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An identity relating Catalan numbers to tangent numbers with arithmetic applications

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Pith's one-line read The paper proves a Catalan-tangent identity via the generating function tanh(x), derives identity (1.3), and uses it plus a q-analog to prove Genocchi and q-tangent divisibility.

desk verdict Proves the Aliniaeifard–Li conjecture with a new Catalan–tangent identity and q-analog; the arithmetic application is true but the printed induction has a repairable sign error. read the letter →

arxiv 2507.20965 v2 pith:N7BJQYGC submitted 2025-07-28 math.CO math.NT

Tongyuan Zhao , Zhicong Lin , Yongchun Zang This is my paper · ORCID
classification math.COmath.NT MSC 05A0505A1911B65
keywords CatalannumberstangentGenocchiq-tangentoddsetcompositionsdivisibilitygeneratingfunctionsq-analogidentities
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

This paper proves a conjecture of Aliniaeifard and Li that connects two classical combinatorial families: for odd n, a signed sum over odd set compositions of {1,...,n} weighted by Catalan numbers equals the tangent number E_n. The proof runs through exponential generating functions: both sides of the identity are shown to have generating function tanh(x). From this the authors extract a second identity involving only tangent numbers and powers of 2, and use it to prove by induction that (n+1)E_{2n+1} is divisible by $2^{{2n}}$ with an odd quotient, the classical Genocchi-number fact. They also find a q-analog of the tangent-number identity and use it to prove Foata's divisibility property of q-tangent numbers, answering Schützenberger's problem.

What carries the argument

The load-bearing objects are the signed Catalan numbers \mu_\ell = (-1)^{\ell/2-1} C_{\ell/2-1} and the exponential generating function tanh(x). Lemma 5.1 packages the odd set compositions: \sum_{n,k} a_{n,k} y^k x^n/n! = y\$\sinh$(x)/(1 - y\$\sinh$(x)); substituting y=2 and inserting the Catalan generating function turns the left side of (1.1) into tanh(x). For the q-analog, the machinery is the set identity U_{2n} = S_0 - S_1 + S_2 - \cdots expressing unimodal permutations as an alternating sum of sets whose parts are a unimodal prefix and an alternating suffix; the q-binomial theorem and the q-tangent interpretation supply the weighted counts.

What would settle it

Check equation (4.1) at n=2: the left side is $2E_1 = 2$ and the right side is $4 - \binom{4}{1}2^1 E_1 = -4$, so the printed rewrite fails; inserting $(-1)^{n-1}$ fixes it, and the corrected equation reproduces the n=2 case of (1.3).

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Extended reading notes

Core claim

The paper's central claim is identity (1.1): for odd n, \sum_{\phi \models [n], \phi \text{ odd}} $2^{{n-\ell(\phi)}}$ \mu_{\ell(\phi)+1} = (-1)^{(n-1)/2} E_n, where \mu_\ell are signed Catalan numbers. It is proved by computing the exponential generating function of the left side via the compositional formula, which gives tanh(x), and matching the known generating function of the odd-index Euler numbers. A direct corollary, identity (1.3), is \sum_{k=0}^{n-1} \binom{2n}{2k+1} $2^{{2n-2k}}$ (-1)^k E_{2k+1} = $2^{{2n+1}}$. The paper presents a purely combinatorial proof of (1.3) by decomposing permutations into a unimodal prefix and an alternating suffix, and this proof produces the q-analog (2.2); the q-version is then used to prove Foata's divisibility theorem for q-tangent numbers. An analogous secant identity (6.2) and its q-analog (6.4) are also proved by the same combinatorial decomposition.

Load-bearing premise

The divisibility proof of Corollary 1.4 depends on equation (4.1) being a correct algebraic rewrite of identity (1.3), but as printed it omits the factor $(-1)^{n-1}$ and fails for even $n$: at $n=2$ it says $2 = -4$.

Editorial extensions

If this is right

  • Conjecture 1.1 is settled: signed Catalan numbers are the Möbius coefficients connecting odd set compositions to tangent numbers in the peak-algebra setting.
  • Identity (1.3) gives an elementary induction proving that (n+1)E_{2n+1} is divisible by 2^{2n} and that the quotient G_{2n+2} is odd.
  • The q-analog (2.2) provides a new proof of Foata's divisibility of q-tangent numbers, addressing the problem Schützenberger raised.
  • The same decomposition yields a secant analog (6.2) and its q-analog (6.4), extending the Catalan-tangent connection to Euler (secant) numbers.

