The free and parking quasi-symmetrizing actions

Adrien Segovia

REVIEW 4 major objections 6 minor 16 references

The free and parking quasi-symmetrizing actions

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Two symmetric-group actions on words have FQSym* and PQSym* as their invariant spaces, and a parameter r interpolates between them as nested Hopf subalgebras.

desk verdict New invariant-theoretic actions yield conditional but credible results; the missing proof of the orbit classification is the main gap. read the letter →

arxiv 2502.07926 v1 pith:HE3JASDU submitted 2025-02-11 math.CO math.RA

Adrien Segovia This is my paper · ORCID

classification math.COmath.RA MSC 05E0516T3005A1505C05

keywords parkingfunctionsfreequasi-symmetricHopfalgebrassymmetricgroupactionsparkizationr-bi-wordsrootedlabeledtrees

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 establishes that two classical Hopf algebras built from words can be defined by symmetric-group actions: the free quasi-symmetrizing action has the elements of $\mathbf{FQSym}^{*}$ as its invariants, and the parking quasi-symmetrizing action has the elements of $\mathbf{PQSym}^{*}$ as its invariants. For the parking action, the orbit of a word is exactly the set of words with the same parkization, so the basis elements of $\mathbf{PQSym}^{*}$ are sums over orbits. The paper then generalizes the parking action by a parameter $r$, swapping adjacent columns of a bi-word only when one column has length smaller than $r$. The invariant spaces form a nested chain of graded Hopf subalgebras $\mathbf{PQSym}^{*}=\mathbf{PQSym}^{*}_{1}\supseteq\mathbf{PQSym}^{*}_{2}\supseteq\cdots\supseteq\mathbf{PQSym}^{*}_{\infty}$, with explicit basis, Hilbert series, and product and coproduct formulas. At $r=\infty$ the dimensions count rooted labeled non-planar trees whose maximal decreasing subtree is a chain, giving a bijective proof of known enumerative formulas.

What carries the argument

The load-bearing encoding is the bijection $\varphi$ between words and bi-words: a two-row array whose top row is a set composition and whose bottom row entries are prime parking functions, with empty columns recording the gap between a word and its parkization. The parking quasi-symmetrizing action swaps adjacent columns when one is empty, and the $r$-action swaps adjacent columns when one has length below $r$. The basis formula expresses $G^r_{(I,\lambda)}$ as the sum of $G_K$ over all shuffles of the large columns $I$ with permutations of the small columns $\lambda$; this orbit-and-basis mechanism carries the Hopf algebra structure and the tree enumeration.

What would settle it

Enumerate the $r=2$ orbits of all words of length 4 by applying the column-swap rule and check whether each orbit contains exactly one canonical r-bi-word; a single orbit containing zero or two such representatives would falsify the orbit classification and with it the basis, Hilbert series, and product and coproduct formulas.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that $\mathbf{FQSym}^{*}$ and $\mathbf{PQSym}^{*}$ are invariant-theoretic objects: an infinite symmetric group acts on words, and the fixed word-sums are precisely the basis elements $G_{\sigma}$ (for standardization) and $G_u$ (for parkization). The parking action is then refined to an $r$-action for $r\in(\mathbb{N}\setminus\{0\})\cup\{\infty\}$; the $r$-invariants, indexed by $r$-bi-words, form nested graded Hopf subalgebras of $\mathbf{PQSym}^{*}$, with $\mathbf{PQSym}^{*}_{1}=\mathbf{PQSym}^{*}$ and $\mathbf{PQSym}^{*}_{\infty}$ cocommutative. In the limit $r=\infty$, the dimensions of homogeneous components coincide with the number of rooted labeled non-planar trees on $[n]$ whose maximal decreasing subtree is a chain, via a bijection through prime parking functions and forests of minimal rooted trees.

Load-bearing premise

The whole r-parameter construction rests on the orbit classification of Proposition 43, stated without proof, that each orbit under the r-parking action contains exactly one r-bi-word, meaning columns shorter than r can be permuted arbitrarily while columns of length at least r keep their relative order.

Editorial extensions

If this is right

$\mathbf{FQSym}^{*}$ and $\mathbf{PQSym}^{*}$ gain invariant-theoretic definitions on words, parallel to the classical description of quasi-symmetric functions as invariants of a quasi-symmetrizing action.
For every $r$, the homogeneous component of degree $n$ of $\mathbf{PQSym}^{*}_r$ has dimension $1+\sum_{k=1}^{n}A^r_{n,k}$, with $A^r_{n,k}$ counting partitions of $[n]$ into $k$ parts weighted by factorials and prime parking function numbers.
Each $\mathbf{PQSym}^{*}_r$ is a Hopf subalgebra of $\mathbf{PQSym}^{*}$, so products and coproducts of invariant elements can be computed inside the chain using the given shuffle formulas.
At $r=\infty$, the dimension of the degree-$n$ component equals the number of rooted labeled non-planar trees on $[n]$ whose maximal decreasing subtree is a chain, yielding a bijective proof of earlier closed formulas for these trees.
$\mathbf{PQSym}^{*}_{\infty}$ is cocommutative while the other members of the chain are noncommutative and noncocommutative, so the chain interpolates between $\mathbf{PQSym}^{*}$ and a cocommutative tree-enumerated algebra.

