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Combinatorics on bi-$\gamma$-positivity of $1/k$-Eulerian polynomials

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Pith's one-line read A family of ordered labeled forests counts the bi-gamma coefficients of every $1/k$-Eulerian polynomial, giving the first purely combinatorial interpretation for all $k$.

desk verdict A promising forest model that is likely correct, but the printed proof omits Proposition 5.11—the bijection carrying the b-side of Theorem 2.4—so the main theorem is conditional as written. read the letter →

arxiv 2501.12055 v1 pith:W5XCDYKH submitted 2025-01-21 math.CO

Sherry H.F. Yan , Xubo Yang , Zhicong Lin This is my paper · ORCID
classification math.CO MSC 05A0505A1905C0505E18
keywords Stirlingpermutationsgamma-positivitybi-gamma-positivity1/k-Eulerianpolynomialsincreasingprunedevenk-aryforestssymmetricdecompositioninvolutionactionontreesorderedlabeled
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

The paper claims to give the first combinatorial interpretation, valid for every positive integer $k$, of the bi-$\gamma$-coefficients of the $1/k$-Eulerian polynomials $A_n^{(k)}(x)$, which at $k=1$ reduce to the classical Eulerian polynomials. Bi-$\gamma$-positivity of these polynomials was known by algebraic methods, but the coefficients themselves had no combinatorial meaning. The new content is a model: the two summands in the symmetric decomposition of $A_n^{(k)}(x)$ are written as weighted sums over increasing pruned even $k$-ary forests, and the $\gamma$-expansion is obtained by counting forests with a single distinguished statistic, the number of old leaves. A reader would care because combinatorial interpretations of $\gamma$-coefficients turn abstract positivity statements into concrete enumerative meaning and typically open the door to refined structural properties such as unimodality and real-rootedness.

What carries the argument

The central objects are increasing pruned even $k$-ary forests: ordered forests whose trees have even-level nodes of degree $k$, pruned so no even-level node has only leaf children, with labels increasing along root-to-leaf paths and left-to-right among children. In such a forest a labeled leaf is old if it has the largest label among all grandchildren of its grandparent, and young otherwise; a singleton is a one-node tree. The paper's machinery consists of three pieces: a bijection from $k$-Stirling permutations to these forests matching the longest-ascent-plateau statistic to the number of labeled leaves minus singletons; a generalized involution action on trees that swaps an old internal node with a young leaf and leaves the rest of the tree unchanged, so each orbit has exactly one representative with no young leaves; and two forest transformations that re-root singletons and remove or insert removable leaves, which the authors use to convert the weighted forest sum into the claimed $\gamma$-expansion.

What would settle it

Enumerate, for a small case such as $n=4$ and $k=3$, all forests in $\mathcal{F}_n(k)$ and $\widehat{\mathcal{F}}_n(k)$, compute the weighted sums $\sum_F x^{\mathrm{lleaf}(F)-\mathrm{si}(F)}$ on each family, and compare them with the symmetric decomposition of $A_4^{(3)}(x)$ obtained from the known recurrence or from the ascent-plateau formula; any mismatch refutes Theorem 2.4. To target the omitted step directly, list $\widehat{Y}_n(k)$ and $\widehat{\mathcal{F}}_n(k)$ for the same small case and verify that the maps $\Gamma$ and $\Gamma'$ are mutual inverses on those lists.

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Extended reading notes

Core claim

The central claim is Theorem 2.4. It states that if $A_n^{(k)}(x)=a_n^{(k)}(x)+xb_n^{(k)}(x)$ is the symmetric decomposition, then $a_n^{(k)}(x)=\sum_{F\in\mathcal{F}_n(k)}x^{\mathrm{lleaf}(F)-\mathrm{si}(F)}$ expands as $\sum_i\gamma_{n,k,i}x^i(1+x)^{n-1-2i}$, and $xb_n^{(k)}(x)=\sum_{F\in\widehat{\mathcal{F}}_n(k)}x^{\mathrm{lleaf}(F)-\mathrm{si}(F)}$ expands as $\sum_i\widehat{\gamma}_{n,k,i}x^i(1+x)^{n-2i}$. Here $\gamma_{n,k,i}$ and $\widehat{\gamma}_{n,k,i}$ count forests in the respective families with exactly $i$ old leaves and with no young leaves and no removable old leaves. The proof constructs a bijection between $k$-Stirling permutations and these forests that sends the ascent-plateau statistic to a leaf statistic, introduces an involution action on the trees that collapses each orbit to a unique forest without young leaves, and then applies two transformations that move singletons and removable leaves in a weight-preserving way. The result is the first purely combinatorial description of the bi-$\gamma$-coefficients for all $k$.

