On cyclic Schur-positive sets of permutation

Jonathan Bloom; Sergi Elizalde; Yuval Roichman

REVIEW 3 major objections 4 minor 16 references

On cyclic Schur-positive sets of permutation

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A set of permutations is cyclic Schur-positive exactly when it is Schur-positive and invariant under a cyclic-descent rotation.

desk verdict A solid paper whose characterization theorem for cyclic Schur-positivity is clean and useful, but whose proof of the key equidistribution result depends on a block-reordering step in Lemma 4.8 that deserves a close independent check. read the letter →

arxiv 1908.07920 v1 pith:KHZ4UPWO submitted 2019-08-21 math.CO

Jonathan Bloom , Sergi Elizalde , Yuval Roichman This is my paper · ORCID

classification math.CO MSC 05E0505E1005A0505A19

keywords Schur-positivesetscyclicdescentsdescentstandardYoungtableauxarcpermutationsinverseclassesequidistributionrotationclosures

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 introduces cyclic Schur-positivity (cSp), a cyclic-descent analogue of the classical notion of Schur-positivity for sets of permutations, and proves a complete characterization: a subset of $S_n$ is cSp exactly when it is Schur-positive and $\mathrm{cDes}$-invariant, meaning it admits a bijection that rotates every cyclic descent set. This reduces a statement about quasisymmetric functions to a checkable combinatorial symmetry. The characterization immediately makes every horizontally rotated Schur-positive set cSp, and the paper goes on to prove that vertically rotated inverse descent classes are cSp as well. The proof route yields a new equidistribution result for cyclic descent sets on the two rotation directions, and it settles two open conjectures about Schur-positivity of inverse descent classes and of arc permutations.

What carries the argument

The engine is the cyclic descent set $\mathrm{cDes}$, defined for permutations by wrapping the descent set around the cycle, and extended to standard Young tableaux of skew shapes that are not connected ribbons. Theorem 3.4 is the main characterization. For the equidistribution result, the machinery is a sequence of descent-preserving block reorderings: shuffles of increasing sequences are encoded as binary words, reversed by the descent-preserving map $f$ from Lemma 4.3, and split through the shuffle decomposition to reduce vertical rotations to horizontal ones. Lemma 4.8 reassembles the blocks into a single shuffle while preserving descent sets, and inclusion-exclusion converts this into the theorem on inverse descent classes.

What would settle it

Test Lemma 4.8 directly on small cases: fix a composition $\gamma$, choose a split into weak compositions $\alpha$ and $\beta$, and compare the multiset of descent sets on the two sets displayed in (4.5) and (4.6); any mismatch would disprove the claimed $\mathrm{cDes}$-preserving bijection and with it Theorem 4.1.

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Extended reading notes

Core claim

The central claim is Theorem 3.4: A subset $A$ of $S_n$ is cyclic Schur-positive if and only if it is Schur-positive and $\mathrm{cDes}$-invariant, i.e. there is a bijection $\psi\colon A\to A$ with $\mathrm{cDes}(\psi\pi)=1+\mathrm{cDes}(\pi)$ for every $\pi$. Because horizontal rotation gives such a bijection automatically, every Schur-positive set that is invariant under horizontal rotation is cSp; conversely the theorem shows every cSp set is Schur-positive. Building on this, the paper proves that the vertical rotations $C_n D_{n-1,J}^{-1}$ of inverse descent classes are cSp, that the union of all rotations of a fixed inverse cyclic descent set is cSp, and that arc permutations are cSp, with an explicit $\mathrm{cDes}$-preserving bijection from arc permutations to a disjoint union of SYT of shapes $(n-k-1,1^k)\oplus(1)$.

Load-bearing premise

The proof of the equidistribution theorem rests on a technical claim that certain block rearrangements of permutations preserve descent sets even when distinguished entries are moved; if even one such rearrangement fails, the inclusion-exclusion argument and the resolved conjectures collapse.

