cell is encoded as an fc-pair. The coefficient $r_{\gamma,\mu}(q)=\sum_{P}q^{\operatorname{fc}_\gamma(P)}$ over placements of type μ is what carries the argument, since it supplies the Hall–Littlewood coefficients up to the factor $q^{\operatorname{area}(\gamma)-n(\mu)}\prod_i[m_i(\mu)]_q!$. The proof mechanism is the modular-law criterion for functions on Dyck paths, together with explicit bijections that keep track of free-cell counts when rows or columns are exchanged, and a multiplicativity bijection for concatenating a Dyck path with $N^kE^k$.

What would settle it

Compare both sides of Theorem 3.6 coefficient by coefficient in the Hall–Littlewood basis for a small non-complete Dyck path such as $\gamma=N^3E^3$; any mismatch localizes an error in the modular-law bijections. For Theorem 3.8, evaluate the stated expansion at $\gamma=N^2E^2$ and check whether the identity holds; this single path settles the validity of the plethystic application.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 3.6, is that $X_\gamma(x;q)=\sum_{\mu\vdash n} q^{\operatorname{area}(\gamma)-n(\mu)}r_{\gamma,\mu}(q)\left(\prod_i [m_i(\mu)]_q!\right)P_\mu(x;q)$, where $r_{\gamma,\mu}(q)$ is the generating function over linked rook placements of type μ weighted by the number of free cells. The linked rook placement is a chain of rooks $(a_1,b_1),\ldots,(a_\ell,b_\ell)$ with $b_i=a_{i+1}$, and the free cells are defined through an extended placement that adds diagonal cells and an extra row; the statistics are tracked by ranks and fc-pairs. The paper proves the identity by verifying the modular law, multiplicativity under concatenation with $N^kE^k$, and the complete graph value $[n]_q!e_n$, and then invoking the modular-law criterion to conclude the right-hand side coincides with $X_\gamma$. The $q=1$ specialization recovers the classical rook expansion, and the coefficients refine $q$-rook polynomials. A further theorem derives unicellular LLT expansions from the plethystic relation; the paper presents these as consequences of the same coefficients.

Load-bearing premise

The proof of the main expansion hangs on the modular-law and multiplicativity bijections for linked rook placements; separately, the LLT corollary hangs on the plethystic relation $X_\gamma(x;q)=(q-1)^{-n}\mathrm{LLT}_\gamma[(q-1)X;q]$, whose application appears to break down for the Dyck path $\gamma=N^2E^2$.

Editorial extensions

If this is right

  • At $q=1$ the formula recovers the ordinary rook-counting monomial expansion of $X_\gamma$, so the main theorem is a genuine $q$-analogue of that classical identity.
  • Summing the linked-rook coefficients by length gives $R_{n-k}(\gamma;q)=\sum_{\ell(\mu)=k} r_{\gamma,\mu}(q)$, so the new coefficients refine $q$-rook polynomials by partition type.
  • The principal specialization $X_\gamma(1,q,\ldots,q^{\alpha-1};q)=q^{\operatorname{area}(\gamma)}\prod_i[\alpha-a_i(\gamma)]_q$ is recovered, connecting the coefficients to $q$-rook and $q$-hit polynomial data.
  • Where the plethystic relation is valid, the same linked-rook coefficients give a combinatorial description of unicellular LLT polynomials in the modified transformed Hall–Littlewood basis.
  • The modular-law proof provides an independent construction of a function on Dyck paths determined by complete-graph values, so the same coefficients can be tested against other expansions of chromatic quasisymmetric functions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the main expansion is correct for every Dyck path, the coefficients $r_{\gamma,\mu}(q)$ form a partition-type refinement of the $q$-rook polynomial, and the principal-specialization comparison suggests a natural refinement of $q$-hit polynomials that the paper does not construct.
  • The apparent failure of the LLT corollary on $\gamma=N^2E^2$ indicates that the plethystic relation, or its application after substituting $X/(q-1)$, may require correction; because the main theorem is proved independently through the modular law, the two results stand or fall separately.
  • A direct bijection between linked rook placements and the pairs of P-tableaux and semistandard Young tableaux that appear in the Schur expansion would give a transparent proof of the Hall–Littlewood coefficients and would test whether the free-cell statistic has a simpler equivalent description.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper introduces linked rook placements and defines a generating function r_{\gamma,\mu}(q) over them. It claims in Theorem 3.6 that the chromatic quasisymmetric function X_\gamma(x;q) of a natural unit interval order has a Hall--Littlewood expansion whose coefficients are q^{area(\gamma)-n(\mu)} r_{\gamma,\mu}(q) \prod_i [m_i(\mu)]_q!. The proof follows the Abreu--Nigro modular-law criterion, with the bulk of the work in Lemmas 4.6 and 4.7, which are verified by diagrams. Applying the Carlsson--Mellit relation (Proposition 2.5), the paper then derives Theorem 3.8, a modified Hall--Littlewood expansion for unicellular LLT polynomials. The paper also gives a refinement of q-rook polynomials and poses several open problems.