Load-bearing premise

The load-bearing premise is that the unproved bijection between the auxiliary pairs $\widehat{Y}_n(k)$ and the forests $\widehat{\mathcal{F}}_n(k)$ (Proposition 5.11) holds exactly as stated; the paper says it follows by the same reasoning as an earlier proved case and omits the details. If that bijection fails, the claimed combinatorial interpretation of the coefficients $\widehat{\gamma}_{n,k,i}$ for the $xb_n^{(k)}(x)$ summand is unsupported.

Editorial extensions

If this is right

  • The bi-$\gamma$-coefficients of every $1/k$-Eulerian polynomial are nonnegative integers with an explicit enumerative meaning, rather than merely an existence statement from algebra.
  • Specializing $k=1$ gives a forest model for the bi-$\gamma$-expansion of the classical Eulerian polynomials.
  • The generalized involution action gives a new $\gamma$-positivity proof for the longest ascent-plateau polynomials over $k$-Stirling permutations starting with 1, independently of the bi-$\gamma$ statement.
  • Because bi-$\gamma$-positivity implies the alternating-increasing property, the forest model gives a direct combinatorial witness to that structural property of these polynomials.
  • The two forest transformations provide a template that may apply to other polynomials whose symmetric decompositions are known.

Reading between the lines

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

  • A natural extension the authors do not pursue is a $q$-refinement tracking the labels of old leaves or internal nodes, which would give a bivariate refinement of the bi-$\gamma$ expansions and might connect to known $q$-Eulerian polynomials.
  • Because the omitted proof of Proposition 5.11 is the only step supporting the $\widehat{\gamma}$-side, a computer check for small $n$ and $k$ would be a cheap way to validate the second half of the theorem before relying on it.
  • The same pruning and leaf statistics might transfer to multiset or colored Stirling permutations, where analogous bi-$\gamma$ questions are open.
  • The generalized involution action could be profitably compared with descent-set cyclic sieving phenomena, suggesting the forest model may carry finer homological or representation-theoretic meaning.
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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

4 major / 5 minor

Summary. The paper studies the 1/k-Eulerian polynomials A_n^{(k)}(x) of Savage and Viswanathan. Its main result, Theorem 2.4, asserts that in the symmetric decomposition A_n^{(k)}(x)=a_n^{(k)}(x)+x b_n^{(k)}(x), the polynomials a_n^{(k)}(x) and x b_n^{(k)}(x) have γ-expansions whose coefficients count increasing pruned even k-ary forests with a prescribed number of old leaves and no young leaves or removable old leaves. The proof proceeds by bijecting k-Stirling permutations to such forests (Section 3), introducing a generalized Foata–Strehl action on the trees (Section 4), and then constructing two transformations Θ and Γ on forests that are meant to prove the γ-expansions (Section 5).

Significance. If Theorem 2.4 is correct, it gives the first combinatorial interpretation of the bi-γ-coefficients of A_n^{(k)}(x) for all k, resolving an open problem noted in the paper. The forest model is natural, the γ-coefficients are defined by explicit forest statistics rather than fitted algebraically, and Theorem 4.1 is a useful standalone γ-positivity result. The paper also credits prior work accurately. The main caveat is proof completeness: the b-side expansion depends on Proposition 5.11, whose proof is omitted.

