An identity for second Eulerian numbers via lattice-point counting

Jack Boncompagni

arxiv: 2605.24586 · v1 · pith:TVFBZ4A7new · submitted 2026-05-23 · 🧮 math.CO

An identity for second Eulerian numbers via lattice-point counting

Jack Boncompagni This is my paper

Pith reviewed 2026-06-30 12:57 UTC · model grok-4.3

classification 🧮 math.CO

keywords Bernoulli numberssecond Eulerian numbersStirling permutationsEhrhart theorylattice-point countingdescent enumerator

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The pith

Lattice-point counting on Stirling permutations yields identities between Bernoulli and second Eulerian numbers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives a geometric proof of two identities that connect the Bernoulli numbers to the second Eulerian numbers. It does so by treating the descent enumerator of Stirling permutations as the lattice-point counting function of a polytope and applying Ehrhart theory to it. A sympathetic reader would care because the argument supplies a direct combinatorial and volumetric reason for algebraic relations that otherwise appear only through generating functions or recurrence relations.

Core claim

The descent enumerator of Stirling permutations, when viewed as the Ehrhart counting function of the associated polytope, satisfies two explicit identities with the Bernoulli numbers that follow immediately from the evaluation properties of that counting function.

What carries the argument

Ehrhart-theoretic lattice-point counting applied to the descent enumerator of Stirling permutations, which directly produces the stated identities with the Bernoulli numbers.

Load-bearing premise

Ferroni's Ehrhart-theoretic idea applies directly to the descent enumerator of Stirling permutations and produces the stated identities with the Bernoulli numbers.

What would settle it

An explicit enumeration of lattice points inside the relevant polytope for small Stirling permutations that fails to match the numerical value predicted by either of the two claimed identities.

Figures

Figures reproduced from arXiv: 2605.24586 by Jack Boncompagni.

**Figure 1.** Figure 1: Comb Poset Cn. 4.1. The Ehrhart polynomial. We want to compute the Ehrhart polynomial EhrPn (k). By definition, for every non-negative integer k, this equals the number of lattice points in the k-th dilate of Pn. That is, EhrPn (k) = #(kPn ∩ Z 2n ) = #{(x1, . . . , xn, y1, . . . , yn) ∈ [0, k] 2n | x1 ≤ · · · ≤ xn, yi ≤ xi ∀i}. Proposition 4.1. For all non-negative integers n, k with k ≥ 1, the Ehrhart pol… view at source ↗

**Figure 2.** Figure 2: Bicomb over two chains: {x < y} and {z} [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 4.** Figure 4: Comb over the antichain {x, y} [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: The Stirling poset P (r) k [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

read the original abstract

The second Eulerian numbers are defined via the descent enumerator of Stirling permutations, a class of permutations introduced by Gessel and Stanley. We give a simple and conceptual proof of two identities relating the Bernoulli numbers and the second Eulerian numbers. We rely on a recent Ehrhart-theoretic idea of Ferroni.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a geometric proof of two second-Eulerian/Bernoulli identities by applying Ferroni's Ehrhart method to Stirling-permutation descents; the value is in the proof technique if the polytope identification holds exactly.

read the letter

The main thing to know is that this short paper supplies a new proof of two identities linking the second Eulerian numbers (from descents on Stirling permutations) to Bernoulli numbers, obtained by feeding the descent enumerator into Ferroni's recent lattice-point counting setup. The identities themselves may already be known in some form, but the contribution is the geometric derivation rather than algebraic manipulation.

The paper does what it sets out to do on its own terms: it takes an external Ehrhart idea and applies it directly to produce the claimed relations. That is a clean move when it works, and it avoids reinventing the algebraic wheel for these particular quantities.

The soft spot is exactly the one flagged in the stress-test note. The load-bearing step is showing that the descent statistic on Stirling permutations corresponds to the lattice-point enumerator of the specific polytope (or alcove) in Ferroni's framework without extra weighting, a non-standard triangulation, or additional lemmas that are not automatic from the general method. The abstract states the claim cleanly, but the full argument must be checked for that precise match; if the correspondence is off by even a small factor, the identities would not follow directly. No other red flags appear in the setup or citations.

This is a standard enumerative-combinatorics paper aimed at readers who care about generating functions for permutation statistics and geometric proofs via Ehrhart theory. Someone working on similar identities or looking for extensions of Ferroni's technique would get concrete value from it. It is coherent enough on its own terms to deserve a serious referee who can verify the polytope details and the resulting identities.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to give a simple and conceptual proof of two identities relating the Bernoulli numbers and the second Eulerian numbers (defined via the descent enumerator of Stirling permutations) by applying a recent Ehrhart-theoretic idea of Ferroni.

