REVIEW 2 major objections 3 minor 3 references
A new refinement of Euler numbers on counting alternating permutations
T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The paper proves that the alternating-permutation identity $E^{\nearrow}_{2n} - E^{\nwarrow}_{2n} = E_{2n-2}$ comes from a positional split of the second-largest entry.
desk verdict A modest but genuinely new counting proof of Heneghan–Petersen's Euler-number identity, with a small circular slip in the displayed derivation that is easily repaired—worth refereeing, but not as clean as the authors claim. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is a positional refinement of alternating permutations. The old refinement sorts up-down permutations by the relative order of $1$ and $n$; the paper's new refinement sorts them by the location of $n-1$: $E^{\uparrow}$ when $n-1$ is in the upper row, $E^{\downarrow}$ when it is in the lower row. The structural fact that $n-1$ cannot occupy an interior valley, because no two larger entries exist to surround it, forces a rigid local pattern around $n-1$ and $n$; this pattern reduces $E^{\downarrow}_{2k}$ to $E_{2k-2}$. On the other side, the transposition $n-1\leftrightarrow n$ shows the upper-row class is always even, and its three-block count satisfies exactly the recurrence of $2E^{\nwarrow}_{2n}$. Matching those recurrences is what carries the argument: the identity stops being a coincidence of series and becomes a comparison of two ways to partition the same set of permutations.
What would settle it
Enumerate the 16 up-down permutations of degree 6 and classify each by whether 5 sits in the upper row or the lower row. The argument requires the five lower-row permutations to be exactly those beginning $5,6$, with the remaining four entries forming an up-down permutation of degree 4; one lower-row occurrence of 5 elsewhere would break Lemma 3.8 and, with it, the proof of Theorem 4.1.
Extended reading notes
Core claim
For $n\ge 1$ the paper proves $E^{\nearrow}_{2n} - E^{\nwarrow}_{2n} = E_{2n-2}$, where $E^{\nearrow}$ counts up-down permutations in which $1$ appears before $n$ (min-max) and $E^{\nwarrow}$ counts those in which $n$ appears before $1$ (max-min). The new proof introduces a second split: $E^{\uparrow}_{n}$ counts up-down permutations with $n-1$ in the upper row, and $E^{\downarrow}_{n}$ counts those with $n-1$ in the lower row. Because $n-1$ can occupy only the upper row or an extremal lower position, with $n$ immediately beside it, the lower-row case is forced to start $n-1,n$ and contributes $E^{\downarrow}_{2k}=E_{2k-2}$; the upper-row case is matched with twice the max-min count through the same three-block factorial decomposition, giving $E^{\uparrow}_{2n}=2E^{\nwarrow}_{2n}$. Substituting these into $E_{2n}=E^{\nearrow}_{2n}+E^{\nwarrow}_{2n}=E^{\uparrow}_{2n}+E^{\downarrow}_{2n}$ yields the theorem. Thus a relation previously derived from formal power series is re-proved by a decomposition of the permutations themselves.
Load-bearing premise
The load-bearing premise is the unproved structural observation that in every alternating permutation the second-largest number must sit either at a peak of the zigzag or at an end of the valley row, with the largest number adjacent; if that placement rule failed, the split into $E^{\uparrow}$ and $E^{\downarrow}$ would be incomplete and Lemma 3.8 would not hold.
Editorial extensions
If this is right
- The identity $E^{\nearrow}_{2n} - E^{\nwarrow}_{2n} = E_{2n-2}$ follows from counting where $n-1$ sits, with no reference to the secant and tangent series.
- For even degrees the lower-row class reproduces the previous Euler number exactly: $E^{\downarrow}_{2k}=E_{2k-2}$; for odd degrees it doubles the previous one: $E^{\downarrow}_{2k+1}=2E_{2k-1}$.
- The new refinement has closed exponential generating functions, $E^{\uparrow}(x)=2\tan^2 x(\sec x+\tan x)$ and $E^{\downarrow}(x)=\sec x+2\tan x$.
- Because both refinements partition the same set $E_{2n}$, the identity becomes an algebraic consequence of two partitions rather than a separate series identity.