major comments (4)
  1. [Section 5.2, Proposition 5.11] This proposition states that Γ and Γ′ induce a bijection between ̂Y_n(k) and ̂F_n(k), but the proof is omitted: the text says it 'can be verified by the same reasoning as in the proof of Proposition 5.10'. This is load-bearing: Proposition 5.13 composes this bijection with Proposition 5.6 to prove (5.4), the b-side of Theorem 2.4. In the hat case the rightmost tree does not have its first k−1 children as leaves, so the behavior of Ψ_x in case (ii) differs and the order of removable leaves under β must be re-verified. A full proof is required.
  2. [Section 5.2, Proposition 5.10] The inverse identities Ψ_y(Ψ_x(F′))=F′ are asserted as 'routine to check' twice, once in each direction of the inverse proof. These identities are not local formalities: Ψ_x has four cases and β chooses the smallest removable leaf, so the proof must establish that the singleton produced by α and the removable leaf chosen by β have the correct relative order. The same identities are needed in the omitted Proposition 5.11. Please write out the case analysis or provide a precise invariant that makes the cancellation immediate.
  3. [Section 5.1, Proposition 5.6] The proof that Θ′ maps ̂Y_n(k) into ̂X_n(k) is skeletal. The assertion 'Since F does not contain any removable young leaves, we have Φ_S2(F)∈̂F_n(k)' and the final 'similar arguments' for rleaf(F′)=0 hide exactly the verification that the hat condition is preserved. Because this proposition is one of the two ingredients of Proposition 5.13, the b-side expansion is not fully supported without a detailed argument.
  4. [Section 5.2, Observation 5.9] This observation is stated as 'can be checked routinely', but it is used in Proposition 5.10 to conclude that y is the greatest element of S′∪{y}, which is what allows α to choose the correct singleton on the next step. Please supply a proof; the observation is not a mere remark.
minor comments (5)
  1. [Section 3, Proposition 3.1] Bijectivity of ξ is justified by 'It is apparent that the construction of ξ is reversible'; please state the inverse reconstruction explicitly, since the map is used to establish the first equalities in (2.1) and (2.2).
  2. [Section 5.1, Proposition 5.4] Only the equality Θ′(Θ(F,S))=(F,S) is shown; the reverse equality is delegated to 'similar arguments'. Please include the reverse direction or explain the symmetry.
  3. [Lemma 4.2] Commutativity of Φ_x and Φ_y is asserted after the listed properties; since the orbit method in Theorem 4.1 depends on it, a short proof or a more explicit justification would improve readability.
  4. [Section 5.3, equations (5.5) and (5.9)] The extension of Lemma 4.3 from trees to forests deserves a sentence, because singleton components require separate treatment: a singleton contributes one to n but zero to oint and oleaf, which is exactly why si(F) appears in the forest identity.
  5. [Throughout] The submitted text contains many encoding artifacts (e.g., '1/slash.left k', '/summation.disp', '/parenleft.alt3'), which make the paper difficult to read; please ensure the published version uses a clean TeX rendering.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bi-gamma forest interpretation is derived through explicit bijections and orbit decompositions, with no fitted parameter or self-citation chain doing the work.

full rationale

The derivation chain is not circular. The starting points are external results: (1.2) from Ma-Mansour and Proposition 1.2 from Ma-Ma-Yeh-Yeh identify A_n^(k)(x) with ascent polynomials over k-Stirling permutations and give the symmetric decomposition. The paper then constructs explicit bijections (Propositions 3.1-3.3) between k-Stirling permutations and increasing pruned even k-ary forests, proving the first equalities in (2.1)-(2.2). The gamma expansions are obtained by defining a generalized Foata-Strehl action and two transformations Gamma and Gamma-prime; the gamma coefficients are not fitted parameters but are defined as forest counts and derived from orbit decompositions. The cited prior work by the present author [16] is used only as inspiration or background for the generalized action, not as the target result, and the action is proved independently in Section 4. The only notable weakness is Proposition 5.11, whose proof is omitted with the text saying it 'can be verified by the same reasoning as in the proof of Proposition 5.10 and the proof is omitted here'; an omitted proof is a completeness gap, not a circular reduction. Likewise, the 'routine' inverse identities in Proposition 5.10 are asserted rather than displayed, but an asserted routine check is not equivalence-to-input. No equation is defined in terms of the quantity it purports to derive, and no fitted value is relabeled as a prediction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on imported theorems (the ascent-plateau formula, the symmetric decomposition of A_n^{(k)}, and the general symmetric-decomposition formalism) that the paper does not re-derive. No free parameters or invented physical entities are introduced; the forest definitions are new mathematical objects but are fully specified rather than postulated.

assumptions (3)
  • domain assumption A_n^{(k)}(x) equals both the excedance sum over permutations and the ascent-plateau sum over k-Stirling permutations.
    Identities (1.1) and (1.2) from Savage-Viswanathan [26] and Ma-Mansour [24] are used as the starting point for the forest interpretation.
  • domain assumption The symmetric decomposition of A_n^{(k)} is given by sums over Q_n(k) and hat Q_n(k).
    External result of Ma-Ma-Yeh-Yeh [23] that converts the polynomial decomposition into a statistic sum; the paper's bijections then represent those sums.
  • standard math Every polynomial has a unique symmetric decomposition a(x)+xb(x), and bi-gamma-positivity is equivalent to both pieces being gamma-positive.
    Standard decomposition of a polynomial into symmetric pieces, invoked to set up the gamma expansions in Theorem 2.4.

