On 102-avoiding inversion sequences

Heesung Shin; JiSun Huh; Sangwook Kim; Seunghyun Seo

REVIEW 2 major objections 4 minor 12 references

On 102-avoiding inversion sequences

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The authors construct an explicit bijection between 102-avoiding inversion sequences and peak-free, valley-free 2-Schröder paths ending in a diagonal step.

desk verdict Genuine progress on the Seo–Shin bijection problem, but the inverse of ψ has an unproved uniqueness claim and the headline statistic is off by one. read the letter →

arxiv 2506.02985 v1 pith:GDXHPZ7I submitted 2025-06-03 math.CO

JiSun Huh , Sangwook Kim , Seunghyun Seo , Heesung Shin This is my paper

classification math.CO MSC 05A0505A1505A19

keywords 102-avoidinginversionsequences2-SchröderpathsUVDlabeledF-pathspatternavoidancerankstatisticbijectionrefinedenumeration

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 resolves an open problem in bijective combinatorics by constructing an explicit bijection between the set of inversion sequences of length n that avoid the pattern 102 and the set of 2-Schröder paths of semilength n that have no peaks, no valleys, and end with a diagonal step. To build the bijection, the authors introduce two intermediate families, UVD paths and labeled F-paths, and establish bijections among all four families. A natural statistic, called rank on inversion sequences and defined analogously on the other families, is shown to be preserved by the bijections, and this leads to a closed-form enumeration of 102-avoiding inversion sequences by rank. The paper also gives rank-refined counts for sequences avoiding 102 together with each of eight other length-3 patterns.

What carries the argument

The key object is the labeled F-path: a lattice path from the origin that never goes below the line $y=x$, whose up-steps carry a label $(a;1)$ and whose down-steps carry a list $(a;b_1,\dots,b_k)$ of nonpositive integers summing to the step's vertical displacement. The machinery is a pair of recursive bijections centered on this object. Reading a labeled F-path from left to right, the map $\phi$ inserts copies of a new maximum entry into a working inversion sequence at positions prescribed by the step's label, while the map $\psi$ splices blocks of the form $u\sigma u$ and vertical steps into a UVD path at returns determined by the labels; both maps preserve a height statistic $\mathrm{ht}$ on the F-path, which becomes the rank statistic on inversion sequences and the valley count $\mathrm{vox}$ on UVD paths. The inverse constructions are recursive: from a 102-avoiding inversion sequence or a UVD path one locates the inserted entries or spliced blocks, removes them, and recurses on the shorter object. This recursive bijectivity is what carries the whole argument, turning an equality of generating functions into an explicit one-to-one correspondence.

What would settle it

For semilengths $n\le 6$, enumerate all UVD paths and, for each path whose second-to-last step is vertical, list every decomposition of the form $S=\alpha u\sigma_1u\cdots u\sigma_k u\,\tau v^k\beta$ satisfying the conditions of Theorem 3.2 Case 2; if any path admits two distinct decompositions, the inverse construction would produce two distinct labeled F-paths with the same image under $\psi$, breaking the claimed bijection.

Watch

Extended reading notes

Core claim

The central claim is that the composed map $\phi\circ\psi^{-1}\circ M$ is a bijection from $\mathrm{SP}_n$, the set of 2-Schröder paths of semilength $n$ having neither peaks $\mathrm{NE}$ nor valleys $\mathrm{EN}$ and ending with a diagonal step, to $\mathrm{IS}_n(102)$, the set of 102-avoiding inversion sequences, and that for every path $P$ the identity $\mathrm{block}(P)=\mathrm{rank}(\phi\circ\psi^{-1}(P))$ holds. The map $M$ is a linear transformation sending the path steps $N,E,H$ to the UVD steps $u,v,d$. The map $\psi$ takes a labeled F-path of semilength $n$ to a UVD path of semilength $n+1$ by cutting the path at chosen return points and inserting blocks $u\sigma u$ and vertical steps according to the labels, and the map $\phi$ takes the same labeled F-path to a 102-avoiding inversion sequence by inserting copies of a new maximum entry at positions determined by the labels. Because both $\phi$ and $\psi$ are proved bijective by recursive inverse constructions, the composition is the desired explicit bijection.