Significance. If correct, the linked-rook description would be a natural q-analogue of the Stanley--Stembridge rook formula and would provide a new combinatorial model for Hall--Littlewood coefficients of chromatic quasisymmetric functions. The modular-law strategy is reasonable and the paper is self-contained. However, the manuscript contains internal counterexamples to its advertised theorems: Theorem 3.8 fails for a two-vertex path under the paper's own definitions, and a direct n=2 computation indicates that Theorem 3.6 itself is false as stated. The proofs are not machine-checked, and the main combinatorial lemmas rely on unformalized figure-based case checks. The claimed results therefore cannot be accepted in their current form.

major comments (3)
  1. [Section 4, Proposition 4.3 and Theorem 3.6, Eq. (3.1)] Proposition 4.3 is false for n=2 under the definitions as written. For \gamma=N^2E^2, the singleton linked rook ((1,2)) is a valid linked rook placement; its unique extension is the extended linked rook (1,1),(1,2),(2,2),(2,3), which covers both diagonal cells, so its type is (2). There are no cells above \gamma, hence no free cells, and r_{\gamma,(2)}=1. This directly contradicts the assertion in the proof of Proposition 4.3 that LRP(\gamma,\mu)=\emptyset unless \mu=(1^n). Substituting into (3.1), with area(N^2E^2)=1, n((2))=0, n((1,1))=1, P_{(2)}=h_2-q e_2 and [2]_q! P_{(1,1)}=(1+q)e_2, gives q(h_2-q e_2)+(1+q)e_2, whereas Definition 2.1 gives X_\gamma=(1+q)e_2. Thus the main theorem fails for n=2 unless a different notion of type or extension is intended, which is not stated.
  2. [Section 2.6 and Section 3, Proposition 2.5 and Theorem 3.8] Theorem 3.8 is false for \gamma=N^2E^2. Definition 2.4 gives LLT_\gamma=h_2+q e_2 for this path, while Definition 2.1 gives X_\gamma=(1+q)e_2. Since both are homogeneous of degree 2, the plethysm in (2.6) would force X_\gamma=h_2+q e_2, a contradiction. Independently, the proof of Theorem 3.8 inserts the factor (1-q)^{n-\ell(\mu)} outside the sum over \mu without justification: using [m]_q!=(q;q)_m/(1-q)^m and homogeneity of P_\mu, the factors (q-1)^n and (q-1)^{-n} cancel globally, and no \mu-dependent power of (1-q) remains. The statement is also ill-formed because the exponent n-\ell(\mu) depends on the summation variable \mu. For n=2, the first identity predicts h_2 and the second predicts e_2, both disagreeing with Definition 2.4.
  3. [Section 4.1, Lemmas 4.6 and 4.7] The proofs of Lemmas 4.6 and 4.7 are not formal. In each case, after defining a bijection, the equality of free-cell generating functions is asserted by reference to Figures 8--17 (e.g., "by Figure 8, we have...") without a written argument that cells away from the displayed local configuration are unaffected or that the displayed ranks determine the counts. Since these two lemmas constitute the entire proof of Proposition 4.1, the modular law for Y_\gamma is not established as a verifiable proof. This is a rigor gap independent of the counterexamples above.
minor comments (3)
  1. [Section 2.3, Eq. (2.5)] Equation (2.5) is written with the product \prod_{i=1}^{\ell(\mu)}(q;q)_{m_i(\mu)}; this indexing is inconsistent with the factors needed, since for \mu=(3,1) it would omit the multiplicity m_3. The product should be over all i with m_i(\mu)>0 or over all i\ge 1. The identity should also be checked for consistency with (2.4), since for \mu=(2) the two sides appear to differ.
  2. [Section 3, Theorem 3.8] The first displayed equation in Theorem 3.8 places (1-q)^{n-\ell(\mu)} outside the summation over \mu, but \ell(\mu) is not defined until the summation variable is specified; the factor must either be moved inside the sum or the theorem restated. The same issue appears in the proof.
  3. [Throughout] There are several smaller presentation issues: the abstract and introduction refer to Theorem 1.4 while the body states the result as Theorem 3.8; the phrase "row counted from the left" in the definition of a_i(\gamma) is ambiguous; and Definition 3.2 allows cells (i,j) with i\le j but then places u_{2\ell} in the extended row, whose coordinates have j=n+1, so the formal scope of j should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.6 is proved from independent external inputs, and the LLT corollary is a derived consequence rather than an assumed conclusion.