Reading between the lines

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

  • Editorial inference: the tanh(x) generating-function route suggests the identity is one member of a family obtained by replacing the Catalan series with other algebraic generating functions; testing coefficients of tanh^m(x) would give similar signed-count identities.
  • Editorial inference: the proof of (2.2) via an alternating sum of permutation sets is close to an involution proof; a sign-reversing involution on the union of the sets S_k would give a bijective proof of (1.1), which the authors note is still missing.
  • Editorial inference: the secant identities (6.2) and (6.4) may imply arithmetic divisibility or parity facts about Euler secant numbers analogous to the Genocchi result, though the paper does not pursue that direction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper proves a conjecture of Aliniaeifard and Li relating signed Catalan numbers summed over odd set compositions to tangent numbers. The main identity (1.1) is proved in Section 5 by showing that both sides have exponential generating function tanh(x). From (1.1) the paper derives identity (1.3) for tangent numbers and then proves a q-analog, Theorem 2.1, by a combinatorial argument using unimodal and alternating permutations. The q-analog is applied in Section 3 to give an alternative proof of Foata's divisibility theorem for q-tangent numbers. Sections 4 and 6 contain arithmetic applications, in particular Corollary 1.4 on the divisibility of (n+1)E_{2n+1} by 2^{2n} and the oddness of the Genocchi quotient, as well as a secant analog of the main identity.

Significance. The paper contains several attractive and mostly self-contained arguments. The generating-function proof of Conjecture 1.1 is clean, and the combinatorial proof of the q-identity (2.2) is a genuine contribution that also yields (1.3) at q=1. The application to Foata's divisibility theorem is a useful simplification of an existing result. However, the proof of Corollary 1.4, which is presented as a key arithmetic application, contains a sign error in the rewriting of (1.3); as printed, the induction in Section 4 is invalid. The error is local and easily repairable, so the main mathematical claims are likely correct, but the submitted text does not establish the arithmetic application as written.

major comments (2)
  1. [§4, Eq. (4.1)] The rewrite of identity (1.3) omits the factor (-1)^{n-1}. Isolating the k=n-1 term in (1.3) gives 8n(-1)^{n-1}E_{2n-1} on the left, so after dividing by 8 the equation should read n(-1)^{n-1}E_{2n-1} = 2^{2n-2} - sum_{k=0}^{n-2} binom(2n,2k+1)2^{2n-2k-3}(-1)^k E_{2k+1}. As printed, (4.1) is false for every even n; for example n=2 gives 4 = -4. The subsequent induction for the divisibility of nE_{2n-1} therefore does not follow from the written equation, and the derivation must be corrected.
  2. [§4, Eq. (4.3)] Equation (4.3) inherits the same missing sign and should read (-1)^{n-1}G_{2n} = 1 - sum_{k=0}^{n-2} (1/(2n+1)) binom(2n+1,2k+2)(-1)^k G_{2k+2}. The parity conclusion is unaffected because (-1)^{n-1} is congruent to 1 modulo 2, so the proof is repairable, but the printed equation is false for even n and the parity induction is not valid as stated.
minor comments (4)
  1. [§2, around (2.6)] The set identity T_{i-1}=S_i - T_i is stated without explanation. Since the alternating decomposition is central to the proof of Theorem 2.1, a sentence describing the bijection or the reason for the equality would improve readability.
  2. [§1–§6 (typography)] There are several typographical artifacts: the symbol [n] is rendered as 'rns' in multiple places, reference [20] spells 'Cambriage' for 'Cambridge', and 'Oberwohlfach' should be 'Oberwolfach'.
  3. [§2, Theorem 2.1] The proof of Corollary 1.3 cites [1, Theorem 11.5], but Theorem 2.1 already gives (1.3) at q=1; the paper should state this explicitly so that the main identity (1.3) does not appear to depend on an external preprint.
  4. [§3, first paragraph] The opening sentence of Section 3 says 'Since by Corollary 1.4 we have ...' but the proof of Theorem 3.1 does not use Corollary 1.4. Clarifying that Corollary 1.4 is only motivation would remove any appearance of circular dependence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Catalan–tangent identity, its q-analog, and the Foata divisibility proof are derived from independent generating-function and combinatorial arguments.

