Improvements on Permutation Reconstruction from Minors

Wenjie Zhong; Xiande Zhang; Yiming Ma

REVIEW 3 major objections 5 minor 25 references

Improvements on Permutation Reconstruction from Minors

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that the minimum deck size for reconstructing any permutation of length $n$ is between $\exp(\Omega(\sqrt{\ln n}))$ and $O(\sqrt{n\ln n})$, and improves the bounds on $N_d$ from quadratic to $d+O(\sqrt{d\ln d})$.

desk verdict The lower bound and the bivariate basis lemma are solid, but the upper bound rests on a false orthogonality claim and needs a major fix before it can be trusted. read the letter →

arxiv 2411.12718 v1 pith:QRKYSGCJ submitted 2024-11-19 math.CO math.OC

Yiming Ma , Wenjie Zhong , Xiande Zhang This is my paper · ORCID

classification math.COmath.OC MSC 05A05

keywords permutationreconstructionk-minorsk-deckHahnpolynomialssequencefromminorsasymptoticbounds

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 asks how many order-preserving subsequences ($k$-minors) of a permutation must be collected before the whole permutation is uniquely determined. It proves that this minimum $k$, written $s(n)$, is at least $\exp(\Omega(\sqrt{\ln n}))$ and at most $2\lceil\sqrt{(n-2)\ln(n-3)}\rceil+2$, so the reconstruction threshold is sublinear in $n$. The same result transfers to the older parameter $N_d$, the length beyond which every permutation is determined by its $(n-d)$-minors: the new bounds $d+\exp(\Omega(\sqrt{\ln d}))

What carries the argument

The load-bearing machinery is a polynomial basis for reconstruction. For a permutation $z\in S_k$, the indicator $z_{ij}$ records whether entry $i$ is less than entry $j$; summing these indicators over all minors gives linear statistics $S_{ij}(x)$ of the $k$-deck. The paper proves that the bivariate polynomials $f_{ij}(x,y)=\binom{x-1}{i-1}\binom{y-x-1}{j-i-1}\binom{n-y}{k-j}$, for $1\le i<j\le k$, form a basis of all bivariate polynomials of degree at most $k-2$, and that two permutations with equal $k$-decks force $\sum_{x<y}\delta_{xy}\phi(x,y)=0$ for every such polynomial $\phi$, where $\delta_{xy}$ compares the two permutations' inversion patterns. This is converted into a univariate certificate: if some polynomial $\phi$ of degree at most $k-2$ satisfies $\phi(-1)>(n-2)\sum_{x=0}^{n-3}|\phi(x)|$, then every permutation is $k$-reconstructible. The paper constructs $\phi=p_k^2$ using Hahn polynomials $T_i$ orthogonal on $\{0,\ldots,n\}$, with $p_k=\sum T_i(-1)/d_i\,T_i(x)$, and computes $\Delta(p_k^2)=(n+1-F_k)F_k$, where $F_k=\sum T_i(-1)^2/d_i$. The imported estimate $F_k>e^{(k+1)k/(n+1)}-1+1/(n+1)$ then yields $F_k>n+1$ when $k\ge\sqrt{(n+1)\ln n}$, making $\Delta$ negative and completing the certificate.

What would settle it

Compute $F_k=\sum_{i=0}^k T_i(-1)^2/d_i$ exactly for the Hahn polynomials used in the paper, with $k=\lceil\sqrt{(n+1)\ln n}\rceil$ for increasing $n$, and check whether $F_k>n+1$; a single $n$ where the inequality $F_k>e^{(k+1)k/(n+1)}-1+1/(n+1)$ or $F_k>n+1$ fails would break the upper-bound proof. Independently, a brute-force search could falsify the stated $s(n)$ upper bound by finding two distinct permutations of length $n$ with the same $k$-deck for $k=2\lceil\sqrt{(n-2)\ln(n-3)}\rceil+2$; for $n=11$ this is already within the paper's own algorithmic range.