full rationale

The central derivation is self-contained against independent external results. Theorem 3.6 defines the linked-rook coefficient r_{γ,μ}(q) purely combinatorially (Definition 3.5), then proves the expansion by verifying the Abreu–Nigro modular-law criterion (Theorem 2.3): Proposition 4.1 checks the modular law for Y_γ, Proposition 4.2 checks multiplicativity via the Pieri rule and the bijection in Lemma 4.12, and Proposition 4.3 checks the complete-graph case using only P_{(1^n)}=e_n and the definition of LRP. No parameter is fitted to X_γ, and no occurrence of the target expansion is used as an input. The Hall–Littlewood identity (2.5) and the Pieri rule are standard external facts. The LLT corollary (Theorem 3.8) is obtained by applying the cited Carlsson–Mellit relation (2.6) to Theorem 3.6; it is a transformation of the same expansion, not a circular reuse. The only self-citation visible is [HOY25] in Section 5.2, a concluding remark on q-rook refinements that is not load-bearing. The noted failure of Theorem 3.8 for γ=N^2E^2 is a substantive correctness problem in the plethystic computation or the quoted relation, but it is not an instance of a conclusion reducing to its premises by construction, so it does not change the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces linked rook placements, extended linked rooks, and free cells as new combinatorial objects; these are definitions rather than unexplained physical entities. The central claim rests on several external theorems (modular law, CM relation), one of which appears misapplied.

assumptions (5)
  • domain assumption Abreu-Nigro theorem: a function on Dyck paths satisfying the modular law is determined by its values on disjoint unions of complete graphs (Theorem 2.3)
    Used to conclude Y_γ = X_γ from the modular law, multiplicativity, and complete graph values.
  • domain assumption The chromatic quasisymmetric function X_γ satisfies the modular law (from AN21)
    Needed for the uniqueness criterion to apply.
  • domain assumption Carlsson-Mellit relation X_γ(x;q) = (q-1)^{-n} LLT_γ[(q-1)X;q] (Proposition 2.5)
    Used to derive the LLT expansion; this relation appears to be misapplied or misstated, yielding a false Theorem 3.8.
  • standard math Pieri rule for Hall-Littlewood polynomials
    Used in the proof of multiplicativity.
  • standard math Kostka-Foulkes expansion (2.5)
    Used to relate products (q;q)_{m_i} P_μ to modified Schur functions.

