{"id":"34a18609-fb76-4f54-b33d-8600063211ab","arxiv_id":"2505.10763","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Odd shifted parking functions are defined and claimed to recover SH_n via V_{shape(p)}, resolving Stanley's open problem, but the proof has a gap.","lead":"This paper introduces odd shifted parking functions, triples (p,sigma,tau), and uses them to give a combinatorial and representation-theoretic realization of the V-basis expansion of Stanley's shifted parking function symmetric function SH_n. If correct, it resolves the main open problem in Stanley's recent paper, but the proof as written contains a false intermediate proposition.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.14 is false: the fiber over garage ((1,1),(+1,0)) is empty, so the proof of Theorem 1.2 does not go through.","rationale":"The reader identified exactly this false proposition. I verified the counterexample: for the garage ((1,1),(+1,0)) with n=2, the only possible odd preimage would need τ={(1,2)} and p=(1,2), but condition σ_1=-σ_2 forces the merged sign σ'_1 = σ_1 σ_2 = -1, so σ'_1=+1 is impossible; the fiber is empty, contradicting R_2≠0. Since Theorem 1.2's proof is explicitly 'By Theorem 3.9 and Proposition 3.14', the central claim is not established. The theorem may well be true — the n=2 example shows the deficit is compensated by multiplicity elsewhere — but the proof as written fails. I recommend no change to the reader's REJECT verdict.","tokens_in":14275,"tokens_out":15756,"duration_ms":126713,"concrete_test":"Enumerate all six size-2 garages and, using Definition 3.12, compute φ_o^{-1}(p,σ) for each. Check that φ_o^{-1}((1,1),(+1,0)) is empty while φ_o^{-1}((1,1),(-1,0)) has two preimages; this directly contradicts Proposition 3.14's per-garage character formula. Also compare the total character over all fibers with SH_2 to see whether the global identity survives despite the false proposition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.14 claims that for every garage (p,σ) with shape λ, the fiber φ_o^{-1}(p,σ) has Frobenius character R_λ. This is false. In Definition 3.10, condition 1 requires σ_a = -σ_b for every matched pair (a,b)∈τ. Under the map φ_o (Definition 3.12), a and b are merged into a, so the garage sign is σ'_a = σ_a σ_b = -1. Thus any garage with σ_a = +1 at a left endpoint of its matching τ(L) cannot lie in the image of φ_o. For n=2, the garage ((1,1),(+1,0)) has matching τ(L)={(1,2)} and is a valid garage, but its fiber is empty: the only possible odd preimage would have p=(1,2) and τ={(1,2)}, which forces σ'_1 = -1, not +1. Yet R_{(2)} = P_2 = p_1^2 ≠ 0. Hence Proposition 3.14 is false as stated, and the proof of Theorem 1.2, which invokes it after Theorem 3.9, is invalid. The n=2 totals still match because the other garage ((1,1),(-1,0)) has a two-element fiber, but the per-garage identity used in the proof is false.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces odd shifted parking functions, triples (p,σ,τ) where p is a parking function of odd shape, σ is a sign vector, and τ is a noncrossing matching of the integers appearing in p. The main theorem (Theorem 1.2) asserts that the Frobenius character of the S_n-action on these objects is the shifted parking function symmetric function SH_n, and that SH_n equals the sum, over sorted odd shifted parking functions, of V_{shape(p)}. This would resolve Stanley's open problem of finding a combinatorial model for the V-basis expansion of SH_n. The proof strategy is to group naive shifted parking functions into 'garages' (Theorem 3.9), then define a map φ_o from odd shifted parking functions to garages and prove that each garage fiber has Frobenius character R_λ (Proposition 3.14). The paper also gives exterior-algebra and Clifford-algebra interpretations of shiftification, leading to a spin-character realization of a scalar multiple of SH_n, and concludes with q,t-analogues and connections to Schröder paths.","tokens_in":14582,"tokens_out":49858,"duration_ms":465803,"significance":"If the main theorem is correct, it resolves a question explicitly raised by Stanley and gives the first combinatorial objects whose Frobenius character is the V-basis expansion of SH_n. The representation-theoretic reformulation of shiftification via exterior and Clifford algebras is a useful contribution in its own right. The paper does not