Watch

Extended reading notes

Core claim

The paper's central claim is that the least $k$ for which every permutation of length $n$ has a distinct $k$-deck satisfies $s(n)=\exp(\Omega(\sqrt{\ln n}))$ and $s(n)=O(\sqrt{n\ln n})$. For large $n$ the explicit inequalities are $s(n)\ge 30.811\,\log_3^{1/2}(n+1)$ and $s(n)\le 2\lceil\sqrt{(n-2)\ln(n-3)}\rceil+2$. The proof of the lower bound embeds binary sequences into permutations, so a pair of sequences with the same $k$-deck produces a pair of permutations with the same $k$-deck; the lower bound for binary sequence reconstruction then transfers verbatim. The proof of the upper bound shows that equal $k$-decks force all low-degree bivariate polynomial sums over the order-comparison array to vanish, and then exhibits a squared Hahn polynomial whose value at $-1$ dominates its total mass elsewhere; this certifies $k$-reconstructibility once $k\ge\sqrt{(n+1)\ln n}$. From $s(n)$ the paper derives $N_d<d+3\sqrt{d\ln d}$ for large $d$, and $N_d>d+\exp(\Omega(\sqrt{\ln d}))$, which together improve both sides of the previous bounds.

Load-bearing premise

The load-bearing premise is the unproved lower bound $F_k>e^{(k+1)k/(n+1)}-1+1/(n+1)$ on a quantity $F_k$ built from orthogonal polynomials; if that estimate fails at the deck sizes considered, the paper gives no other way to certify reconstructibility.

Editorial extensions

If this is right

For large $n$, a deck of about $2\sqrt{n\ln n}$ minors is enough to reconstruct every permutation of length $n$, a sublinear amount of data.
For any fixed deficiency $d$, all permutations of length at least $d+O(\sqrt{d\ln d})$ are determined by their $(n-d)$-minors, improving the earlier quadratic upper bound on $N_d$.
No universal reconstruction guarantee can hold with decks of size below $\exp(\Omega(\sqrt{\ln n}))$, so the threshold cannot be pushed to a fixed fractional power of $n$ by this type of argument.
The exact small values are now $s(5)=s(6)=s(7)=4$ and $s(8)=s(9)=s(10)=5$, with $s(11)$ the first undetermined case, lying between 5 and 8.

Reading between the lines

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

The gap between $\exp(\Omega(\sqrt{\ln n}))$ and $O(\sqrt{n\ln n})$ is very wide; if the true $s(n)$ is closer to the lower bound, then $N_d-d$ would be subpolynomial in $d$, which is a testable conjecture by improving the $F_k$ estimate.
The same Hahn-polynomial certificate might transfer to matrix or hypermatrix reconstruction, where the analogue of the $k$-deck is a multiset of submatrices; the condition $F_k>n+1$ gives a concrete sufficient bound one could check numerically in those settings.
The characteristic-vector algorithm implies that certifying non-reconstructibility for larger $n$ can be done without enumerating all $n!$ permutations; applying it to $n=11$ is a feasible next step that would decide whether $s(11)$ is 5, 6, 7, or 8.

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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

Bottom line: the lower bound is probably correct and the bivariate basis lemma is a nice piece of work, but the headline upper bound does not follow from the arguments as written. Eq. (3.2) is false: for n=4, T_1(x)=1-8x, and the unweighted sum over x=0..4 is -75, not 0. The orthogonality claim is missing the binomial weight (or the normalization is off), and Eq. (3.3) collapses with it. Since F_k > n+1 is the whole engine behind Theorem 3.2, the O(sqrt(n log n)) upper bound is unsupported in this draft.

