New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations

Arthemy V. Kiselev; Kian C. Shah

REVIEW

New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations

Kian C. Shah, Arthemy V. Kiselev

Number Theory · Submitted 6 Oct 2026 · 24 references

Pith's verdict: Genuinely new sequence with exact values to p=18 via a clean subset recursion; the math holds up, the abstract over-claims an unpublished growth bound, and the tail c(15), c(18) need an independent run.

T0 · AI review

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T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

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arXiv ID2610.08636 (version 1)

Pith IDT5U3K4LY, Pith's permanent record of this paper and its review

CitationArthemy V. Kiselev, Kian C. Shah. "New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations." arXiv:2610.08636 (2026). https://pith.science/paper/T5U3K4LY

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Editor's note

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What you should know: the core deliverable is real. The paper reduces the alternating composition of N=2p weighted p-th-order differential operators, evaluated on monomial weights, to a signed sum over a small set Phi_p of late-growing permutations. Formula (8) computes c(p) as that signed sum of falling-factorial products over Phi_p, normalized by the monomial Wronskian product of k!. The subset recursion (10) is a genuine insight: the running exponent depends only on the set of labels already placed, not on their order, so the memoization state collapses to a bit mask. From there they get exact values c(4) through c(18), matching the known c(1)=1, c(2)=2, c(3)=90; old and new implementations agree wherever both ran (p<=14), and Buring's C/FLINT code independently verified p<=7. The sequence is registered as OEIS A392714, the factorization of c(14) yields a certified 206-digit prime, and code and primality certificates are on github. The derivation is direct and, as far as I checked, correct.

The soft spots are presentation and verification, not the displayed math. First, the abstract states the growth-rate bound b>=2.6744 as if derived here; the body attributes it to Zaboronski's unpublished note (footnote 3, Claim 5.2). That is an attribution fix, not an error, but the abstract needs it. Second, Table 6's p=6 row prints 227886133184567796056800, which is the digit count 22 welded onto the true 22-digit value 7886133184567796056800 from Appendix C. The stress-test note reads this as evidence that silent corruption is possible; I don't buy that inference. It is a transcription typo, and Appendix C's value is consistent with its factorization, as is c(18)'s decimal with its own. What the typo does justify is insisting on an independent run of the tail: c(15) and c(18) rest solely on Algorithm 2. c(16) and c(17) had a second route through Petrella's Julia port, which helps, but the headline tail values still need a fresh implementation.

Third, the identification of Phi_p with late-growing permutations and the alternating parity difference are deferred to the to-appear [12]; Table 2's evidence is suggestive but the proofs are not here. Minor: the falling-factorial range bound is imported from the authors' own preprint [11], worth a self-contained sentence in revision. The Appendix A universality point -- that the coefficient C_k depends only on p -- is a sketch, but Theorem 1.1 is cited from published [5,7], so evaluating on monomials is a legitimate way to extract the constant.

Who this serves: people working with the Wronskian constant, SH-Lie realizations by differential operators, or the new OEIS entry; also a clean worked example of CRT-based exact computation. My recommendation: send it to peer review. The tail recomputation, abstract re-attribution, and Table 6 fix are conditions, not blockers -- the paper is honest and reproducible, with an over-claimed abstract and an under-checked tail.

Referee report

Significance

If the reported values are correct, this is a solid, self-contained computational contribution: formula (8) is derived explicitly and is parameter-free, the p=2 worked example checks out, and the stated bound M_p genuinely protects the CRT reconstruction. The small values are cross-verified at several levels (p≤3 symbolic, p≤6 direct, p≤7 by Buring's independent C/FLINT run, p≤14 by agreement of two code paths), the code is public, and the primality certificate for the 206-digit prime of c(14) is posted. The gaps are as follows: the tail values c(15)–c(18) rest on a single code path (per the AI-use statement, agreement between code paths was checked only to p≤14), and a concrete transcription failure inside Table 6 (the p=6 row welds the digit-count "22" onto the true value) shows that silent corruption is a live risk on exactly this table; the abstract's quantitative growth bound b≥2.6744 is supported only by an unpublished private communication; and the universality step in Appendix A that turns the monomial computation into the universal constant is asserted rather than demonstrated.

