{"id":"28d57bdb-3699-45f6-ab7a-f19f62d376cb","arxiv_id":"2502.07926","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New symmetric group actions make FQSym* and PQSym* into invariant spaces, and their r-parameter deformations give a nested chain of Hopf subalgebras with explicit bases and Hilbert series.","lead":"This paper defines two new actions of the infinite symmetric group on words, the free and parking quasi-symmetrizing actions, whose invariant sums are exactly the elements of the Hopf algebras FQSym* and PQSym*. It then builds a family of nested Hopf subalgebras PQSym*_r and connects the r=infinity case to a tree enumeration problem.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 43's orbit classification is asserted without proof and is the load-bearing premise for the entire PQSym*_r construction; a small-case orbit enumeration or an independent derivation would settle whether it holds.","rationale":"I read the construction in good faith. The invariant characterizations in Theorems 19 and 34 are well supported by the bijections f,g and phi and by the elementary orbit descriptions for the r=1 actions. The genuinely soft part is the r-family. The most load-bearing unsupported statement is Proposition 43's orbit classification, exactly as the reader identified. My own inspection of the adjacent-swap rule suggests the classification is very likely correct: heavy columns preserve their relative order, and light columns can be bubbled past them, so the claimed canonical representatives are plausible. However, because no proof is supplied and the paper explicitly derives the Hilbert series, basis, and product/coproduct formulas from this classification, a conditional verdict is appropriate. I found no internal inconsistency or a concrete counterexample to the main claims, and the r=infinity enumeration matches independent tree-counting formulas, which is supporting evidence. The proposed computational check would either expose a false orbit description or increase confidence in the missing proof.","tokens_in":18161,"tokens_out":14868,"duration_ms":143753,"concrete_test":"Implement the r-parking action on bi-words for small lengths n <= 6 and r in {2,3,infinity}: represent all words of length n as bi-words via phi, generate orbits by repeatedly applying the adjacent swaps of Proposition 35, and compare the number of orbits and the size of each orbit with the count of r-bi-words of length n from Proposition 44. A single mismatch for any tested (n,r) would refute Proposition 43; if all small cases match, the classification is supported in the tested range and the remaining issue is the absence of a written proof for all n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 43 asserts, without proof, that the orbits of the r-parking action on bi-words are indexed by r-bi-words: columns of length at least r (the I part) keep their relative order, while columns of length strictly less than r (the lambda part) can be permuted arbitrarily and are canonically ordered by first-row minimum. This classification is the exact premise used to define the basis G(r)_{(I,lambda)}, to count dimensions in Proposition 44, to expand basis vectors as shuffles in Proposition 45, and to derive the product and coproduct formulas in Propositions 47 and 50. The easy half of the statement is visible from the adjacent-swap rule in Proposition 35: an allowed swap never exchanges two columns of length at least r, so the relative order of the I-columns is preserved. What is not shown is that the allowed adjacent transpositions generate every placement of the lambda-columns relative to the I-columns, and that no two distinct r-bi-words lie in the same orbit. If the orbit set were finer or coarser than claimed, the basis would fail and the Hilbert series and Hopf-algebra conclusions would collapse. The paper gives no proof or reference for this classification, and Proposition 45's shuffle formula is equally unsupported. No counterexample appears in the text; the gap is a missing argument rather than an observed contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18379,"tokens_out":16414,"duration_ms":143940,"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":[{"comment":"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":"Section 4, Proposition 43"},{"comment":"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":"Section 4, Proposition 45"},{"comment":"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":"Section 4, Proposition 50"},{"comment":"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.","section":"Section 4, Proposition 47"}],"minor_comments":[{"comment":"The word 'intergers' should be 'integers'.","section":"Section 1.1"},{"comment":"Example 33 is typeset as a large matrix of cases; aligning the bi-words in a table would improve readability.","section":"Section 3, Example 33"},{"comment":"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":"Section 4, Definition 41"},{"comment":"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":"Section 4, Proposition 44"},{"comment":"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":"Section 2, Remark 20"},{"comment":"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.","section":"Section 5, Proposition 57"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of math.CO and the programme is attractive. My main concern is the missing proof of the orbit classification in Proposition 43, which affects the basis and all structure formulas for PQSym*_r. I do not see an internal inconsistency, and the r = ∞ results are a valuable contribution. I recommend major revision rather than rejection, because the gap appears fixable within the manuscript's scope. The author should also double-check the coefficient formula in Proposition 50, which as stated looks problematic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nIf 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.