What is genuinely new and good: the threshold s(n), the clean embedding proof of s(n) >= s'(n), and the bivariate basis lemma (Lemma 3.2) with the resulting sufficient condition in Theorem 3.1. That theorem is a useful reduction. The lower bound s(n) >= 30.811 log_3^{1/2}(n+1) follows from Dudik-Schulman and is fine. The small-n algorithm (Algorithm 1) is a reasonable heuristic, but Lemma 4.3 has no code or certificate, so the exact values s(9)=s(10)=5 should be treated as computational claims.

The other soft spots are minor by comparison: the Mathematica verification in Theorem 2.2 is not reproducible, and the 'left grows faster' step in Theorem 4.1 could use a few lines of proof. None of these are fatal on their own.

Where does this leave us? The paper deserves a serious referee because the framework and the lower bound are worth engaging with, and the upper bound may be repairable with a corrected Hahn-polynomial argument. But as submitted, the central asymptotic upper bound should not be accepted. I would send it to review with the expectation of major revision: fix the orthogonality/normalization and re-derive (3.3), or find another polynomial that satisfies Theorem 3.1. If that cannot be done, Theorem 3.2 should be removed or scaled back.

Referee Report

3 major / 5 minor

Summary. The paper studies the permutation reconstruction problem from k-minors, defining s(n) as the smallest k such that every permutation in S_n is reconstructible from its k-deck. It claims lower and upper bounds s(n) >= 30.811*sqrt(log_3(n+1)) for n >= 16 and s(n) <= 2*ceil(sqrt((n-2)ln(n-3))) + 2 for n >= 7, giving exp(Omega(sqrt(ln n))) <= s(n) <= O(sqrt(n ln n)) asymptotically. It then translates these bounds into improved bounds on N_d, the smallest number such that any permutation of length n >= N_d is (n-d)-reconstructible, obtaining d + exp(Omega(sqrt(ln d))) < N_d < d + O(sqrt(d ln d)). The paper also provides an algorithm and exact or small-interval values of s(n) for n <= 10.

Significance. The claimed improvement over the previous bounds N_d > d + log_2 d and N_d < d^2/4 + 2d + 4 is significant if correct. The reduction of permutation reconstruction to binary-sequence reconstruction (Theorem 2.1), the bivariate polynomial basis lemma (Lemma 3.2), and the sufficient condition in Theorem 3.1 are clean and mostly rigorous. However, the proof of the upper bound relies on an orthogonality statement that is false as printed, so the central upper-bound result is currently unsupported. If the Hahn-polynomial normalization is corrected and the external estimate from [11] is properly matched, the approach may be salvageable.

major comments (3)

[Eq. (3.2)] The claimed orthogonality of the polynomials T_i on [0,n] is false as stated. For n=4, i=1, the displayed formula gives T_1(x)=1-8x, and the sum over x=0..4 of T_1(x) equals -75, which is nonzero and therefore contradicts Eq. (3.2) for j=0. Consequently, the derivation of Eq. (3.3) - in particular the identity (n+1)*sum_x p_k(x)^2 = (n+1)F_k - is invalid, and the subsequent conclusion 'F_k > n+1' in Theorem 3.2 has no proof. The authors must correct the definition of T_i (for example by supplying the missing weight or the correct Hahn-polynomial normalization) or replace the Hahn-polynomial argument entirely.
[Theorem 3.2] The upper bound depends entirely on the imported estimate F_k > e^{(k+1)k/(n+1)} - 1 + 1/(n+1) from [11, Section 3.2.1], which is not proved in the paper. Because the definition of F_k here uses the T_i and d_i whose orthogonality fails, it is not established that the quantity bounded in [11] coincides with the F_k used in this paper. The authors need to state the exact external theorem with matching notation and either prove the estimate in their setting or give a precise reference that supports the inequality for the specific F_k defined in Section 3.
[Theorem 2.2] The proof of the lower bound relies on the unstated Mathematica assertion that f(30.811*sqrt(log_3(n+1))) <= n for all n >= 16, where f is defined in the proof. This computer check is load-bearing for the specific constant 30.811, and no code, certificate, or rigorous argument is provided. In addition, k is used as a real number in the expression s'(n) >= k+1, although k must be an integer; the statement and proof should use floor or ceiling of the real expression. The authors should either provide the verification or replace the assertion with an analytic argument, and handle the integrality of k throughout.