Load-bearing premise

The inverse step of the path construction assumes that every lattice path of the relevant kind whose second-to-last step is vertical has exactly one way to be cut into the blocks the paper specifies, and this uniqueness is asserted but not justified.

Editorial extensions

If this is right

Every statistic on 102-avoiding inversion sequences now transfers through the bijection to a statistic on 2-Schröder paths of the restricted type, so enumeration problems can be attacked on whichever side is more convenient.
The identity $\mathrm{block}(P)=\mathrm{rank}(\phi\circ\psi^{-1}(P))$ gives a new geometric interpretation of the rank statistic as the number of returns of the corresponding path.
The rank-refined formula in Theorem 4.1 provides a closed count of 102-avoiding inversion sequences of length $n$ with a fixed rank $t$.
For the eight second patterns $\tau$ analyzed, the paper supplies explicit rank-refined formulas, including Fibonacci-number and tiling-based expressions for the cases $\tau=011$ and $\tau=012$.

Reading between the lines

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

The same template—labeled F-paths mediating between an inversion-sequence family and a path family—may extend to other pattern pairs whose generating functions coincide, such as families of inversion sequences avoiding other length-3 patterns that are known to be counted by the same algebraic equations.
Because the bijection preserves the rank/return statistic, the rank of a 102-avoiding inversion sequence may correspond to a known geometric invariant of 2-Schröder paths, such as area or bounce; testing this on small cases could yield a new equidistribution result.
The open doubly-avoiding cases $\tau\in\{000,010,100\}$ lie exactly where the maximum entry of the inversion sequence no longer follows the predictable insertion pattern used by $\phi$, so extending the method likely requires a different labeling rule rather than a simple tweak.
The recursive description of $\phi$ and $\psi$ suggests an immediate algorithmic implementation that converts between an inversion sequence and its corresponding 2-Schröder path in time proportional to the length of the sequence.

Share X Bluesky LinkedIn Reddit HN

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

The paper gives an explicit bijection between 102-avoiding inversion sequences and the restricted 2-Schröder paths, a problem Seo and Shin left open. The route through UVD paths and labeled F-paths is new, and the rank-refined counts look right. But the proof has a real gap in the inverse direction of ψ, and the paper's stated equality block(P)=rank(...) is off by one.

What is actually new: the explicit bijection itself, the intermediate objects, and the refined rank enumerations. The forward constructions are recursive and well-motivated, and the small cases check out. The equinumerosity was already known via generating functions, so the contribution is the bijection and the statistic transfer.

The soft spot is Theorem 3.2, Case 2. The inverse of ψ asserts a unique decomposition S=αuσ₁u...uσ_k u τ v^k β with β of the form v...vd, but no proof is given. The terminal run of v's in S does not mark the split between the inserted v^k and β's own initial v's, so uniqueness is not obvious. I could not find a counterexample, and the constraints may force the split, but the paper needs to show that. Secondary, the paper says block(P)=rank(φψ^{-1}(P)); tracing the definitions gives block(P)=rank(e)+1 (e.g., for n=1, SP_1={NH} has block 1 while rank((0))=0). That is a simple indexing slip but should be fixed. There are also several 'straightforward to verify' pattern-avoidance claims in Sections 3 and 5 that should be written out, especially in the doubly-avoiding cases.

This deserves a serious referee. The gap is fillable, not a demonstrated failure. The off-by-one is cosmetic. If the uniqueness proof is supplied, the paper is a solid contribution. I would send it to peer review and ask the authors to expand the proof of Theorem 3.2 and correct the statistic equality. For a reading group, it is a good paper to work through.

Referee Report

2 major / 4 minor

Summary. The paper constructs an explicit bijection between inversion sequences of length n avoiding the pattern 102 and 2-Schröder paths of semilength n having neither peaks NE nor valleys EN and ending with a diagonal step, thereby resolving an open problem posed by Seo and Shin. The bijection is built through two intermediate families: UVD paths and labeled F-paths. The authors define recursive maps φ from labeled F-paths to 102-avoiding inversion sequences and ψ from labeled F-paths to UVD paths, together with the linear map M sending 2-Schröder paths to UVD paths, and they claim statistic preservation rank = ht = vox = block − 1. The same framework is used to give refined enumerations of 102-avoiding inversion sequences by rank and to treat eight doubly-avoiding cases. Small cases agree with the known Schröder counts.