Reading between the lines

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

If the orbit classification holds, $\mathbf{PQSym}^{*}_{\infty}$ should be understood as the parking analogue of symmetric functions, with a basis indexed by multisets of columns rather than ordered columns; the paper stops short of naming such an analogue.
The same bi-word machinery should adapt to other word-like families equipped with a parkization-type algorithm, producing parameterized Hopf subalgebras whose $r=\infty$ limits are enumerated by forests or trees.
Computing the low-degree Hilbert series for $r=2,3$ by explicit orbit enumeration would test whether the chain's intermediate algebras are genuinely distinct from the endpoints or collapse onto previously known algebras.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The real news here is that the free and parking quasi-symmetrizing actions give invariant-theoretic definitions of FQSym* and PQSym*, and the r-parameter family produces a nested chain of Hopf subalgebras interpolating from PQSym* down to PQSym*_∞. That is a genuine extension of Hivert's r-QSym and r-WQSym constructions, and the r=∞ enumeration gives a new bijective proof of known Seo-Shin and Rattan counts. The main theorems (19 and 34) are well supported by the explicit bijections f/g and φ, and I see no circularity: the enumerative benchmarks at r=∞ are used as checks, not as inputs. The paper is worth referee time.

The weaknesses are real but concentrated. Proposition 43, which says that r-bi-words index the orbits of the r-parking action, is stated without proof. That classification is load-bearing: it drives the basis, the Hilbert series, and the product and coproduct formulas. The easy half—that columns of length at least r never swap—follows directly from the adjacent-swap rule, but the paper does not show that all placements of the short columns are reachable, nor that distinct r-bi-words are in distinct orbits. The stress-test note is on target. Proposition 45's shuffle formula is also stated without proof, and the subalgebra proof for Proposition 50 leans on a one-sentence appeal to Lemma 48. These are missing arguments, not observed contradictions; I found no counterexample to the claimed orbit structure.

If those gaps are filled, the construction of PQSym*_r should stand. The paper's proof skeletons are otherwise clear, the examples are plentiful, and the connections to trees are well explained. The main value is for algebraic combinatorists working with Hopf algebras of words and parking functions; the reader gets a useful new framework plus several concrete formulas.

My recommendation: send it to a serious referee. It should not be desk-rejected, but the referee report should ask for a full proof of Proposition 43 (or an explicit reference), and for details behind Proposition 45 and the subalgebra argument. Those are fixable gaps, and the central idea is sound.

Referee Report

4 major / 6 minor

Summary. The paper defines two actions of the infinite symmetric group on words over positive integers, called the free and parking quasi-symmetrizing actions, and proves that their invariant spaces are FQSym* and PQSym*, respectively (Theorems 19 and 34). It then introduces an r-parameter family of actions, r in N ∪ {∞}, and claims that the invariant spaces PQSym*_r form a nested chain of graded Hopf subalgebras of PQSym* (Theorem 52), with bases indexed by r-bi-words (Proposition 43), Hilbert series (Proposition 44), and explicit product and coproduct formulas (Propositions 47 and 50). The final section specializes to r = ∞ and gives a bijection between the basis elements and rooted labeled non-planar trees whose maximal decreasing subtree is a chain, yielding a bijective proof of enumerative formulas of Seo-Shin and Rattan.

Significance. The constructions are natural and, if the missing arguments are supplied, the paper would give invariant-theoretic characterizations of FQSym* and PQSym* in the spirit of Hivert's actions for QSym and WQSym, together with an interpolating family PQSym*_r that is a noncommutative analogue of Hivert's r-QSym. The r = ∞ enumeration is a concrete positive result: the bijections in Propositions 56 and 57 connect the dimension of PQSym*_∞ to known tree statistics and give a bijective proof of a formula of Seo and Shin. The paper is clearly written and the examples are helpful. The main caveat is that the orbit classification underlying the general r-construction is asserted without proof; the paper's central claims are therefore conditional on a missing combinatorial argument.

major comments (4)