minor comments (5)

[Proof of Theorem 3.2] The exponent in the displayed inequalities is sometimes written as sqrt(ln n)/(n+1) and sometimes as sqrt(ln n/(n+1)); these are different quantities, and the inequalities should be re-checked and made consistent.
[Algorithm 1, Step 3] The statement that the matrices S and P are 'exactly the same' for the reverse characteristic vectors is not self-evident; the relationship between the row index i, the values ar{S}_j, and the forward characteristic vectors should be spelled out more explicitly.
[Section 2] Theorem 2.2 is stated as s(n) >= 30.811 * sqrt(log_3(n+1)), but s(n) is an integer; the statement should use a floor or ceiling of the real expression to be formally correct.
[Throughout] There are minor typos and formatting issues, such as 'parame ter' in the abstract, 'assistant of computers' in Section 4, and inconsistent spacing around 'For' in Example 2.1; these do not affect the mathematics.
[Table II] The bounds for s(n) in Table II for n=11..14 are derived from the previous N_d results of Raykova; it would improve clarity to explicitly indicate which bound comes from which source.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning found; both bounds reduce to independent prior results ([12] and [11]) and the paper's own combinatorial derivations, not to its conclusions.

full rationale

The derivation chain is not circular. The lower bound is obtained by an explicit mapping Ψ from binary sequences to permutations that preserves k-deck equality, proving s(n) ≥ s'(n); the subsequent lower bound then invokes the independent binary-sequence reconstruction result of Dudík and Schulman [12]. No fitted parameter or target conclusion is assumed in this reduction. The upper bound is a genuine combinatorial argument: Lemma 3.1 derives linear equations for the order-indicator differences from deck equality, Lemma 3.2 proves that the displayed bivariate polynomials form a basis, Corollary 3.1 converts these equations into a vanishing condition for all low-degree bivariate polynomials, and Theorem 3.1 reduces the existence of a good univariate polynomial to a concrete polynomial inequality. The paper then imports the Hahn-polynomial construction and the estimate F_k > e^{(k+1)k/(n+1)} − 1 + 1/(n+1) from the independent prior work of Foster and Krasikov [11]. These are external published results, not self-citations, and they are used as inputs rather than as renamed versions of the permutation reconstruction claim. The small-n table values are obtained by computer search, not by fitting parameters that are later called predictions. The paper contains one correctness-sensitive imported step: the unweighted orthogonality displayed in Eq. (3.2) appears nonstandard and the F_k estimate is not re-proved in the text. However, those are verification concerns, not circularity, because neither step defines the conclusion in terms of itself. Accordingly, the circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central asymptotic claims rest on two external estimates: the binary sequence reconstruction bound of Dudik and Schulman [12] and the Hahn polynomial lower bound from Foster and Krasikov [11]. The paper adds two computer-verified assertions without artifacts: a Mathematica inequality in Theorem 2.2 and the exhaustive search in Lemma 4.3. No new entities are postulated.

free parameters (2)

constant 30.811 in lower bound s(n) >= 30.811 log_3^{1/2}(n+1) = 30.811 (approx)
Chosen so the inequality f(k) <= n holds for n >= 16; asserted by Mathematica, not derived analytically.
constant 3 in N_d upper bound = 3
Chosen to make the inequality (n-d-2)^2 > 4n ln(n-1) hold for all n >= n0 for large d; any constant above 2 works asymptotically.