Significance. If the constructions are correct, the paper resolves the explicitly posed Seo–Shin bijection problem by an elementary, self-contained recursive bijection, and it introduces two intermediate objects that are likely to be useful for further refined enumeration. The rank statistic ties the main bijection to previously studied statistics on (102,101)-avoiding sequences, and the doubly-avoiding results extend several known Wilf-equivalence refinements. The proofs are constructive and do not appear to reduce the central result to the equinumerosity results of [SS23], so the claimed resolution is not circular. However, a key injectivity/uniqueness step in the inverse construction of ψ is asserted rather than proved, and a displayed statistic identity is off by one; these issues must be repaired before the central bijection can be regarded as fully established.

major comments (2)

[Section 3.2, Theorem 3.2, Case (2)] The proof of the inverse direction of ψ asserts, without proof, that every UVD path S whose second-to-last step is vertical admits a unique decomposition S = α uσ₁u ⋯ uσ_k u τ v^k β, with αβ, σ₁, …, σ_k, τ UVD paths and β of the form v⋯vd. This uniqueness is load-bearing: ψ is the middle map in the composed bijection φ∘ψ^{-1}∘M, so any ambiguity or missing existence case would break the resolution of the Seo–Shin problem. The concern is real because the forward construction inserts v^k immediately before β, and β itself may begin with vertical steps; in fact the paper's own construction allows β = d (for example in the worked path S(18) of Section 3.2), so the terminal run v^{k+r}d carries no marker separating v^k from β. The paper states 'there exists a unique positive integer k' but supplies no argument for existence or uniqueness. A rigorous proof of the decomposition is required; the current text also should clarify that β may be d, i.e. that the phrase 'of the form v⋯vd' means r ≥ 0.
[Section 3, after Theorem 3.2] The stated equality block(P) = rank(φ∘ψ^{-1}(P)) is off by one. From the paper's own identities vox(S) = block(P) − 1 (Section 2.2), ht(Q) = vox(ψ(Q)) (Theorem 3.2), and rank(e) = ht(φ^{-1}(e)) (Theorem 3.1), one obtains rank = vox = block − 1, hence block(P) = rank + 1, not block(P) = rank. The n = 1 case already contradicts the displayed claim: SP₁ = {NH} has block = 1, while the unique 102-avoiding inversion sequence (0) has rank = 0. This should be corrected throughout, or the definitions of rank/block should be consistently adjusted; as written, the advertised statistic preservation is false.

minor comments (4)

[Proposition 5.15] The displayed definition c(j,k) appears to contain a typo: the binomial coefficient is written as binom(2j+k, n), but it should be binom(2j+k, j), as the final equality c(j,k) = [x^j] C^k indicates.
[Section 3.2, Case (1)] The treatment of empty β in the decomposition α u β d would benefit from an explicit statement of block(∅) and vox(∅), since the text uses vox(∅) = −1 but still writes identities such as vox(α) = h − a and vox(β) = a − 1 without spelling out how the empty case is handled.
[Section 4, equations for D and D_t] The derivation of D_t = D_0^{t+1} is stated without a decomposition argument; adding one sentence explaining how a UVD path splits at its t+1 returns would improve readability.
[Figure 4] Figure 4 is visually dense and the v-runs and returns are hard to distinguish at printed scale; annotating the first few returns or using a step-style drawing would help.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the three bijections are constructed from scratch; the only self-citation (Proposition 5.1) is a peripheral restatement, and the flagged uniqueness assertion in Theorem 3.2 is an omitted proof, not a circular reduction.

