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arxiv: 2605.15415 · v1 · pith:MBJQSZ53new · submitted 2026-05-14 · 🧮 math.CO

Boolean--Eulerian numbers

Mikl\'os B\'ona , Vincent Vatter This is my paper

Pith reviewed 2026-05-19 15:03 UTC · model grok-4.3

classification 🧮 math.CO
keywords decreasing binary treesEuler numbersalternating permutationsordered set partitionsgamma-positivityBoolean-Eulerian polynomialsreal-rooted polynomials
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The pith

The number of red- or blue-colored decreasing binary trees equals 2^{n-1} times the nth Euler number.

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

The paper examines decreasing binary trees where each vertex with two children is colored red or blue. It constructs two bijections, one linking the trees to ordered set partitions into odd-sized blocks arranged as alternating permutations and the other to nonplane decreasing 1-2 trees with binary labels on non-root vertices. These establish that the trees are counted by 2^{n-1} times the nth Euler number with exponential generating function 1/(1 - tan z). Refining the count by the number of right edges defines Boolean-Eulerian polynomials as an explicit algebraic transform of the classical Eulerian polynomials. The model recasts the Foata-Strehl orbit decomposition to give a combinatorial proof of gamma-positivity while the transform carries real-rootedness and zero interlacing from the Eulerian polynomials.

Core claim

Decreasing binary trees with red or blue colors on each vertex with two children are enumerated by 2^{n-1} times the nth Euler number, as shown by a bijection to nonplane decreasing 1-2 trees with binary labels on non-root vertices. The exponential generating function is 1/(1 - tan z) via a bijection to ordered set partitions into odd blocks each forming an alternating permutation. When refined by the number of right edges the resulting Boolean-Eulerian polynomials arise as an algebraic transform of the Eulerian polynomials. The Foata-Strehl orbit decomposition recast in the decreasing-binary-tree model gives a direct combinatorial proof of gamma-positivity, and the algebraic transform also

What carries the argument

The decreasing binary tree model with red or blue colors on vertices that have two children, refined by the number of right edges, which carries the two bijections and supports the algebraic transform.

If this is right

  • The exponential generating function of the colored trees is 1/(1 - tan z).
  • The Boolean-Eulerian polynomials arise from an explicit algebraic transform of the classical Eulerian polynomials.
  • Gamma-positivity of the Boolean-Eulerian polynomials admits a direct combinatorial proof via the Foata-Strehl orbit decomposition in the tree model.
  • The algebraic transform preserves real-rootedness and interlacing of zeros from the Eulerian polynomials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The decreasing binary tree model with vertex colors could be adapted to supply combinatorial proofs of positivity for other families of polynomials arising from permutation statistics.
  • The algebraic transform between the two polynomial families may extend to additional root-related properties or to refined statistics not considered in the paper.
  • Recasting orbit decompositions in tree models might connect to similar positivity questions in nearby combinatorial enumerations such as other labeled tree families.

Load-bearing premise

The two constructed bijections correctly match the colored trees to the target objects while preserving all required statistics including odd block sizes, alternating permutations, right edges, and binary labels.

What would settle it

Directly enumerating the red-blue colored decreasing binary trees for a small n such as n=5 and checking whether the total equals 2^{4} times the 5th Euler number.

Figures

Figures reproduced from arXiv: 2605.15415 by Mikl\'os B\'ona, Vincent Vatter.

Figure 1
Figure 1. Figure 1: The permutation p = 1653724 and its decreasing binary tree T(p). The peaks of p are at positions 2 and 5, with values 6 and 7; these correspond to the shaded vertices of T(p), namely the vertices with two children. The descents of p are at positions 2, 3, 5, corresponding to the right edges of T(p). Binary plane trees on n unlabeled vertices are counted by the Catalan numbers, and refining by the number of… view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of Theorem 3.1, with red for 0 and blue for 1. The bicolored decreasing binary tree on the left has its full vertices (7, 6) colored. On the right is the corresponding pair (Q, ε): the same vertex set drawn as a nonplane decreasing 1-2 tree, with each non-root vertex carrying a binary label. The label on the the child with smaller label among the children of its parent records the left-righ… view at source ↗
read the original abstract

We study decreasing binary trees in which every vertex with two children is colored red or blue. We construct two bijections. The first, to ordered set partitions into odd-sized blocks each arranged as an alternating permutation, shows that the exponential generating function of these trees is $1/(1-\tan z)$. The second, to nonplane decreasing 1-2 trees paired with a binary label on each non-root vertex, proves combinatorially that the count equals $2^{n-1}$ times the~$n$th Euler number. Refining by the number of right edges yields the Boolean--Eulerian polynomials, which are an explicit algebraic transform of the classical Eulerian polynomials. The Foata--Strehl orbit decomposition, recast in the decreasing-binary-tree model, gives a direct combinatorial proof of gamma-positivity, and the algebraic transform carries real-rootedness and interlacing of zeros from the Eulerian polynomials to the Boolean--Eulerian polynomials.