assumptions (4)

standard math Lower bound on F_k: F_k > e^{(k+1)k/(n+1)} - 1 + 1/(n+1), from [11, Section 3.2.1].
Upper bound Theorem 3.2 depends on this estimate; the paper quotes it without proof.
standard math Upper bound S(k) <= 1.2 Gamma(log_3 k) 3^{(3/2) log_3^2 k - (1/2) log_3 k} for k >= 5 from [12].
Lower bound Theorem 2.2 relies on this external bound for binary sequence reconstruction.
ad hoc to paper Mathematica verification that f(30.811 log_3^{1/2}(n+1)) <= n for n >= 16.
The proof asserts this is verified by Mathematica but gives no code, data, or analytic argument.
ad hoc to paper Algorithm 1 returns 'No solution!' for (n,k) = (9,5) and (10,5).
Computer-assisted claim underlying s(9) <= 5 and s(10) <= 5; no code or output supplied.

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Pith. "Pith review of Improvements on Permutation Reconstruction from Minors." pith.science (2026). https://pith.science/paper/QRKYSGCJ

@misc{pith2026241112718,
  author       = {Pith},
  title        = {Pith review of: Improvements on Permutation Reconstruction from Minors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QRKYSGCJ}},
  note         = {Machine review of arXiv:2411.12718}
}

abstract

We study the reconstruction problem of permutation sequences from their $k$-minors, which are subsequences of length $k$ with entries renumbered by $1,2,\ldots,k$ preserving order. We prove that the minimum number $k$ such that any permutation of length $n$ can be reconstructed from the multiset of its $k$-minors is between $\exp{(\Omega(\sqrt{\ln n}))}$ and $O(\sqrt{n\ln n})$. These results imply better bounds of a well-studied parameter $N_d$, which is the smallest number such that any permutation of length $n\ge N_d$ can be reconstructed by its $(n-d)$-minors. The new bounds are $ d+\exp(\Omega(\sqrt{\ln d}))<N_d<d+O(\sqrt{d\ln d})$ asymptotically, and the previous bounds were $d+\log_2 d<N_d<d^2/4+2d+4$.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [11]

An improvement of a Borwein– Erd´ elyi–K´ os result,

W. Foster and I. Krasikov, “An improvement of a Borwein– Erd´ elyi–K´ os result,”Methods and Applications of Analysis , vol. 7, no. 4, pp. 605–614, 2000

work page 2000

[2] [1]

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J. Acharya, H. Das, O. Milenkovic, A. Orlitsky, and S. Pan , “String reconstruction from substring compositions,” SIAM Journal on Discrete Mathematics , vol. 29, no. 3, pp. 1340–1371, 2015

work page 2015

[3] [2]

Reconstr ucting strings from random traces,

T. Batu, S. Kannan, S. Khanna, and A. McGregor, “Reconstr ucting strings from random traces,” in SODA, vol. 4, 2004, pp. 910–918

work page 2004

[4] [3]

Finding approximate patterns in strings,

E. Ukkonen, “Finding approximate patterns in strings,” Journal of Algorithms , vol. 6, no. 1, pp. 132–137, 1985

work page 1985

[5] [4]

The gapp ed k-deck problem,

R. Golm, M. Nahvi, R. Gabrys, and O. Milenkovic, “The gapp ed k-deck problem,” in 2022 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2022, pp. 49–54

work page 2022

[6] [5]

Portable an d error-free DNA-based data storage,

S. H. T. Yazdi, R. Gabrys, and O. Milenkovic, “Portable an d error-free DNA-based data storage,” Scientiﬁc Reports , vol. 7, no. 1, p. 5011, 2017. 10

work page 2017

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Unique reconstruction of c oded strings from multiset substring spectra,

R. Gabrys and O. Milenkovic, “Unique reconstruction of c oded strings from multiset substring spectra,” IEEE Transactions on Information Theory , vol. 65, no. 12, pp. 7682–7696, 2019

work page 2019

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A collection of mathematical problems,

S. M. Ulam, “A collection of mathematical problems,” Interscience, 1960

work page 1960