Reading between the lines
- Editorial inference: the recurrence proof of $E^{\uparrow}_{2n}=2E^{\nwarrow}_{2n}$ suggests that an explicit bijection should exist between the permutations counted by $E^{\uparrow}_{2n}$ and two disjoint copies of those counted by $E^{\nwarrow}_{2n}$; constructing it would make the paper's 'bijective explanation' label fully literal.
- Editorial inference: the same positional criterion can be applied to down-up permutations, and the companion sequences $D^{\uparrow}_n,D^{\downarrow}_n$ mentioned in the final remarks should be evaluable by the same secant-tangent convolution, likely giving closed forms parallel to $E^{\uparrow}(x)$ and $E^{\downarrow}(x)$.
- Editorial inference: if both refinements inherit the exponential growth of the Euler numbers, the ratio $E^{\downarrow}_n/E^{\uparrow}_n$ posed as an open question should tend to $1$, although the paper does not prove this.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines a new refinement of Euler numbers by splitting the set of up-down alternating permutations according to whether the second-largest element n−1 appears in the upper row (E↑_n) or the lower row (E↓_n). It proves recurrences for these numbers (Lemma 3.6 and Lemma 3.8), derives their exponential generating functions (Section 5), and uses them to give a combinatorial proof of Heneghan–Petersen's identity Eր_{2n} − Eտ_{2n} = E_{2n−2}. The central claim is that this identity, previously proved by power series, now has a bijective explanation, with the key comparison coming from Lemma 4.2 (E↑_{2n} = 2Eտ_{2n}) and Lemma 3.8 (E↓_{2n} = E_{2n−2}).
Significance. If the proof is repaired, the paper answers an open question raised by Heneghan–Petersen by producing a genuinely combinatorial proof of a nontrivial relation between refinements of Euler numbers. The new refinement E↑,E↓ is natural and comes with clean generating functions (2 tan^2 x(sec x + tan x) and sec x + 2 tan x). The argument is elementary and should be accessible to a broad audience. The manuscript includes explicit examples and tables that support the recurrences.
major comments (2)
- [Section 4, final displayed derivation] The proof of Theorem 4.1 is circular as printed. In the chain E_{2n} = Eր_{2n} + Eտ_{2n} = (Eտ_{2n} + E_{2n−2}) + Eտ_{2n}, the middle equality substitutes Eր_{2n} = Eտ_{2n} + E_{2n−2}, which is exactly the identity being proved. No prior statement justifies this replacement. The intended non-circular argument is available and should replace the displayed chain: by Lemma 4.2 and Lemma 3.8, E_{2n} = E↑_{2n} + E↓_{2n} = 2Eտ_{2n} + E_{2n−2}; comparing this with E_{2n} = Eր_{2n} + Eտ_{2n} yields Eր_{2n} − Eտ_{2n} = E_{2n−2}. This repair is short, but the manuscript must be corrected.
- [Section 3, Observation 3.3] Observation 3.3, which states that n−1 appears either in the upper row or at an extremal lower position and that n is adjacent to it, is load-bearing: Lemma 3.8 and the final proof of Theorem 4.1 rest on it. The paper gives only a verbal sketch ('This is because there does not exist two numbers in {1,...,n} which are strictly greater than n−1'). I recommend adding a rigorous proof, for instance by showing that if n−1 were in a non-extremal lower position, then one of its two upper-row neighbors would be smaller than n−1 and the other would be equal to the unique larger element n, forcing an impossible descent/ascent pattern.
minor comments (3)
- [Section 4, after the displayed derivation] The sentence 'Hence Eր_{2n} − Eտ_{2n} = E_{2n−2} was equivalent to ths idea of our refinement for E_{2n}' contains a typo: 'ths' should be 'this'.
- [Section 4, proof of Lemma 4.2] The comparison of the two recurrences is compressed. The statement 'replacing n by 2n and k by 2k and interchanging j and k' is correct but should be spelled out with the explicit substitution so that the matched coefficients are transparent.
- [Section 5, generating functions] The derivation of E↓(x) = sec x + 2 tan x is correct but stated very quickly; adding one line showing the split into even and odd indices would improve readability.