include machine-checked proofs or code, so the assessment rests on the written arguments. The stress-test counterexample does not, on reading the paper literally, land: Definition 3.12 says σ' is the restriction of σ to the integers appearing in p', not the product of the signs of the merged positions. Under the literal restriction reading, the garage ((1,1),(+1,0)) does have the preimage ((1,2),(+,-),{(1,2)}). However, the current text is sufficiently ambiguous that a reader can misread it as the product convention, under which Proposition 3.14 would be false; the proof must be rewritten to remove this ambiguity and to supply the missing verifications.","major_comments":[{"comment":"The definition of φ_o is ambiguous in a load-bearing way. Definition 3.10 uses σ both as a position-indexed tuple and as an integer-indexed aggregated vector (the σ_k in the parenthetical), and Definition 3.12 says only that σ' is the restriction of σ to the integers appearing in p'. The intended meaning must be stated explicitly: the output garage sign at a left endpoint a equals the input aggregated sign σ_a, not the product σ_a σ_b over the merged positions. If the latter convention were used, Proposition 3.14 would fail already for n=2 because the garage ((1,1),(+1,0)) would have empty fiber while R_{(2)}=P_2≠0. Please introduce separate notation for the position-indexed and integer-indexed sign vectors and state exactly which one is restricted in the definition of φ_o.","section":"§3.2, Definition 3.12"},{"comment":"The proof of Proposition 3.14 is too terse to be verified. It asserts that each choice of split of a matched pair gives exactly one valid sorted odd shifted parking function, citing Lemma 3.4, but it does not prove that the simultaneous choices for all matched pairs yield a parking function, nor does it verify conditions 2 and 3 of Definition 3.10 for the resulting triple. In particular, condition 2 requires that every integer between the endpoints of an arc appears in p, which is not automatic from Lemma 3.4 alone. Since Proposition 3.14 is the key step linking Theorem 3.9 to Theorem 1.2, this gap needs a complete proof rather than the current one-sentence appeal.","section":"§3.2, Proposition 3.14"},{"comment":"Example 3.15 is incorrect as printed. Several of the listed preimages have even multiplicities in p (for instance, the second preimage has four 1's), which violates the requirement that the shape of p be odd. In the same example, the displayed σ-vectors assign 0 to integers that appear with odd multiplicity, contradicting Definition 3.10 (e.g., the first preimage has a 0 at the integer 3, which appears once). The shapes and the final character computation are correct for the intended four preimages, but the example must be recomputed with the correct compressed sign vectors and with all parts odd. As written, the example obscures the restriction convention and cannot be used to understand the construction.","section":"§3.2, Example 3.15"}],"minor_comments":[{"comment":"The displayed formula defines OKrew(λ) with a stray factor V_λ inside the definition of the coefficient; the formula should define OKrew(λ) as a scalar and then state SH_n = ∑ OKrew(λ)V_λ.","section":"§2.2, Theorem 2.2"},{"comment":"The notation σ∈{-1,1}^n is used for the position-indexed tuple, while the conditions refer to the aggregated signs σ_a for integers a. These two objects should be denoted by different symbols, or the statement should explicitly say that the triple for a sorted odd shifted parking function carries the aggregated vector.","section":"§3.2, Definition 3.10"},{"comment":"The displayed formula for OKrew(μ) contains garbled binomial factors; please correct the expression and check the algebra preceding it.","section":"§4.1, Proposition 4.2"},{"comment":"The claimed bijection from NShPf to OShPf is stated without proof. If it follows from the fiber decomposition in Proposition 3.14, that should be said; otherwise a short proof or an explicit inverse should be supplied.","section":"§4.1, equation (14)"},{"comment":"The abstract says SH_n is interpreted as the spin character of a projective representation, but Corollary 3.16 gives the spin character as 2^{n/2}SH_n; the scalar factor should be acknowledged.","section":"Abstract and Corollary 3.16"},{"comment":"The caption says steps are labelled by υ with σ in superscripts, but several