full rationale

The central claim is that the composition φ∘ψ⁻¹∘M is a bijection from the restricted 2-Schröder paths to 102-avoiding inversion sequences. The map M is explicitly defined by a linear transformation, and φ and ψ are defined recursively with inverse reconstructions argued from the pattern-avoidance and path conditions. No fitted parameter is introduced, and no target quantity is used as an input to the construction. The paper's own identities vox(S) = block(P) − 1 and ht(Q) = rank(φ(Q)) directly track statistics, and the final equality block(P) = rank(φψ⁻¹(P)) is presented as a consequence of these identities rather than as an assumption; note, however, that the paper's own equations imply block(P) = rank(e) + 1, so the headline equality is off by one. Equinumerosity from [SS23] is cited as background and motivation, not as a load-bearing step. Proposition 5.1 is explicitly a restatement of [HKSS24], but it appears in the peripheral doubly-avoiding section and is not used to prove the main bijection. The only substantive flagged issue is in Theorem 3.2, Case 2 of the proof of ψ's bijectivity, where the paper asserts 'there exists a unique positive integer k' decomposing S as α uσ₁u uσ₂u ... uσₖu τ vᵏβ, with no proof of existence or uniqueness; in particular, the split of the terminal v-run between vᵏ and β is not marked in the word itself. This is a missing proof and a potential correctness gap, not a circularity by construction or a fitted-input reduction, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No free parameters are fitted. The derivations rely on standard tools plus cited enumeration theorems from [YL21], [CMSW16], [Ges16] and standard Catalan/Dyck facts. The two new path families are introduced as tools and are validated by bijections rather than assumed.

assumptions (6)

domain assumption Yan and Lin [YL21, Theorem 3.1]: |IS_n(102,011)| = |IS_n(102,012)| = F_{2n-1}
Used in Sections 5.2 and 5.3 to supply total counts from which rank-refined counts are derived.
domain assumption Yan and Lin [YL21, Theorem 4.10]: |IS_n(102,120)| = 1 + Σ_{i=1}^{n-1} binom(2i, i-1)
Used in the proof of Lemma 5.8(2) and Proposition 5.9.
domain assumption Corteel et al. [CMSW16] characterize 001-avoiding inversion sequences as e1<...<ek ≥ e_{k+1}≥...≥en
Basis for Proposition 5.2.
standard math Lemma 5.6: the number of Dyck paths of semilength n ending with ud^k equals k/n binom(2n-k-1, n-1); cited as well-known and from OEIS A33184
Used in Propositions 5.7, 5.10, and 5.12.
standard math Lagrange inversion (Gessel [Ges16])
Used in Section 4 to extract coefficients of E(y).
standard math The generating function identity for Catalan numbers C(x)=1+xC(x)^2, with [x^j]C^k = k/(2j+k) binom(2j+k, j)
Used in the generating function manipulations of Sections 5.7 and 5.8.

invented entities (2)

UVD paths independent evidence
purpose: Intermediate combinatorial family in the bijection between restricted 2-Schröder paths and labeled F-paths
Defined in Section 2.2; the linear map M from SP_n and the recursive bijection ψ give independent checks, and their enumeration matches the large Schröder numbers.
Labeled F-paths independent evidence
purpose: Intermediate family carrying insertion labels that encode multiple insertions of the maximum in inversion sequences
Defined in Section 2.3; bijections φ and ψ and the worked example provide falsifiable content, and sizes match |IS_{n+1}(102)|.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On 102-avoiding inversion sequences." pith.science (2026). https://pith.science/paper/GDXHPZ7I

@misc{pith2026250602985,
  author       = {Pith},
  title        = {Pith review of: On 102-avoiding inversion sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GDXHPZ7I}},
  note         = {Machine review of arXiv:2506.02985}
}

abstract

In this article, we provide a bijection between the set of inversion sequences avoiding the pattern 102 and the set of 2-Schr\"{o}der paths having neither peaks nor valleys and ending with a diagonal step. To achieve this, we introduce two intermediate objects, called UVD paths and labeled $F$-paths, and establish bijections among all four families. For each of these combinatorial objects, we define a natural statistic and enumerate the corresponding structures with respect to this statistic. In addition, we study inversion sequences avoiding 102 and another pattern of length 3, providing refined enumerations according to the same statistic.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

12 extracted references · 11 canonical work pages

[1]

Martinez, Carla D

Sylvie Corteel, Megan A. Martinez, Carla D. Savage, and Michael Weselcouch. Patterns in inversion sequences I . Discrete Math. Theor. Comput. Sci. , 18(2):Paper No. 2, 21, 2016

work page 2016
[2]