Major objections (4)

[§4.2, Algorithm 2; §5; Appendix C] The exact values c(15)–c(18) rest on a single code path. The AI-use statement on p. 22 says the re-calculated values "agree with those of the original Python program wherever both were run (p≤14)", which explicitly excludes p≥15; Appendix C tells us c(15) and c(18) were first obtained by the authors and c(16), c(17) by G. Petrella, but it is not documented whether Petrella's Julia implements Algorithm 2 or the exact-arithmetic path, so even the two runs of c(16), c(17) are not established as independent. The headline deliverables — the OEIS entries at p≥15, the Table 8 fits with p_max=18, and the Table 9 extrapolations — all depend on these tail values. That silent corruption is a live risk is demonstrated inside the paper: the p=6 row of Table 6 prints "227886133184567796056800", which is the digit-count "22" welded onto the true 22-digit value 7886133184567796056800 of Appendix C, with the digit-count column dropped for that row. Internal consistency (c(18)'s decimal matching its Appendix C factorization) is reassuring but does not replace an independent computation. Please provide an independent recomputation of c(15)–c(18) (a second implementation, or residue-level cross-checks), or explicitly flag those entries as carrying only single-implementation support.
[Abstract; Claim 5.2; footnote 3; §6] The headline bound b≥2.6744 (and the exact leading coefficient α=2) in formula (13) is quoted from O. Zaboronski's unpublished "work in progress" note cited in footnote 3, and Claim 1.3 (positivity and monotonicity of c(p)) likewise rests on that note. The fitted β values in Table 8 (−2.6696 to −2.6725 on the segments shown) do not by themselves establish the claimed bound. Please either include the argument establishing b≥2.6744 and α=2 within this manuscript, or explicitly state these as external conjectures with the published-data evidence, so that the abstract's quantitative claim is not stronger than what the manuscript itself supports.
[Appendix A, closing step of the proof of Theorem 1.1 (p. 25)] After the derivative-budget and antisymmetry arguments force the unique profile {k_1,...,k_N}={0,1,...,N-1} (the equality case in (14)), the proof asserts that the coefficient C_k "arises purely from the combinatorics of the iterated Leibniz expansion and depends only on p, not on f or on the weight functions." This universality is what justifies specializing to the monomial weights w_j=x^{j-1} and identifying the number computed by formula (8) with the universal constant in (2); if the coefficient were weight-dependent, the monomial evaluation would not recover c(p). The step is plausible and agrees with the p≤3 symbolic checks, but as written it is asserted in a single sentence. A short expansion — e.g., counting, for the fixed derivative budget, the number of Leibniz placements that assign k_i derivatives to w_i — would close the gap.
[Table 6, p=6 row] This is the concrete internal corruption risk: the row reads "6 227886133184567796056800", which merges the digit-count "22" (c(6) has 22 digits) with the true value 7886133184567796056800 given in Appendix C and consistent with the factorization 2^5·3^6·5^2·11·23·223·239·1002820739. The fix is one line, but the authors should audit Table 6 (and the other tables of exact values) for similar merges of the digit-count column into the value column before the OEIS entries are presented as definitive.

Minor objections (6)

[Conjecture 3.1, p. 7] The convergence target is printed as "converges from above to 1 + 0. 8"; footnote 8 and Remark 3.3 indicate the intended limit is 1, so the stray "8" should be removed and the conjecture restated cleanly.
[Table 7, p. 18] The rows for the primes q=2,3,5,7,11,13,17,19 run together in the typesetting (e.g., the q=2 row reads "211 5 2 5 4 17 ..."). The per-prime valuations v_q(c(p)) and the p-indexing should be reformatted so the table is legible without cross-checking every entry against Appendix C; I verified several entries against the Appendix C factorizations and they are consistent.
[Lemma 4.4, p. 15] The base case is mislabeled: the sentence "The base case A={2,...,N}\{e} is the definition" describes the induction step at |A|=N−1, not the base case, which is A=∅ (where Ψ_q(∅)=π_q(1) is the definition). Correcting the label would remove a genuine point of confusion for readers checking the induction.
[Notation, (7)–(8) and §4.2 table F[E]] The glyph E^p_k in (7)–(8) denotes the falling factorial (E_k)!/(E_k−p)!, while in Algorithm 2 the same notation F[E]←E^p mod q denotes the falling factorial of E tabulated over p≤E≤p(p+1)/2. The two meanings should be distinguished (e.g., subscripts for falling factorials) to avoid ambiguity in a paper whose central formula is exactly this product.
[§5.2 and the Method subsection, p. 2] The statement "the observed growth rate is c(p)∼p^{2p²}" (and the similar sentence in the Method paragraph) is precise only up to subexponential factors given the fitted form log c(p)=2p² log p+βp²+...; the text should say "up to exp(O(p²)) factors" or phrase the claim in terms of log c(p).
[Table 1 and §1, p. 5] The paper states the values c(1), c(2), c(3) are "previously known" and verifies c(3)=90 by Maple; this is fine, but the provenance of c(2)=2 and c(3)=90 (attributed to [5,7] via Theorem 1.1) should be stated once explicitly so readers do not mistake them for new results of this paper.