superscripts are 0 on steps with nonzero υ; clarify what the superscripts represent.","section":"Figure 1 caption"}],"recommendation":"major_revision","confidential_remarks":"The central construction appears sound and, if properly written up, would be a significant contribution. The reader's high-confidence rejection is based on a misreading of Definition 3.12: the 'restriction' convention does give a nonempty fiber for the n=2 garage ((1,1),(+1,0)). Nevertheless, the manuscript is not ready in its current form because the definition of φ_o is ambiguous, the proof of Proposition 3.14 is incomplete, and Example 3.15 is seriously wrong. These are fixable within the scope of the paper, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: this paper has a genuinely new combinatorial model and a plausible answer to Stanley's open question, but the central proof is currently not valid. Proposition 3.14 is false, and Theorem 1.2 is not established as written.\n\nWhat's genuinely new: the odd shifted parking functions are a natural definition, and Proposition 1.1—the exterior-algebra realization of shiftification—is a clean observation that holds up. The Clifford-algebra extension to spin characters is a nice addition, and the connections to Schröder numbers and Haglund's q,t-Schröder theorem are worth exploring. The writing is clear, and the authors are honest about what remains open.\n\nThe soft spot is serious. Proposition 3.14 claims that for every garage (p,σ) of shape λ, the fiber φ_o^{-1}(p,σ) has Frobenius character R_λ. That is false. For n=2, the garage ((1,1),(+1,0)) is valid, but any odd shifted parking function mapping to it would need a matched pair (1,2) merging to position 1. Condition 1 of Definition 3.10 forces σ_1 = -σ_2, so the merged sign becomes σ_1σ_2 = -1, not +1. The fiber is empty, while R_{(2)} is nonzero. So the per-garage identity used after Theorem 3.9 breaks. The n=2 totals still match because the other garage of shape (2) has a two-element fiber, but that is not the claimed proposition. The bijection in (14) is also under-specified, and the proof of Proposition 3.14 hand-waves the sign multiplicities when splitting R_{2k}.\n\nThis is not a fundamentally bad idea. The main theorem is likely true, and the framework is promising. But as submitted, the proof is incomplete, so the main result should not be relied upon. The paper definitely deserves a serious referee; the flaw is localized enough that a revision could fix it, perhaps via a coarser grouping of garages or a corrected per-garage statement. The exterior-algebra and Clifford-algebra parts stand on their own.\n\nIf I were editing, I'd send it back for major revision, not desk-reject. For a reading group, it's worth one session to try to patch the proof.\n\nBest.","headline":"Fresh combinatorial objects and a plausible main theorem, but Proposition 3.14 is false as stated, so the proof of Theorem 1.2 doesn't go through.","tokens_in":15040,"tokens_out":6280,"would_cite":true,"duration_ms":59585,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","20C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Odd shifted parking functions give a combinatorial and representation-theoretic description of the $V$-basis expansion of the shifted parking function symmetric function $SH_n$, resolving the main open problem in the paper that introduced…","keywords":["shifted parking functions","odd shifted parking functions","V-basis","Frobenius character","spin representation","shiftification","noncrossing matchings","Schröder paths"],"falsifier":"Enumerate the $\\varphi_o$-fiber of the size-2 garage $(p,\\sigma)=((1,1),(+1,0))$: the parity condition $\\sigma_a=-\\sigma_b$ on the only matched pair forces $\\sigma_a=-1$, so no odd shifted parking function maps to this garage, while $R_{(2)}=P_2$ is nonzero. This directly tests Proposition 3.14, the load-bearing step of the proof of Theorem 1.2.","tokens_in":14099,"feed_emoji":"🚗","tokens_out":12578,"duration_ms":118164,"temperature":0.7,"pith_summary":"The paper introduces odd shifted parking functions, triples $(p,\\sigma,\\tau)$ in which $p$ is a parking function of odd shape, $\\sigma$ is a sign vector obeying parity constraints, and $\\tau$ is a noncrossing matching of the values appearing in $p$. Its central claim is that the symmetric group action on these triples has Frobenius character $SH_n$, the