\n\nMy 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.","headline":"New invariant-theoretic actions yield conditional but credible results; the missing proof of the orbit classification is the main gap.","tokens_in":18976,"tokens_out":1251,"would_cite":true,"duration_ms":15021,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","16T30","05A15","05C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["parking functions","free quasi-symmetric functions","parking quasi-symmetric functions","Hopf algebras","symmetric group actions","parkization","r-bi-words","rooted labeled trees"],"falsifier":"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.","tokens_in":17899,"feed_emoji":"🅿️","tokens_out":14039,"duration_ms":107664,"temperature":0.7,"pith_summary":"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.","feed_headline":"Parking actions make word-sums into nested Hopf algebras","feed_subtitle":"A parameter r interpolates between parking quasi-symmetric functions and a tree-enumerative limit.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["$\\mathbf{FQSym}^{*}$ 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."],"supporting_citations":[{"why":"Defines the free quasi-symmetric function algebra on permutations whose grading dual carries the $G_{\\sigma}$ basis used here.","marker":"[9]"},{"why":"Introduces the Hopf algebra of parking functions and the parkization procedure realizing $\\mathbf{PQSym}^{*}$ as word sums.","marker":"[10]"},{"why":"Gives the dendriform and coproduct structures of $(\\mathbf{PQSym},\\mathbf{PQSym}^{*})$ used to compute the coproduct formulas.","marker":"[11]"},{"why":"Introduces the quasi-symmetrizing action paradigm of describing quasi-symmetric functions as invariants, which the paper adapts to free and parking settings.","marker":"[5]"},{"why":"Provides the $r$-parameter interpolation of quasi-symmetrizing actions whose structure the paper mirrors for $\\mathbf{PQSym}^{*}$.","marker":"[6]"},{"why":"Supplies the bijection between parking functions and rooted labeled non-planar forests used in the $r=\\infty$ enumeration.","marker":"[2]"},{"why":"Relates prime parking functions to minimal rooted trees, the bridge from $\\infty$-bi-word columns to tree forests.","marker":"[12]"},{"why":"States the tree-counting formulas for maximal decreasing subtrees that the $r=\\infty$ dimensions reproduce bijectively.","marker":"[13]"}],"fun_headline_variants":["Parking actions yield nested Hopf subalgebra chain","Word-sums fixed by symmetric group form Hopf algebras","r-actions interpolate to tree-enumerative limit","Invariants of parking actions create nested Hopf chain","From parking invariants to tree counts via r-actions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Parking actions yield nested Hopf subalgebra chain","Word-sums fixed by symmetric group form Hopf algebras","r-actions interpolate to tree-enumerative limit","Invariants of parking actions create nested Hopf chain","From parking invariants to tree counts via r-actions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1253,"prompt_tokens":917,"completion_tokens":336,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":256}},"tokens_in":533,"tokens_out":336,"duration_ms":4507,"temperature":1.0,"reasoning_tokens":256,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T11:25:45.130648+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Duality between quasi- symmetrical func- tions and the Solomon descent algebra","cited_arxiv_id":null,"evidence_quote":"Defines the free quasi-symmetric function algebra on permutations whose grading dual carries the $G_{\\sigma}$ basis used here."},{"cited_title":"Hopf algebras and dend riform structures aris- ing from parking functions","cited_arxiv_id":null,"evidence_quote":"Gives the dendriform and coproduct structures of $(\\mathbf{PQSym},\\mathbf{PQSym}^{*})$ used to compute the coproduct formulas."},{"cited_title":"Combinatoire des fonctions quasi-sym´ etri ques","cited_arxiv_id":null,"evidence_quote":"Introduces the quasi-symmetrizing action paradigm of describing quasi-symmetric functions as invariants, which the paper adapts to free and parking settings."},{"cited_title":"Local action of the symmetric group and gener aliza- tions of quasi-symmetric functions","cited_arxiv_id":null,"evidence_quote":"Provides the $r$-parameter interpolation of quasi-symmetrizing actions whose structure the paper mirrors for $\\mathbf{PQSym}^{*}$."},{"cited_title":"Mappings of acyclic and parking functions","cited_arxiv_id":null,"evidence_quote":"Supplies the bijection between parking functions and rooted labeled non-planar forests used in the $r=\\infty$ enumeration."},{"cited_title":"Permutation factorizations and prime park ing functions","cited_arxiv_id":null,"evidence_quote":"Relates prime parking functions to minimal rooted trees, the bridge from $\\infty$-bi-word columns to tree forests."},{"cited_title":"On the enumeration of rooted trees wi th fixed size of maximal decreasing trees","cited_arxiv_id":null,"evidence_quote":"States the tree-counting formulas for maximal decreasing subtrees that the $r=\\infty$ dimensions reproduce bijectively."}],"review_version":1}