Ira M. Gessel. Lagrange inversion. J. Combin. Theory Ser. A , 144:212--249, 2016

work page 2016
[3]

Bijections on pattern avoiding inversion sequences and related objects

JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin. Bijections on pattern avoiding inversion sequences and related objects. Adv. in Appl. Math. , 161:Paper No. 102771, 40, 2024

work page 2024
[4]

Length-four pattern avoidance in inversion sequences

Letong Hong and Rupert Li. Length-four pattern avoidance in inversion sequences. Electron. J. Combin. , 29(4):Paper No. 4.37, 15, 2022

work page 2022
[5]

Patterns in permutations and words

Sergey Kitaev. Patterns in permutations and words . Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg, 2011. With a foreword by Jeffrey B. Remmel

work page 2011
[6]

An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences

Ilias Kotsireas, Toufik Mansour, and G\" o khan Y ld r m. An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences. J. Symbolic Comput. , 120:Paper No. 102231, 18, 2024

work page 2024
[7]

Pattern avoidance in inversion sequences

Toufik Mansour and Mark Shattuck. Pattern avoidance in inversion sequences. Pure Math. Appl. (PU.M.A.) , 25(2):157--176, 2015

work page 2015
[8]

Patterns in inversion sequences II : inversion sequences avoiding triples of relations

Megan Martinez and Carla Savage. Patterns in inversion sequences II : inversion sequences avoiding triples of relations. J. Integer Seq. , 21(2):Art. 18.2.2, 44, 2018

work page 2018

Show all 12 references

[9]

The O n- L ine E ncyclopedia of I nteger S equences, 2025

OEIS Foundation Inc. The O n- L ine E ncyclopedia of I nteger S equences, 2025. Published electronically at http://oeis.org

2025
[10]

On D elannoy paths without peaks and valleys

Seunghyun Seo and Heesung Shin. On D elannoy paths without peaks and valleys. Discrete Math. , 346(7):Paper No. 113399, 12, 2023

2023
[11]

Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701v3, 2024

Benjamin Testart. Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701v3, 2024

2024
[12]

Inversion sequences avoiding pairs of patterns

Chunyan Yan and Zhicong Lin. Inversion sequences avoiding pairs of patterns. Discrete Math. Theor. Comput. Sci. , 22(1):Paper No. 23, 35, [2020--2021]

2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Martinez, Carla D

Sylvie Corteel, Megan A. Martinez, Carla D. Savage, and Michael Weselcouch. Patterns in inversion sequences I . Discrete Math. Theor. Comput. Sci. , 18(2):Paper No. 2, 21, 2016

work page 2016

[2] [2]

Ira M. Gessel. Lagrange inversion. J. Combin. Theory Ser. A , 144:212--249, 2016

work page 2016

[3] [3]

Bijections on pattern avoiding inversion sequences and related objects

JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin. Bijections on pattern avoiding inversion sequences and related objects. Adv. in Appl. Math. , 161:Paper No. 102771, 40, 2024

work page 2024

[4] [4]

Length-four pattern avoidance in inversion sequences

Letong Hong and Rupert Li. Length-four pattern avoidance in inversion sequences. Electron. J. Combin. , 29(4):Paper No. 4.37, 15, 2022

work page 2022

[5] [5]

Patterns in permutations and words

Sergey Kitaev. Patterns in permutations and words . Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg, 2011. With a foreword by Jeffrey B. Remmel

work page 2011

[6] [6]

An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences

Ilias Kotsireas, Toufik Mansour, and G\" o khan Y ld r m. An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences. J. Symbolic Comput. , 120:Paper No. 102231, 18, 2024

work page 2024

[7] [7]

Pattern avoidance in inversion sequences

Toufik Mansour and Mark Shattuck. Pattern avoidance in inversion sequences. Pure Math. Appl. (PU.M.A.) , 25(2):157--176, 2015

work page 2015

[8] [8]

Patterns in inversion sequences II : inversion sequences avoiding triples of relations

Megan Martinez and Carla Savage. Patterns in inversion sequences II : inversion sequences avoiding triples of relations. J. Integer Seq. , 21(2):Art. 18.2.2, 44, 2018

work page 2018