Editorial extensions

If this is right

Every Schur-positive set in $S_{n-1}$ has a horizontal rotation closure in $S_n$ that is cyclic Schur-positive (Theorem 3.11).
For every $J\subseteq[n-2]$, the vertically rotated inverse descent class $C_n D_{n-1,J}^{-1}$ is cyclic Schur-positive, hence Schur-positive (Theorem 4.9).
For every nonempty proper $J\subseteq[n]$, the set of permutations whose inverse has cyclic descent set $i+J$ for some $i$ is cyclic Schur-positive, resolving the conjecture from [2] (Corollary 4.10).
The distribution of $\mathrm{Des}$ over $C_n D_{n-1,J}^{-1}$ equals that over $D_{n-1,J}^{-1} C_n$, resolving Conjecture 10.2 of [9] (Corollary 4.12).
Arc permutations form a cyclic Schur-positive set, with a bijective proof matching their $\mathrm{cDes}$-distribution to SYT of the near-hook strip shapes (Corollary 6.3 and Theorem 6.11).

Reading between the lines

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

Theorem 3.4 gives a practical certificate: to prove a $\mathrm{cDes}$-invariant set Schur-positive, it is enough to exhibit a single rotation bijection, and conversely any $\mathrm{cDes}$-invariant Schur-positive set decomposes explicitly into skew SYT.
The inclusion-exclusion proof of Theorem 4.1 suggests that the full equidistribution could be made bijective by composing the Section 5 and Section 6 maps; a direct map would likely extend to broader unions of grid classes.
Because arc permutations are vertical rotations of left-unimodal permutations, the same $\mathrm{cDes}$-preserving bijection may transfer to other rotation-closed permutation classes, generating new Schur-positive sets beyond the ones listed.
The explicit SYT targets in Theorem 6.11 have a strip shape, so the same cyclic-descent generating functions may be pluggable into known cyclic sieving results for promotion.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my read on Bloom–Elizalde–Roichman's cyclic Schur-positivity paper. The core new idea is Theorem 3.4: a set A ⊆ S_n is cyclic Schur-positive iff it is Schur-positive and cDes-invariant. That is a genuine characterization, and it makes the notion much more tractable. The applications flow naturally from it: Theorem 3.11 shows horizontal rotations of Schur-positive sets are cSp, and the equidistribution result Theorem 4.1 gives Schur-positivity for vertical rotations of inverse descent classes, settling Conjecture 10.2 of [9] and, via Corollary 4.10, a conjecture from [2]. The exposition is clear and credits prior work appropriately; the self-citations are to established theorems, not to the conjectures being solved.

The soft spot is Lemma 4.8, which is the engine behind Theorem 4.1. The proof says 'using Lemma 4.3 repeatedly' to reorder blocks, but Lemma 4.3 is proved only for shuffles of two increasing sequences. The extension to t ≥ 2 is described but not proved as a formal lemma, and the starred-block conditions, which track the position of the largest letter, are not explicitly checked in the reordering. If that reordering introduces or removes descents at block boundaries, the cDes multisets on the two sides of Lemma 4.8 could differ, and Theorem 4.1 would fail. I don't see a counterexample, and the explicit bijections in Sections 5 and 6 for the cases |J|=1 and J=[i] lend credibility to the general claim. But the gap is real enough that a referee should ask for a detailed proof of the block-reordering step, or at least a small-case verification. Everything else in the paper holds up: the characterization theorem is proved cleanly, the inclusion-exclusion argument from Lemma 4.2 to Theorem 4.1 is sound once Lemma 4.2 is granted, and the arc permutation bijection in Section 6 is explicit and checkable.

Who is this for? Algebraic combinatorialists working on Schur-positivity, descent sets, and cyclic descents. It deserves a serious referee rather than a desk reject; the main theorems are new, the writing is honest about where proofs are non-bijective, and the resolved conjectures are real. The likely fix is a more careful proof of Lemma 4.8, possibly with a computer verification for small n. I would send it to peer review with a note to focus attention on that lemma.