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.

Referee Report

2 major / 1 minor

Summary. The paper introduces Boolean-Eulerian numbers via colored decreasing binary trees (red/blue on binary vertices). It constructs two explicit bijections: the first maps to ordered odd-block partitions whose blocks are alternating permutations, yielding EGF 1/(1-tan z); the second maps to nonplane decreasing 1-2 trees with binary labels on non-root vertices, proving the count equals 2^{n-1} times the nth Euler number. Refining by right edges produces Boolean-Eulerian polynomials as an algebraic transform of the classical Eulerian polynomials. The Foata-Strehl orbit decomposition is recast in the tree model to give a combinatorial proof of gamma-positivity, and the transform is shown to carry real-rootedness and zero interlacing.

Significance. If the bijections hold, the work supplies new combinatorial models connecting colored binary trees to alternating permutations and Euler numbers, together with a polynomial refinement and a direct combinatorial proof of gamma-positivity. The algebraic transfer of real-rootedness and interlacing is a clear strength, as is the explicit, machine-checkable nature of the claimed bijections.

major comments (2)
  1. [Abstract / first bijection description] Abstract and the paragraphs describing the first bijection: the mapping from colored decreasing binary trees to ordered odd-block partitions with alternating-permutation blocks is asserted to be bijective and to preserve the required statistics, but the manuscript must supply the explicit forward and inverse mappings (including how colors and right edges are tracked) to confirm surjectivity and the EGF identity.
  2. [Abstract / second bijection description] Abstract and the paragraphs describing the second bijection: the claim that this bijection proves the count equals 2^{n-1} E_n and refines correctly by right edges (for the Boolean-Eulerian polynomials) rests on the mapping to nonplane decreasing 1-2 trees with binary labels preserving the right-edge statistic; an explicit verification of injectivity, surjectivity, and statistic preservation is load-bearing for all subsequent polynomial and positivity results.
minor comments (1)
  1. [Introduction / polynomial definition] The algebraic transform relating Boolean-Eulerian polynomials to Eulerian polynomials is stated but would benefit from an explicit formula or generating-function equation displayed early in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on the bijections. We address the major comments point by point below and will revise the manuscript to incorporate additional explicit details where helpful.

read point-by-point responses

  1. Referee: [Abstract / first bijection description] Abstract and the paragraphs describing the first bijection: the mapping from colored decreasing binary trees to ordered odd-block partitions with alternating-permutation blocks is asserted to be bijective and to preserve the required statistics, but the manuscript must supply the explicit forward and inverse mappings (including how colors and right edges are tracked) to confirm surjectivity and the EGF identity.

    Authors: The manuscript constructs the bijection in Section 2 and verifies it preserves the necessary statistics to obtain the EGF 1/(1-tan z). To address the request for greater explicitness, the revised version will include fully detailed forward and inverse maps, with explicit tracking of vertex colors and right edges, together with a self-contained proof of bijectivity. revision: yes

  2. Referee: [Abstract / second bijection description] Abstract and the paragraphs describing the second bijection: the claim that this bijection proves the count equals 2^{n-1} E_n and refines correctly by right edges (for the Boolean-Eulerian polynomials) rests on the mapping to nonplane decreasing 1-2 trees with binary labels preserving the right-edge statistic; an explicit verification of injectivity, surjectivity, and statistic preservation is load-bearing for all subsequent polynomial and positivity results.

    Authors: Section 3 presents the second bijection and uses it to establish the enumeration 2^{n-1} E_n together with the refinement by right edges. We will expand this section in revision to give a complete, self-contained argument for injectivity and surjectivity, along with a direct check that the right-edge statistic is preserved, thereby reinforcing the foundation for the polynomial and positivity results that follow. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds via explicit bijections and algebraic transform.

full rationale

The paper establishes its central enumerative claims through two explicitly constructed bijections (one to odd-block alternating partitions yielding the EGF 1/(1-tan z), the other to labeled 1-2 trees yielding the 2^{n-1} E_n count) together with a direct algebraic substitution defining the Boolean-Eulerian polynomials from the classical Eulerian polynomials. Gamma-positivity follows from recasting the external Foata-Strehl decomposition inside the decreasing-binary-tree model. None of these steps reduces to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation; the bijections and transform are independent of the target identities and rest on standard external facts about Euler numbers and alternating permutations. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces the new objects (colored decreasing binary trees) and relies on standard combinatorial definitions; no free parameters, no ad-hoc axioms, and no invented physical entities appear.

axioms (1)
  • standard math Standard definitions and known enumerative properties of Euler numbers, alternating permutations, and the Foata-Strehl orbit decomposition hold.
    The abstract invokes these established objects to state the bijections and the gamma-positivity proof.

pith-pipeline@v0.9.0 · 5685 in / 1440 out tokens · 72378 ms · 2026-05-19T15:03:19.108996+00:00 · methodology

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Reference graph

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