[Section 4, Proposition 43] The statement that the orbits of the r-parking action are indexed by r-bi-words is load-bearing for the entire PQSym*_r construction, but no proof is given. The easy half is that an allowed adjacent swap never exchanges two columns of length at least r, so the relative order of the I-columns is an invariant. What is missing is a proof that every placement of the lambda-columns among the I-columns and every permutation of the lambda-columns is reachable by allowed swaps, and that two distinct r-bi-words lie in distinct orbits. Since this classification is used to define the basis, to count dimensions in Proposition 44, and to justify the product and coproduct formulas in Propositions 47 and 50, it cannot be left as an exercise. Please provide a complete argument, for example by exhibiting a normal form under the adjacent-swap rewriting system or by an independent orbit-counting argument.
[Section 4, Proposition 45] Formula (8) is stated without proof. It asserts that G(r)_{(I,lambda)} expands in the G-basis as the sum of G_K over all shuffles of the I-columns with arbitrary permutations of the lambda-columns. This expansion is used in the proofs of Propositions 47 and 50 to reduce computations in PQSym*_r to computations in PQSym*, so it is a second load-bearing statement. A proof must show that the orbit described in Proposition 43 has precisely these parkizations and that each occurs with coefficient 1. In addition, the notation G_K is ambiguous because the basis elements of PQSym* are indexed by parking functions, not by arbitrary bi-words; the formula should read G_{Park(K)} (or the convention should be stated explicitly).
[Section 4, Proposition 50] The proof that PQSym*_r is a subalgebra is incomplete at the decisive step. After expanding the product via formula (8), the paper asserts that the coefficients of the G_w are constant on r-orbits, saying only that this is 'a consequence of Lemma 48.' No detailed argument is supplied to show that the columns of length smaller than r in the concatenation of two words behave as claimed, nor that the multiplicities are uniform on each orbit. Moreover, the displayed formula is ambiguous as written: the summation condition 'w=a·b' does not make clear whether multiplicities over pairs (a,b) are counted, and the coefficient 1/|Orb_r(w) ∩ PF| appears to give rational coefficients in the G^r basis, whereas a product of integer sums must have integer coefficients after grouping by orbits. Please restate the formula with explicit multiplicities and give a complete proof of invariance.
[Section 4, Proposition 47] The proof of the coproduct formula contains an equality of sets that is asserted without proof: the set of all splits ([C_1,...,C_i],[C_{i+1},...,C_n]) arising from C in I /A1 lambda-sigma is claimed to equal the set of pairs (J,J') with J in I_1 /A1 K-sigma and J' in I_2 /A1 (lambda\K)-sigma over all decompositions I = I_1·I_2 and K subset of lambda. This identification is exactly what converts the coproduct of PQSym* into the proposed coproduct of PQSym*_r, so it should be proved explicitly. The proof also depends on the unproved formula (8); it should be revisited once Proposition 45 is established.

minor comments (6)

[Section 1.1] The word 'intergers' should be 'integers'.
[Section 3, Example 33] Example 33 is typeset as a large matrix of cases; aligning the bi-words in a table would improve readability.
[Section 4, Definition 41] It would help to state explicitly that each M_i is a prime parking function, rather than referring back to point (2) of Definition 21.
[Section 4, Proposition 44] The proof says 'It is easy to prove' for the count of r-bi-words; a short counting argument would make the paper self-contained.
[Section 2, Remark 20] The assertion that the generalized free actions give subalgebras but not subcoalgebras is made without proof or reference; if it is not needed for the main results, it could be omitted or briefly justified.
[Section 5, Proposition 57] The equality between the second and third expressions for A^∞_{n+1,k+1} is attributed to Theorem 3 of [13]; a parenthetical indication of the identity used would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the actions are defined independently and the main invariants follow from orbit decompositions, with external enumerative results used as benchmarks rather than fitted inputs.

full rationale

The paper's central claims are not circular. The free and parking quasi-symmetrizing actions are defined directly on encodings (E and BW) by explicit elementary transpositions, independently of the invariant spaces FQSym* and PQSym*. The orbit descriptions in Theorems 19 and 34 follow from the bijections f/g and φ together with the standard polynomial realizations (1) and (2); the invariant spaces are then characterized as orbit sums, not assumed. The r-actions (Proposition 35) are likewise new actions, and the basis, Hilbert series, and product/coproduct formulas are stated consequences of the asserted orbit classification (Proposition 43). That classification is not proved in the text, which is a genuine rigor gap and correctness risk, but it is not circular: no parameter is fitted to a target result. The r = ∞ enumerative identities are checked against independent results of Seo-Shin and Rattan, which enter as external benchmarks in a bijective argument, not as inputs that force the conclusion. No load-bearing step reduces by construction to its own conclusion.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities are needed. The paper relies on standard Hopf algebra background, the cited polynomial realizations of FQSym* and PQSym*, the parkization algorithm, and external enumerative results for parking functions and trees.