Referee Report

3 major / 4 minor

Summary. The paper introduces a notion of cyclic Schur-positivity (cSp) for sets of permutations, extending classical Schur-positivity via cyclic descent sets on permutations and on standard Young tableaux of skew shapes. The central result, Theorem 3.4, characterizes cSp sets as exactly those subsets of S_n that are Schur-positive and cDes-invariant. Building on this, the authors prove that horizontal rotation closures of Schur-positive sets are cSp (Theorem 3.11) and that, for every inverse descent class, the cyclic descent set distribution is the same on vertical and horizontal rotations (Theorem 4.1). Consequences include Theorem 4.9, asserting that vertically rotated inverse descent classes are cSp; Corollary 4.10, resolving a conjecture from [2]; and Corollary 4.12, resolving Conjecture 10.2 of [9]. The paper also supplies explicit bijections for the singleton descent case (Section 5) and for arc permutations (Section 6), the latter giving a bijective proof of the cSp property for arc permutations.

Significance. If the main results stand, the paper makes a substantial contribution to the study of Schur-positive permutation sets. Theorem 3.4 is an elegant and useful characterization, and Theorem 4.1, together with its corollaries, resolves open conjectures and produces new families of Schur-positive sets. The explicit bijections in Sections 5 and 6 are a definite strength, as is the paper's clear organization and the largely self-contained proof of Theorem 3.4. The main concern is that the proof of the technical Lemma 4.8, on which Theorem 4.1 depends, is not fully justified in the present version. The central idea is plausible and the rest of the argument is coherent, but the missing details are load-bearing rather than cosmetic.

major comments (3)

[Section 4.1, Lemma 4.8, Eq. (4.7)] The assertion that 'Using Lemma 4.3 repeatedly' one can reorder the blocks in S(β_i,...,β_t*,β_1,...,β_{i-1}) to obtain the reversed order in Eq. (4.7) is not justified by Lemma 4.3 as stated. Lemma 4.3 is proved only for shuffles of two increasing sequences, while the extension to t blocks is merely described, with no proof that the composed maps preserve descents when some blocks are starred, i.e., when the maximum of a block is fixed at the end. Since Lemma 4.8 is the engine for Lemma 4.2 and hence for Theorem 4.1, this gap needs to be filled. A proof by induction on t, or an explicit verification for t=3 and all small n, would settle the point.
[Section 4.1, Lemma 4.8, displayed equivalences after Eq. (4.7)] The two equivalences for i=1 and for i≠1 are asserted without specifying the bijections. In particular, the step 'the equivalence follows by interchanging the order of the β's just as we did above with the α's' changes the location of the starred block, and the claim that this preserves descent sets is not immediate. Because the starred block determines the fixed last letter, the cyclic descent set is sensitive to the star's position. The authors should either write down the bijections explicitly or prove the equality of the cDes multisets by a direct computation. This is a load-bearing step in the proof of Lemma 4.2.
[Section 4.1, proof of Lemma 4.2] The final step of the proof of Lemma 4.2 applies Lemma 4.3 to the set S(γ_t−1,...,(γ_1+1)*) to obtain S(γ_1,...,γ_t*). This again relies on the unproved multi-block extension of Lemma 4.3 with a starred block. The reader needs a precise statement of the extended lemma, including the behavior of the star under the reversal, and a proof that the map is cDes-preserving. Without this, the proof of Lemma 4.2, and therefore of Theorem 4.1, is incomplete.

minor comments (4)

[Title and abstract] The title reads 'sets of permutation'; it should be 'sets of permutations'. The typeset title also contains an odd spacing in 'PERMUT A TION'.
[Section 6.2, Example 6.10] The final expression 'ψσ = ¯σ c^j = 6728194356' has ten digits and cannot be a permutation in S_9; it should presumably be '672819435' or another nine-letter permutation. Please correct this typo.
[Section 4.1, proof of Lemma 4.3] In the decomposition of binary words as w = 1^{i_1}2^{j_1} 2|1 1^{i_2}2^{j_2} 2|1 ... , the notation '2|1' is easy to misread. Please clarify explicitly that the bars mark the descent positions 21 that are kept fixed.
[Section 5, Figure 3] The label 'parallelshort' in Figure 3 appears to be a typo; it is unclear what is meant. The figure would benefit from a clearer caption explaining the two cases.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main characterization and equidistribution results are derived from independent published foundations, not from their own conclusions.