Significance. If the central identification holds, the result supplies a geometric derivation of the identities that avoids direct combinatorial manipulation, potentially strengthening connections between Ehrhart theory, lattice-point enumeration, and permutation statistics; the use of an external technique is credited explicitly as the source of the conceptual simplicity.

major comments (1)

[abstract, paragraph 2] The load-bearing step is the precise matching of the descent statistic on Stirling permutations to the lattice-point enumerator of the polytope (or alcove) in Ferroni's construction; without an explicit definition of this polytope and a verification that the generating function extracts exactly the second Eulerian numbers (rather than a weighted or triangulated variant), the claimed identities rest on an unverified correspondence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for greater explicitness in the central correspondence. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [abstract, paragraph 2] The load-bearing step is the precise matching of the descent statistic on Stirling permutations to the lattice-point enumerator of the polytope (or alcove) in Ferroni's construction; without an explicit definition of this polytope and a verification that the generating function extracts exactly the second Eulerian numbers (rather than a weighted or triangulated variant), the claimed identities rest on an unverified correspondence.

Authors: We agree that the manuscript would benefit from an explicit recall of Ferroni's polytope and a direct verification of the correspondence. The original submission relies on the reader consulting Ferroni's construction for the alcove and the lattice-point count, but does not reproduce the definition or the bijection with Stirling permutations. In the revised version we will insert a new subsection (likely after the introduction) that: (i) states the precise polytope/alcove used, (ii) recalls the lattice-point enumerator, and (iii) proves that this enumerator equals the descent generating function on Stirling permutations, thereby confirming that the second Eulerian numbers appear exactly (with no extra weights or triangulation factors). This addition will make the load-bearing step self-contained while preserving the conceptual simplicity of the argument. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external Ehrhart theory from Ferroni to descent enumerator without self-referential reduction or fitted inputs.

full rationale

The paper presents the identities as following directly from applying Ferroni's Ehrhart-theoretic construction to the descent enumerator of Stirling permutations, with the Bernoulli numbers recovered via the standard Ehrhart-Bernoulli relation. This is an external geometric technique cited from another author, not a self-citation chain, self-definition, or renaming of fitted quantities. The central step is the identification of the relevant polytope whose lattice-point count extracts the descent generating function, but the paper frames this as a direct application rather than a redefinition or internal fit. No load-bearing premise reduces to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a proof that invokes standard Ehrhart theory and the definition of second Eulerian numbers; no free parameters, ad-hoc axioms, or new entities are indicated in the abstract.

axioms (1)

standard math Ehrhart theory supplies a polynomial whose value at positive integers counts lattice points in a dilated polytope, and this polynomial can be related to descent statistics on permutations.
The proof relies on Ferroni's recent application of this theory to permutation enumeration.

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

MR 3410115 [Fer26] Luis Ferroni,From Eulerian to Bernoulli numbers via Ehrhart polynomials, Amer

[BR15] Matthias Beck and Sinai Robins,Computing the continuous discretely, second ed., Undergraduate Texts in Mathematics, Springer, New York, 2015, Available at: https: //doi.org/10.1007/978-1-4939-2969-6. MR 3410115 [Fer26] Luis Ferroni,From Eulerian to Bernoulli numbers via Ehrhart polynomials, Amer. Math. Monthly (2026), to appear. [Fu21] Amy M. Fu,So...

work page doi:10.1007/978-1-4939-2969-6 2015
[2]

[Par94] SeungKyung Park, P -partitions and q-Stirling numbers, J. Combin. Theory Ser. A68 (1994), no. 1, 33–52, Available at https://doi.org/10.1016/0097-3165(94)90090-6. MR 1295782 [Sta86] Richard P. Stanley,Two poset polytopes, Discrete Comput. Geom.1(1986), no. 1, 9–23, Available athttps://doi.org/10.1007/BF02187680. MR 824105 [Sta12] ,Enumerative comb...

work page doi:10.1016/0097-3165(94)90090-6 1994

[1] [1]

MR 3410115 [Fer26] Luis Ferroni,From Eulerian to Bernoulli numbers via Ehrhart polynomials, Amer

[BR15] Matthias Beck and Sinai Robins,Computing the continuous discretely, second ed., Undergraduate Texts in Mathematics, Springer, New York, 2015, Available at: https: //doi.org/10.1007/978-1-4939-2969-6. MR 3410115 [Fer26] Luis Ferroni,From Eulerian to Bernoulli numbers via Ehrhart polynomials, Amer. Math. Monthly (2026), to appear. [Fu21] Amy M. Fu,So...

work page doi:10.1007/978-1-4939-2969-6 2015

[2] [2]

[Par94] SeungKyung Park, P -partitions and q-Stirling numbers, J. Combin. Theory Ser. A68 (1994), no. 1, 33–52, Available at https://doi.org/10.1016/0097-3165(94)90090-6. MR 1295782 [Sta86] Richard P. Stanley,Two poset polytopes, Discrete Comput. Geom.1(1986), no. 1, 9–23, Available athttps://doi.org/10.1007/BF02187680. MR 824105 [Sta12] ,Enumerative comb...

work page doi:10.1016/0097-3165(94)90090-6 1994