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The paper

The paper claims

The alternating composition of 2p weighted differential operators on the line equals c(p) times the Wronskian of the weights, and the paper computes the universal integers c(p) exactly up to p = 18 as the new OEIS sequence A392714.

The reading

The paper establishes an exact bridge between differential operators and combinatorics: on the real line, the fully antisymmetrised composition of $N = 2p$ weighted operators $w_j(x)\,\partial_x^p$ acts on any smooth $f$ as $c(p)\cdot\mathrm{Wronsk}(w_1,\dots,w_N)(x)\,\partial_x^p f(x)$, where $c(p)$ is a universal integer that does not depend on $f$ or on the weights. The paper's contribution is to make that constant computable: feeding in the monomial weights $w_j = x^{j-1}$ reduces the sum over all $(2p)!$ permutations to a much smaller signed sum over the late-growing permutations $\Phi_p \subseteq S_{2p-1}$, expressed through falling factorials. A memoized recursion over subsets, run modulo several primes and reconstructed by the Chinese Remainder Theorem, then yields the exact values $c(p)$ for every $p \leq 18$, the largest being $c(18) \approx 4.881\times 10^{462}$, and these integers turn out to form a sequence new to the OEIS, registered as A392714. The constants are not a curiosity: they are the coefficients of the higher-arity brackets that deform the binary Lie bracket into strongly homotopy Lie algebras, and their super-exponential growth controls how strongly such collective terms can act.

What carries the argument

The load-bearing object is the late-growing permutation set $\Phi_p\subseteq S_{2p-1}$ together with the closed formula (8) that expresses $c(p)$ through it. Late-growing permutations are those whose partial sums never exceed the partial sums of the average permutation; after the monomial substitution they are exactly the permutations whose iterated action keeps the exponent of $x$ nonnegative, so they replace the full group $S_{2p}$ in the alternating sum. Formula (8) reads $c(p)\prod_{k=0}^{2p-1}k! = \sum_{\sigma\in\Phi_p}(-1)^\sigma\prod_{k=1}^{2p-1}(E_k)_p$, where $E_k$ is the running exponent built by the right-to-left operator action and $(E_k)_p$ is the falling factorial. The decisive acceleration is Lemma 4.2: the exponent $E(A) = p + (p+1)|A| - \sum_{j\in A} j$ depends only on the set $A$ of labels still to be placed, never on their order, so the sum obeys the subset recursion $\Psi(A) = \sum_{e\in A,\ E(A\setminus\{e\})\geq p}(-1)^{\#\{f\in A:\ f>e\}}(E(A\setminus\{e\}))_p\,\Psi(A\setminus\{e\})$ with $\Psi(\{2,\dots,2p\}) = c(p)\prod_{k=0}^{2p-1}k!$. Memoising this recursion over bit masks and sweeping it modulo primes, with a Chinese Remainder Theorem reconstruction, is Algorithm 2; the underlying identity (2) is supported by the appendix's argument that antisymmetry plus the derivative budget of $2p^2$ forces the unique surviving Leibniz profile $\{0,1,\dots,N-1\}$, making the alternating sum proportional to the Wronskian.