Circularity Check
Theorem 4.1's displayed proof substitutes the target identity before deriving it; the lemmas themselves are independent, so this is a local circular step, not a global one.
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other
[Section 4, proof of Theorem 4.1, displayed equation after 'It follows from this and Lemma 3.8']
"It follows from this and Lemma 3.8 that E_{2n} = Eր_{2n}+Eտ_{2n} = (Eտ_{2n}+E_{2n−2})+Eտ_{2n} = 2Eտ_{2n}+E_{2n−2} = E↑_{2n}+E↓_{2n}."
The second equality of the chain replaces Eր_{2n} with Eտ_{2n}+E_{2n−2}, which is exactly the theorem's conclusion Eր_{2n}−Eտ_{2n}=E_{2n−2}. At that point no previous line has established this substitution; it can be derived only by combining E_{2n}=Eր+Eտ, E_{2n}=E↑+E↓, Lemma 3.8 (E↓_{2n}=E_{2n−2}) and Lemma 4.2 (E↑_{2n}=2Eտ_{2n}), and then solving. The displayed proof therefore assumes the target identity in the middle step. A valid proof would first write E_{2n}=Eր+Eտ=E↑+E↓=2Eտ_{2n}+E_{2n−2} and then conclude Eր_{2n}=Eտ_{2n}+E_{2n−2}.
full rationale
We found one clear circular step in the proof of Theorem 4.1. The displayed equation in Section 4 substitutes Eր_{2n} = Eտ_{2n}+E_{2n−2} in the middle of the chain, which is exactly the identity being proved. This substitution is not justified by any preceding statement; it can be obtained only after combining the two refinements and Lemmas 3.8 and 4.2. However, the lemmas themselves are derived independently: Lemma 3.8 provides a direct bijection for E↓_{2n}=E_{2n−2}, and Lemma 4.2 matches recurrences to give E↑_{2n}=2Eտ_{2n}. Neither lemma assumes Theorem 4.1. The target identity is not used as an input anywhere else. The proof is easily repaired by rearranging the equations, but as written the manuscript's key step is formally circular. The paper does not rely on self-citation for the main claim beyond citing Heneghan-Petersen for the original definitions and generating functions, which are not the disputed identity. Observation 3.3 is an unproved structural fact but is true and not circular. Overall, the circularity is local to the displayed derivation and does not invalidate the underlying combinatorial result, but it does mean the written proof of the central theorem currently assumes the conclusion.
Assumptions & free parameters
assumptions (3)
- standard math André's theorem: sum over n of E_n x^n / n! equals sec x + tan x.
- standard math The number of alternating segments of length m is E_m, including E_0 = 1.
- domain assumption Definitions of min-max and max-min permutations and the sequences Eր and Eտ follow Heneghan-Petersen [2].
Cite this review
Pith. "Pith review of A new refinement of Euler numbers on counting alternating permutations." pith.science (2026). https://pith.science/paper/CBWIABFS
@misc{pith2026190800701,
author = {Pith},
title = {Pith review of: A new refinement of Euler numbers on counting alternating permutations},
year = {2026},
howpublished = {\url{https://pith.science/paper/CBWIABFS}},
note = {Machine review of arXiv:1908.00701}
}
read the original abstract
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer the combinatorial question on some particular relation of Euler numbers proved by Heneghan-Petersen, Power series for up-down min-max permutations, College Math. Journal, Vol. 45, No. 2 (2014), 83-91.
Reference graph
Works this paper leans on
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[1]
Andr´ e, D´ eveloppement de secx et tan x, C. R. Math. Acad. Sci. Paris 88 (1879), 965-967
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[2]
Heneghan-Petersen, Power series for up-down min-max permu tations, College Math. J., Vol. 45 2 (2014), 83-91
work page 2014
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[3]
Department of Engineering, Kanagawa University, 3-27-1 Ro kkaku-bashi, Yoko- hama 221-8686, Japan
Wilf, generatingfunctionology, 3rd ed., A K Peters, Wellesley MA, 2 006. Department of Engineering, Kanagawa University, 3-27-1 Ro kkaku-bashi, Yoko- hama 221-8686, Japan
Reviewed August 14, 2026 · model on record in the stance chip above.
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