assumptions (5)

standard math The base field K has characteristic 0 and the Hopf algebras considered are graded connected.
Statements about graded duals and Hopf subalgebras throughout Sections 1 to 5 use this background; it is stated in Section 1.1.
domain assumption The polynomial realizations of FQSym* and PQSym* given by Formulas (1) and (2) are valid.
Theorems 19 and 34 identify invariants with these realizations; the realizations are cited to Malvenuto-Reutenauer and Novelli-Thibon and not reproven in this paper.
domain assumption The parkization algorithm and the prime-block decomposition of parking functions have the properties stated in Lemma 25 and Proposition 29.
The bijection phi between words and bi-words, and hence the parking action, depends on these properties; the paper gives an algorithmic proof sketch but relies on prior work for the counting facts.
standard math External enumerative results used in Section 5: prime parking function counts (n-1)^(n-1), the Foata-Riordan bijection, the Seo-Shin formula (9), and Rattan's bijection.
These are cited to [2], [12], and [13] and are used as benchmarks for the r=infinity enumeration.
domain assumption Infinite formal sums of words can be acted on by S-infinity and invariance is checked orbit-wise.
The actions are extended linearly to infinite sums, and invariant elements are identified with finite linear combinations of orbit sums in each degree; the paper does not discuss topology, which is standard in this subfield.

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Cite this review

Pith. "Pith review of The free and parking quasi-symmetrizing actions." pith.science (2026). https://pith.science/paper/HE3JASDU

@misc{pith2026250207926,
  author       = {Pith},
  title        = {Pith review of: The free and parking quasi-symmetrizing actions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HE3JASDU}},
  note         = {Machine review of arXiv:2502.07926}
}

abstract

We define two actions of the infinite symmetric group on the set of words on positive integers, called the free and parking quasi-symmetrizing actions, whose invariants are respectively the elements of the Hopf algebras $\textbf{FQSym}^*$ and $\textbf{PQSym}^*$. We study in depth the parking quasi-symmetrizing action by generalizing it to actions with a parameter $r\in(\mathbb{N}\setminus \{0\} )\bigcup\{\infty\}$. We prove that the spaces of the invariants under these $r$-actions form an infinite chain of nested graded Hopf subalgebras of $\textbf{PQSym}^*$. We give some properties of these Hopf algebras including their Hilbert series, a basis, and formulas for their product and coproduct. Finally we look more closely at the case $r=\infty$, obtaining enumerative results related to trees with maximal decreasing subtrees of given sizes.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

The Hopf algebras of symmet ric functions and quasi-symmetric functions in non-commutative variables a re free and co-free

N. Bergeron and M. Zabrocki. “The Hopf algebras of symmet ric functions and quasi-symmetric functions in non-commutative variables a re free and co-free”. In: Journal of Algebra and Its Applications 8.04 (2009), pp. 581–600

work page 2009

[2] [2]

Mappings of acyclic and parking functions

D. Foata and J. Riordan. “Mappings of acyclic and parking functions”. In: aequationes mathematicae 9 (1973), pp. 299–300

work page 1973

[3] [3]

Bidendriform bialgebras, trees, and free qu asi-symmetric func- tions

L. Foissy. “Bidendriform bialgebras, trees, and free qu asi-symmetric func- tions”. In: Journal of Pure and Applied Algebra 209.2 (2007), pp. 439–459

work page 2007

[4] [4]

r-QSym is free over Sym

A.M. Garsia and N. Wallach. “r-QSym is free over Sym”. In: Journal of Com- binatorial Theory, Series A 114.4 (2007), pp. 704–732

work page 2007

[5] [5]

Combinatoire des fonctions quasi-sym´ etri ques

F. Hivert. “Combinatoire des fonctions quasi-sym´ etri ques”. Th` ese de doctorat dirig´ ee par J.-Y. Thibon, universit´ e de Marne-la-Vall´ ee. PhD thesis. (1999)

work page 1999

[6] [6]

Local action of the symmetric group and gener aliza- tions of quasi-symmetric functions

F. Hivert. “Local action of the symmetric group and gener aliza- tions of quasi-symmetric functions”. In: Preprint, October (2004). url: https://www.lri.fr/˜ hivert/paper.html. 17

work page 2004

[7] [7]

An occupancy discipline and a pplications

A. G. Konheim and B. Weiss. “An occupancy discipline and a pplications”. In: SIAM Journal on Applied Mathematics 14.6 (1966), pp. 1266–1274

work page 1966

[8] [8]

Le Mono ¨ ıde plaxi que

A. Lascoux and M. P. Sch¨ utzenberger. “Le Mono ¨ ıde plaxi que”. In: Quad. del. Ric. Sci. 109 (1981), 129–156

work page 1981