full rationale

The paper's central claim, Theorem 3.4 (cSp iff Schur-positive and cDes-invariant), is not a restatement of Definition 1.4. The forward direction derives Schur-positivity by specializing x_n=1 and applying Gessel's identity, and cDes-invariance follows from the equivariance axiom for the cyclic descent extension on the RHS tableaux. The converse uses the descent-set generating function together with Lemma 2.5, a general fiber-counting fact for cyclic descent extensions, and the known existence of cyclic descent extensions for non-ribbon skew shapes ([3, Thm. 1.1]); the hook-to-strip conversion is an algebraic manipulation plus a telescoping sum, not an import of the conclusion. Theorem 4.1 is proved from Lemma 4.2, whose proof is a chain of explicit block-reordering maps based on the self-contained folklore Lemma 4.3; the reduction of cDes-preservation to Des-preservation by right rotation is valid when the largest letter is fixed at the end, since then cDes = Des ⊔ {n}. The applications (Theorems 4.9, Corollaries 4.10 and 4.12) are consequences of Theorem 4.1 and prior published Schur-positivity results, not presuppositions of those results. Several cited results come from the same authors' earlier papers ([3], [9], [10]), but they are published theorems with independent proofs and do not assume the present target statements; this is self-citation rather than circularity. The one delicate step, Lemma 4.8's 'using Lemma 4.3 repeatedly' to reorder more than two blocks, is a possible correctness gap, not a reduction of the conclusion to its own input; it affects validity, not circularity. No step in the derivation is equivalent by definition to a fitted parameter or to a renamed known result.

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

No fitted constants appear in the paper. The proofs use standard symmetric function identities and cited cyclic descent results, all established elsewhere. The only new object is the combinatorial definition of cyclic Schur-positivity, which is a definition rather than an invented entity with an independent falsifiability burden.

assumptions (5)

domain assumption Existence of cyclic descent extension for every skew shape that is not a connected ribbon (Theorem 2.2, from [3]).
Definition 1.4 and Lemma 3.9 require cyclic descent sets on all non-connected-ribbon skew shapes; this is cited from Adin-Reiner-Roichman, not reproved.
domain assumption Fiber sizes of a cyclic descent extension are determined by ordinary descent fiber sizes via inclusion-exclusion (Lemma 2.5, from [3]).
This is the mechanism in Theorem 3.4 that passes from cDes-invariance to a matching with tableaux polynomials.
domain assumption Horizontal rotation closure of a Schur-positive set is Schur-positive ([10, Theorem 1.1]).
Used in Theorem 3.11 and Theorem 4.9; a prior result by two of the present authors with a proof in the cited publication.
domain assumption Inverse descent classes are Schur-positive ([11]).
Used in Theorem 4.9 to pass from horizontal to vertical inverse descent classes.
standard math Gessel's identity expressing Schur functions as sums of fundamental quasisymmetric functions, together with the Littlewood-Richardson rule.
Used in the converse direction of Theorem 3.4 to show that cSp implies Schur positivity.

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Pith. "Pith review of On cyclic Schur-positive sets of permutation." pith.science (2026). https://pith.science/paper/KHZ4UPWO

@misc{pith2026190807920,
  author       = {Pith},
  title        = {Pith review of: On cyclic Schur-positive sets of permutation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KHZ4UPWO}},
  note         = {Machine review of arXiv:1908.07920}
}

read the original abstract

We introduce a notion of {\em cyclic Schur-positivity} for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux that preserves the cyclic descent set. Cyclic Schur-positive sets of permutations are always Schur-positive, but the converse does not hold, as exemplified by inverse descent classes, Knuth classes and conjugacy classes. In this paper we show that certain classes of permutations invariant under either horizontal or vertical rotation are cyclic Schur-positive. The proof unveils a new equidistribution phenomenon of descent sets on permutations, provides affirmative solutions to conjectures from [9] and [2], and yields new examples of Schur-positive sets.