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Pith. "Pith review of Combinatorics on bi-$\gamma$-positivity of $1/k$-Eulerian polynomials." pith.science (2026). https://pith.science/paper/W5XCDYKH

@misc{pith2026250112055,
  author       = {Pith},
  title        = {Pith review of: Combinatorics on bi-$\gamma$-positivity of $1/k$-Eulerian polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5XCDYKH}},
  note         = {Machine review of arXiv:2501.12055}
}
abstract

The $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ were introduced as ascent polynomials over $k$-inversion sequences by Savage and Viswanathan. The bi-$\gamma$-positivity of the $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ was known but to give a combinatorial interpretation of the corresponding bi-$\gamma$-coefficients still remains open. The study of the theme of bi-$\gamma$-positivities from purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-$\gamma$-coefficients of $A^{(k)}_{n}(x)$ by using the model of certain ordered labeled forests. Our combinatorial approach consists of three main steps: (i) construct a bijection between $k$-Stirling permutations and certain forests that are named increasing pruned even $k$-ary forests; (ii) introduce a generalized Foata--Strehl action on increasing pruned even $k$-ary trees which implies the longest ascent-plateau polynomials over $k$-Stirling permutations with initial letter $1$ are $\gamma$-positive, a result that may have independent interest; (iii) develop two crucial transformations on increasing pruned even $k$-ary forests to conclude our combinatorial interpretation.

Figures

Figures reproduced from arXiv: 2501.12055 by the authors.

Figure 1
Figure 1. A forest F ∈ F10(3). Let us first review some terminology related to trees. A tree is an acyclic connected graph, and a forest is a graph such that every connected component is a tree. An ordered tree is a tree with one designated node, which is called the root, and the subtrees of each node are linearly ordered. In this paper we will assume that all the trees are ordered. In a tree T, the level of a node v in T is … view at source ↗
Figure 2
Figure 2. All 9 forests in F3(2). we simply write TM(k) (resp., FM(k)) as Tn(k) (resp., Fn(k)) for short. In what follows, we will always write a forest F as a sequence (T1, T2, . . . , Tm) for some m, where Ti is the i-th tree of F (counting from left to right) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. All 6 forests of F̂3(2). For the forest in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: An increasing pruned even 3-ary forest (T1, T2, T3). trees in the forest ξ(µj) to the node vj as subtrees from left to right for all j ∈ [k]. Denote by Ti the resulting tree. Set ξ(π) = (T1, T2, . . . , Tm). For example, if we let π = 133377711446664225552888 ∈ Q8(3), …
Figure 5
Figure 5. Figure 5: The transformation Φ4. • If v is an old internal node, then suppose that the node u is the grand parent of v, the nodes µ1, µ2, . . . , µk are all the children of u from left to right, and the nodes ν1, ν2, . . . , νk are all the children of v from left to right. For e…
Figure 6
Figure 6. Figure 6: An example of the transformation ΦS with S = {3, 5}. Given a tree T, we simply write u ∈ T for any node u of T. Definition 5.1 (Removable young leaf) Let F = (T1, T2, . . . , Tm) ∈ Fn(k). A young leaf u ∈ Tm is said to be a removable young leaf of F if u is a grand chi…
Figure 7
Figure 7. Figure 7: An example of a removable young leaf. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: The transformation Ψ2 applying to F1. 5.2 Introducing two transformations on Fn(k) Let F = (T1, T2, . . . , Tm) ∈ Fn(k). Assume that the node u ∈ Ti is labeled by x in F. We define the fundamental transformation Ψx(F) as follows: • If u is a singleton and u ∉ Tm, choos…
Figure 9
Figure 9. Figure 9: The transformation Ψ2 applying to F3. Denote by Ψx(F) the resulting forest. For instance, if we let F1 and F2 be the forests illustrated in [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: An example of the map Γ. 1 3 2 5 4 7 8 6 1 2 5 3 4 7 8 6 F F′ [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: An example of the transformation β. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

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