full rationale

The paper's central derivation chain is self-contained. Conjecture 1.1 (identity (1.1)) is proved in Section 5 by computing both sides' exponential generating functions: the LHS becomes tanh x via the Catalan generating function and the compositional formula for odd set compositions (Lemma 5.1), while the RHS is tanh x by the standard tangent-number EGF. Identity (1.3) is derived from (1.1) using an external theorem of Aliniaeifard and Li and is also given a fully combinatorial proof as the q=1 case of Theorem 2.1, whose proof uses only the q-binomial theorem, Lemma 2.2, and the permutation interpretation of q-tangent numbers. Foata's divisibility theorem is an external result proved in [8]; the present alternative proof uses (2.2) plus two external lemmas ([8, Lemma 2.2] and [13, Theorem 4.1]). Although [13] shares an author with this paper, the cited lemma is a general characterization of the polynomial B_n(q) that is proved independently there and does not assume Foata's theorem or the target identities, so the self-citation is not load-bearing. The proof of Corollary 1.4 does contain an apparent sign error in (4.1), but that is a correctness issue, not circular reasoning. No step reduces a prediction to its input by construction.

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

The central claims rest on standard combinatorial background: exponential generating functions for ordered set partitions, the q-binomial theorem, the inversion-number interpretation of q-tangent numbers, the Catalan generating function, and two external lemmas (Foata's Lemma 2.2 and Lin et al.'s Lemma 3.3). No ad hoc parameters or entities are introduced.

assumptions (6)
  • standard math Exponential formula for ordered set partitions with block weight (Lemma 5.1)
    Used in Section 5 to write the generating function for odd set compositions as y sinh(x)/(1 - y sinh(x)).
  • standard math q-binomial theorem (Stanley, Enumerative Combinatorics, Vol. 1, Prop 1.3.17)
    Used in Lemma 2.2 to evaluate the inversion generating function of unimodal permutations.
  • standard math Combinatorial interpretation of q-tangent numbers as inversion sums over alternating permutations (equation 2.1)
    This theorem from Foata-Han and Stanley is used in the proof of Theorem 2.1 and in Section 3.
  • standard math Catalan generating function identity sum_k C_k x^k = (1 - sqrt(1 - 4x))/(2x)
    Used in Section 5 to evaluate the generating function of the signed Catalan numbers.
  • standard math Foata's Lemma 2.2 on divisibility of q-binomial coefficients by products of Ev polynomials
    External lemma from Foata's paper invoked in the proof of Theorem 3.1 to distribute divisors across q-binomial factors.
  • standard math Lemma 3.3 (Lin, Ma, Wang, Wang) characterizing divisibility by B_n(q) through (1+q^m)^{floor(n/2m)}
    Published theorem used in Lemma 3.4 to show B_{2n}(q) divides (-q; q)_{2n-1}.

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Pith. "Pith review of An identity relating Catalan numbers to tangent numbers with arithmetic applications." pith.science (2026). https://pith.science/paper/N7BJQYGC

@misc{pith2026250720965,
  author       = {Pith},
  title        = {Pith review of: An identity relating Catalan numbers to tangent numbers with arithmetic applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N7BJQYGC}},
  note         = {Machine review of arXiv:2507.20965}
}
abstract

We prove a combinatorial identity relating Catalan numbers to tangent numbers arising from the study of peak algebra that was conjectured by Aliniaeifard and Li. This identity leads to the discovery of the intriguing identity $$ \sum_{k=0}^{n-1}{2n\choose 2k+1}2^{2n-2k}(-1)^{k}E_{2k+1}=2^{2n+1}, $$ where $E_{2k+1}$ denote the tangent numbers. Interestingly, the latter identity can be applied to prove that $(n + 1)E_{2n+1}$ is divisible by $2^{2n}$ and the quotient is an odd number, a fact whose traditional proofs require significant calculations. Moreover, we find a natural $q$-analog of the latter identity with a combinatorial proof. This $q$-identity can be applied to prove Foata's divisibility property of the $q$-tangent numbers, which responds to a problem raised by Sch\"utzenberger.

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Reference graph

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