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Cite this review

Pith. "Pith review of Hall--Littlewood expansions of chromatic quasisymmetric polynomials using linked rook placements." pith.science (2026). https://pith.science/paper/FJNHJPKY

@misc{pith2026250623082,
  author       = {Pith},
  title        = {Pith review of: Hall--Littlewood expansions of chromatic quasisymmetric polynomials using linked rook placements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FJNHJPKY}},
  note         = {Machine review of arXiv:2506.23082}
}
read the original abstract

In this work, we obtain a Hall--Littlewood expansion of the chromatic quasisymmetric function arising from a natural unit interval order and describe the coefficients in terms of linked rook placements. Applying the Carlsson--Mellit relation between chromatic quasisymmetric functions and unicellular LLT polynomials, we also obtain a combinatorial description for the coefficients of the unicellular LLT polynomials expanded in terms of the modified transformed Hall--Littlewood polynomials.

Figures

Figures reproduced from arXiv: 2506.23082 by the authors.

Figure 1
Figure 1. For a Dyck path with column heights m = (2, 3, 5, 6, 6, 6), the corre￾sponding natural unit interval order P(m) and the incomparability graph inc(P) are drawn below. We define the modified Hall–Littlewood polynomials Pµ(x; q) by replacing the modified Schur function by the Schur function in the expansion (2.5): Hµ(x; q) = X λ⊢|µ| Kλµ(q)sλ(x). Lastly, we define the modified transformed Hall–Littlewood polynomials H˜ … view at source ↗
Figure 2
Figure 2. Modular triples of type (1, i) (top) and type (2, i) (bottom). 2.6. LLT polynomials. LLT polynomials are a family of symmetric functions introduced by Lascoux, Leclerc, and Thibon in [LLT97], which naturally arise in the description of the power￾sum plethysm operators on symmetric functions. The original definition of LLT polynomials uses cospin statistic of ribbon tableaux, but Haiman and Bylund found a consistent … view at source ↗
Figure 3
Figure 3. The diagram on the left shows a linked rook ((1, 4),(4, 6),(6, 9)) of the Dyck path γ ∈ Dn corresponding to the partition λ = (7, 5, 5, 4, 2, 2, 1) ⊆ δ9. The diagram on the right shows a linked rook placement {L1, L2, L3} on γ, where L1 = ((1, 4),(4, 6),(6, 9)), L2 = ((2, 5),(5, 7)), and L3 = ((3, 8)). 3.1. Linked rook placements. Let n be a fixed positive integer. Definition 3.1. Let γ ∈ Dn with the associated part… view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: The diagram on the left shows an extended linked rook of length 1, and the diagram on the right shows an extended linked rook of length 4 whose underlying linked rook is the left diagram in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The diagram on the left shows a linked rook placement P of the Dyck path γ ∈ D9 corresponding to λ = (6, 6, 4, 4, 2, 1). The right diagram shows its corresponding extended linked rook placement ext(P), where the ranks of the topmost extended rook in each column and the…
Figure 6
Figure 6. Figure 6: The two cases of fc-pairs, where a is the rank of the topmost extended rook in column i and b is the rank of the leftmost extended rook in row j. 1 q q 2 q [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Linked rook placements of type (3, 2) and their extended linked rook placements. Free cells are denoted by white bullets. Note also that, by definition, the free cells are in bijection with the fc-pairs. Hence, fcγ(P) is equal to the number of fc-pairs of P. Definition…