Extended reading notes

Core claim

The central claim is identity (2): for any smooth $f$ and any weights, $A_p[w_1,\dots,w_N] = c(p)\cdot\mathrm{Wronsk}(w_1,\dots,w_N)\cdot\partial_x^p$, where $A_p$ is the signed sum $\sum_{\sigma\in S_{2p}}(-1)^\sigma w_{\sigma(1)}\partial_x^p\circ\cdots\circ w_{\sigma(N)}\partial_x^p$ and the Wronskian is the determinant of the matrix of iterated derivatives of the weights. The paper computes the universal constants by specialising to the monomials $1,x,\dots,x^{2p-1}$: after that substitution a permutation contributes precisely when it is late-growing, and the constant is the signed sum $c(p)\prod_{k=0}^{2p-1}k! = \sum_{\sigma\in\Phi_p}(-1)^\sigma\prod_{k=1}^{2p-1}(E_k)_p$ of falling factorials $(E)_p = E!/(E-p)! = E(E-1)\cdots(E-p+1)$. Evaluating this formula gives $c(1)=1$, $c(2)=2$, $c(3)=90$, $c(4)=586656$, and so on up to $c(18)\approx 4.881\times10^{462}$, a set of values that matches no existing entry in the OEIS and is therefore registered there as the new sequence A392714. The growth is $\log c(p) \simeq 2p^2\log p - bp^2 + o(p^2)$ with subleading constant $b\geq 2.6744$, and the algorithm that reaches $p=18$ exploits the fact that the recursion's state is just the set of not-yet-placed labels, stored as bit masks, with all arithmetic done modulo primes.

Load-bearing premise

Everything computed inherits from the appendix's assertion, in proving Theorem 1.1, that the coefficient of the unique surviving Leibniz profile {0,1,...,N-1} is a universal integer depending only on p and not on the test function or the weight functions; the appendix shows the profile is unique but does not work out the coefficient's universality in detail.

Editorial extensions

If this is right

The exact integers c(p) for 1 ≤ p ≤ 18 are now the registered sequence A392714, and no existing OEIS entry matched the computed values, so the sequence is new to the registry.
The fitted growth law predicts that c(19) through c(22) have about 532, 606, 685, and 770 decimal digits, so running the same algorithm on larger machines should produce the next exact values.
Since log c(p) ~ 2p^2 log p, the constants outrun the factorial (2p)!; in the strongly homotopy Lie models that motivate identity (2), that super-exponential growth means the collective N-ary brackets at arity N > 4 are numerically suppressed relative to the binary Lie bracket.
The prime factorisations follow a visible pattern — small primes in high powers, very few very large primes — and v_p(c(p)) = p − 1 for every prime p ≤ 17, supporting Conjecture 5.1; the factorisation of c(14) yields a newly confirmed 206-decimal-digit prime.
The counts of even and odd permutations in every Phi_p differ by exactly (−1)^(p+1), a parity law the paper states and announces a proof of in a sequel.

Reading between the lines

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

Extension: because the exponent E(A) in Lemma 4.2 depends on A only through |A| and the sum of its labels, the memoized subset table may collapse into a two-parameter recurrence that produces c(p) without enumerating any subsets; the paper does not pursue this.
Extension: the same template — monomial weights, falling-factorial sums, subset-state recursion, modular reconstruction — should transfer to the multidimensional Wronskian brackets the authors list as an open problem, yielding constants c(p_1,...,p_d) indexed by per-variable differential orders.
Extension: the next term is a cheap joint test — running Algorithm 2 for p = 19 should produce a value of about 532 digits whose 19-adic valuation would be 18 if Conjecture 5.1 holds; a mismatch would expose either the algorithm or the valuation pattern.

Original abstract

The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x^{\,p}$ of strict order $p$ on the line $\mathbb{R} \ni x$ is again an operator of order $p$; its coefficient is the universal constant $c(p)$ times the Wronskian of the weights $w_1,\ldots,w_N$. Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p \geqslant 2)$: e.g., $c(2) = 2$ or $c(3) = 90$. Direct symbolic expansion (over $|S_{2p}| =(2p)!$ permutations) fails for $p \geqslant 4$. Taking the monomials $w_j = x^{j-1}$ reduces the summation to the much smaller set $\Phi_p \subseteq S_{2p-1} \subsetneq S_{2p}$ of late-growing permutations. Expressing $c(p)$ as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to $c(18) = 4.881\ldots \cdot 10^{462}$. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$ for $p\gg 1$, with $b\geqslant 2.6744$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-07: 1 request, 5,499 input tokens.