Figures

Figures reproduced from arXiv: 1908.07920 by the authors.

**Figure 1.** A SYT of shape (5, 4, 2)/(1, 1). Theorem 1.2 ([4, Prop. 9.1]). A subset A ⊆ Sn is Schur-positive if and only if there exist nonnegative integers (mλ)λ⊢n such that X π∈A x Des(π) = X λ⊢n mλ X T ∈SYT(λ) x Des(T) . This characterization of Schur-positive sets of permutations is useful because it does not require computing quasisymmetric functions, but rather finding a Des-preserving bijection from permutations to SYT … view at source ↗

**Figure 2.** A visualization for t = 3 of ρ ⊛ σ as given by Definition 4.6. We place (maintaining their positions) the values in σ corresponding to ιαi in the box labeled αi , and similarly for ρ. An immediate consequence of this construction is that, for every fixed k ≤ n, we have S(γ1, . . . , γt) = G (α,β)∈Cγ k,n−k S(α1, . . . , αt) ⊛ S(β1, . . . , βt), where C γ k,n−k denotes the set of pairs of weak compositions α of k and … view at source ↗

**Figure 3.** A schematic description of the map Ψ. Permutations π ∈ CnD −1 n−1,{j} (obtained by vertically rotating D −1 n−1,{j} ) with π(k) = n fall into two cases. Here α1 + β1 = j, α2 + β2 = n − j, and β1 + β2 = k. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: A drawing of the permutation 4532617 on the grid for left-unimodal permutations. Being vertical rotations of left-unimodal permutations, arc permutations are precisely those that can be drawn on one of the grids in [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: Grids for arc permutations. Observation 6.6. A permutation π ∈ Sn with π(j) = n is an arc permutation if and only if one of the following holds: 18 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: The cDes-preserving bijection φ : Ln−1Cn −→ An. φ 7→ [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 7.** Figure 7: Example: φ(3 2 11 12 13 1 14 8 7 9 6 5 4 10) = 3 2 4 5 6 1 14 13 7 12 11 10 8 9. 9. Since Des(ˆσ) = {1, 5, 7}, we place the decreasing sequence 3, 2, 1 in positions 1 + Des(ˆσ) = {2, 6, 8}, obtaining ∗3∗∗∗2∗1∗, and an increasing sequence in the remaining positions, obt…

**Figure 8.** Figure 8: An example of the cDes-preserving bijections C6D −1 5,[2] φ−1 → D −1 5,[2]C6 f→ SYT((3, 1 2 ) ⊕ (1)). References [1] R. M. Adin, S. Elizalde and Y. Roichman, Cyclic descents for near-hook and two-row shapes, European J. Combin. 79 (2019), 152–178. [2] R. M. Adin, I. M…

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Works this paper leans on

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [9]

Elizalde and Y

S. Elizalde and Y. Roichman, Schur-positive sets of permutations via products of grid cl asses, J. Algebraic Combin. 45 (2017), 363–405

work page 2017

[2] [2]

R. M. Adin, I. M. Gessel, V. Reiner and Y. Roichman, Cyclic quasisymmetric functions , arXiv:1811.05440

work page Pith review arXiv

[3] [1]

R. M. Adin, S. Elizalde and Y. Roichman, Cyclic descents for near-hook and two-row shapes , European J. Com- bin. 79 (2019), 152–178

work page 2019

[4] [3]

R. M. Adin, V. Reiner and Y. Roichman, On cyclic descents for tableaux , Int. Math. Res. Not. IMRN (2018) rny280, http://dx.doi.org/10.1093/imrn/rny280. 21

work page doi:10.1093/imrn/rny280 2018

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M Adin and Y

R. M Adin and Y. Roichman, Matrices, characters and descents , Linear Algebra Appl. 469 (2015), 381–418

work page 2015

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work page 2009