Figure 8
Figure 8. Figure 8: If a > b, then fcγ0 (P) = fcγ1 (P) = fcγ0 (ϕ(P)) − 1. Circles indicate free cells. a b a − 1 b − 1 i i + 1 j fcγ0 (P) a b a − 1 b − 1 i i + 1 j fcγ1 (P) b a b − 1 a − 1 i i + 1 j fcγ0 (ϕ(P)) [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: If a < b, then fcγ0 (P) − 1 = fcγ1 (P) = fcγ0 (ϕ(P)). Circles indicate free cells. a a a − 1 a − 1 i i + 1 j fcγ0 (P) a a a − 1 a − 1 i i + 1 j fcγ1 (P) a a a − 1 a − 1 i i + 1 j fcγ0 (ϕ(P)) [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: If a = b, then fcγ0 (P) = fcγ1 (P) = fcγ0 (ϕ(P)) − 1. Circles indicate free cells. • If a < b, by [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: If a ≥ b1, b2, we have fcγ0 (P) = fcγ1 (P) = fcγ2 (θ(P)). a b1 b2 b2 − 1 b1 − 1 fcγ0 (P) i i + 1 j a b1 b2 b2 − 1 b1 − 1 fcγ1 (P) i i + 1 j a b1 b2 b2 − 1 b1 − 1 fcγ2 (θ(P)) i i + 1 j [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: If a < b1, b2, we have fcγ0 (P) − 1 = fcγ1 (P) = fcγ2 (θ(P)) + 1. a b1 b2 b2 − 1 b1 − 1 fcγ0 (P) i i + 1 j a b1 b2 b2 − 1 b1 − 1 fcγ1 (P) i i + 1 j a b2 b1 b2 − 1 b1 − 1 fcγ2 (θ(P)) i i + 1 j [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: If b1 ≤ a < b2, we have fcγ0 (P) − 1 = fcγ1 (P) = fcγ2 (θ(P)) + 1. It is easy checked that θ is an involution. Hence, to prove (4.8), it suffices to show that for each P ∈ B0(γ2), we have q fcγ1 (P ) + q fcγ1 (P )+1 = q fcγ0 (P ) + q fcγ2 (θ(P ))+1 . (4.11) We verify …
Figure 14
Figure 14. Figure 14: If b2 ≤ a < b1, we have fcγ0 (P) = fcγ1 (P) = fcγ2 (θ(P)). Similarly, we prove the modular law for the other modular triples. Lemma 4.7. Let (γ0, γ1, γ2) ∈ M1,i n and µ ⊢ n. Then (4.4) holds. Proof. Let j = γ2(i)+ 1. Note that LRP(γ2, µ) ⊆ LRP(γ1, µ) ⊆ LRP(γ0, µ). For…
Figure 15
Figure 15. Figure 15: If a > b, then fcγ0 (P) − 1 = fcγ1 (P) = fcγ0 (φ(P)). b + 1 a + 1 b a i j j − 1 fcγ0 (P) b + 1 a + 1 b a i j j − 1 fcγ1 (P) a + 1 b + 1 b a i j j − 1 fcγ0 (φ(P)) [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: If a < b, then fcγ0 (P) = fcγ1 (P) = fcγ0 (φ(P)) − 1. a + 1 a + 1 a a i j j − 1 fcγ0 (P) a + 1 a + 1 a a i j j − 1 fcγ1 (P) a + 1 a + 1 a a i j j − 1 fcγ0 (φ(P)) [PITH_FULL_IMAGE:figures/full_fig_p017_16.png]
Figure 17
Figure 17. Figure 17: If a = b, then fcγ0 (P) = fcγ1 (P) = fcγ0 (φ(P)) − 1. • If a ≥ b1, b2, we have fcγ0 (P) = fcγ1 (P) and fcγ2 (ϑ(P)) + 1 = fcγ1 (P) + 1. • If a < b1, b2, we have fcγ0 (P) = fcγ1 (P) + 1 and fcγ2 (ϑ(P)) + 1 = fcγ1 (P). • If b1 < a ≤ b2, we have fcγ0 (P) = fcγ1 (P) + 1 an…
Figure 18
Figure 18. Figure 18: The left diagram shows ext(P) for a linked rook placement P. Then ϕ(P) = (Q, w), where ext(Q) is shown on the right and w = 0112221302. Therefore, by (4.22) and (4.25), we always have |{(ia, jb) : 0 ≤ i < j, ia ∈ B, jb ∈ Mµ \ B}| = X 0≤i<j (ν ′ i+1 − µ ′ i+1)(µ ′ j − …

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  1. A tableaux formula for $q$-rook numbers

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    A weighted sum over standard Young tableaux computes Garsia-Remmel q-rook numbers, and this reconnects them to LLT function coefficients.

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