What the abstract itself states, as probabilities: claims first 0.35 · states limitation 0.80 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.74 · mentions code or data availability 0.17.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x^{\,p}$ of strict order $p$ on the line $\mathbb{R} \ni x$ is again an operator of order $p$; its coefficient is the universal constant $c(p)$ times the Wronskian of the weights $w_1,\ldots,w_N$. Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p \geqslant 2)$: e.g., $c(2) = 2$ or $c(3) = 90$. Direct symbolic expansion (over $|S_{2p}| =(2p)!$ permutations) fails for $p \geqslant 4$. Taking the monomials $w_j = x^{j-1}$ reduces the summation to the much smaller set $\Phi_p \subseteq S_{2p-1} \subsetneq S_{2p}$ of late-growing permutations. Expressing $c(p)$ as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to $c(18) = 4.881\ldots \cdot 10^{462}$. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$ for $p\gg 1$, with $b\geqslant 2.6744$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x^{\,p}$ of strict order $p$ on the line $\mathbb{R} \ni x$ is again an operator of order $p$; its coefficient is the universal constant $c(p)$ times the Wronskian of the weights $w_1,\ldots,w_N$.	background 0.71	0.39	0.29	asserted 1.00
s1	Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p \geqslant 2)$: e.g., $c(2) = 2$ or $c(3) = 90$.	gap 0.70	0.44	0.07	asserted 0.96
s2	Direct symbolic expansion (over $\|S_{2p}\| =(2p)!$ permutations) fails for $p \geqslant 4$.	limitation 0.52	0.60	0.03	asserted 1.00
s3	Taking the monomials $w_j = x^{j-1}$ reduces the summation to the much smaller set $\Phi_p \subseteq S_{2p-1} \subsetneq S_{2p}$ of late-growing permutations.	method 0.97	0.81	0.07	asserted 1.00
s4	Expressing $c(p)$ as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to $c(18) = 4.881\ldots \cdot 10^{462}$.	method 0.50	0.95	0.17	asserted 1.00
s5	The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$ for $p\gg 1$, with $b\geqslant 2.6744$.	result 0.91	0.92	0.63	asserted 0.98

What kind of paper the classifier reports this is: method or tool 0.85 · evidence from simulation or computation 0.79.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (24)

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Shah K.\,C., Kiselev A.\,V. (2026) The alternating compositions of weighted differential operators yield the weights' Wronskian with which constant\,? Preprint arXiv:2605.11137 [math.CO], 22 p

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[22]

Petrella G. (2026) julia-wronskian: Julia code for computing OEIS A392714,\\ https://github.com/giannipetrella/julia-wronskian

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Proof of Theorem thm:main app:proof The proof relies on the following two lemmas and the corollary they imply. lemma [Antisymmetry] lem:antisym The alternating composition satisfies, for a b , \[ A _p[ , w_a, , w_b, ](f) \;=\; -\, A _p[ , w_b, , w_a, ](f). \] lemma proof Compose each summation index in eq:opdef with the transposition = (a\ b) . As ranges ...

View on Pith 2026

Discussion

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[1] [12]

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Shah K.\,C., Kiselev A.\,V. (2026) New integer sequence OEIS\:A392\,714 counts Wronskians, II. Computing within the permutation group algebra, Proc.\ 36th Int.\ colloquium on group theoretic methods in Physics: Group36 (13--17 July 2026, Valladolid, Spain), 16 p.\ (in preparation)

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The on-line encyclopedia of integer sequences: Sequence A392714,\\ https://oeis.org/A392714, OEIS Foundation Inc., 2026

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Proof of Theorem thm:main app:proof The proof relies on the following two lemmas and the corollary they imply. lemma [Antisymmetry] lem:antisym The alternating composition satisfies, for a b , \[ A _p[ , w_a, , w_b, ](f) \;=\; -\, A _p[ , w_b, , w_a, ](f). \] lemma proof Compose each summation index in eq:opdef with the transposition = (a\ b) . As ranges ...

View on Pith 2026