shifted parking function symmetric function obtained by shiftifying the parking function symmetric function $PF_n$. This yields the first combinatorial interpretation of the $V$-basis expansion $SH_n=\\sum V_{\\operatorname{shape}(p)}$, with the sum over sorted odd shifted parking functions, resolving the open problem that motivated the definition of $SH_n$. The paper further shows that the construction extends to a projective representation whose spin character is $2^{n/2}SH_n$, and that a $t$-graded version matches a Schr\\\"oder-path statistic at $q=1$. A curious reader should care because the result ties parking functions, projective representations of symmetric groups, and Schr\\\"oder combinatorics into one family of objects.","feed_headline":"Odd shifted parking functions settle an open problem","feed_subtitle":"New triples of parking function, signs, and matchings give the V-basis expansion of SH_n.","key_machinery":"The load-bearing identity is the relation $A^2=B^2+B$ between the generating functions $A=\\sum P_{2k-1}t^{2k-1}$ and $B=\\sum P_{2k}t^{2k}$, which gives $R_{2k}:=\\sum_{i=1}^k P_{2i}P_{2k-2i}=\\sum_{i=1}^k P_{2i-1}P_{2(k-i)+1}$. On the combinatorial side, the key objects are garages: sorted naive shifted parking functions $(p,\\sigma)$ whose parity word $\\upsilon\\in\\{0,1,2\\}^n$ satisfies a lattice-word condition, with an associated matching path and noncrossing matching $\\tau(L)$. Garages group naive shifted parking functions into equivalence classes whose Frobenius characters are exactly $R_{\\lambda(p)}$; odd shifted parking functions map to garages by replacing each matched value by its partner, and the splitting identity above rewrites each even part as a sum of odd-part products, matching the choices of how many of each paired value occur. The shiftification operator $sh$ itself, realized as tensoring with the exterior algebra, supplies the bridge from $PF_n$ to $SH_n$ and later to spin characters via the Clifford algebra.","core_discovery":"The shiftification map $sh$ sends $h_k$ to $2P_k$ and kills even power sums, so $SH_n=sh(PF_n)$ lives in the subalgebra $\\operatorname{Sym}^P$ generated by odd power sums. In that algebra the products $V_\\lambda=\\prod_i P_{\\lambda_i}$ with $\\lambda$ odd form a basis, and the paper that introduced $SH_n$ computed the coefficients as odd Kreweras numbers but left open a combinatorial meaning for them. The paper's Theorem 1.2 asserts that if $\\operatorname{OShPf}(n)$ is the set of odd shifted parking functions---parking functions with odd shape, a sign vector, and a noncrossing matching $\\tau$ such that matched values have opposite signs and nested matched pairs have equal signs---then the $S_n$-action by permuting $p$ and $\\sigma$ has Frobenius character $SH_n$, and the sorted objects give $SH_n=\\sum_{(p,\\sigma,\\tau)} V_{\\operatorname{shape}(p)}$. The proof groups naive shifted parking functions into garage classes with character $R_\\lambda$, then maps odd shifted parking functions to garages by collapsing matched pairs; each garage fiber is claimed to have character $R_\\lambda$ after splitting $R_{2k}=\\sum_{i=1}^k P_{2i-1}P_{2(k-i)+1}$. In this way odd shifted parking functions realize the odd Kreweras numbers as counts of combinatorial objects.","pith_inferences":["If the garage-fiber character assertion can be repaired, the bijection in the paper likely transports $q,t$-statistics such as area and bounce from Schr\\\"oder paths to odd shifted parking functions, giving the missing graded bijection for the paper's Problem 4.5.","The Clifford-algebra realization suggests the existence of a genuine super-diagonal-harmonics module whose bigraded Frobenius character specializes to $SH_n$ at $q=t=1$; the paper only notes a resemblance to the bosonic-fermionic coinvariant algebra.","The parity word used in the garage definition may itself define a new statistic on Schr\\\"oder paths that is preserved under the paper's bijection, possibly related to odd parts and noncrossing matchings.","A direct bijective proof of the odd-Kreweras formula would likely reveal additional structure worth studying independently, since the current proof passes through the two expansions of $R_{2k}$."],"forward_implications":["The open problem is closed: the coefficient of $V_\\lambda$ in $SH_n$ is the number of sorted odd shifted parking functions of shape $\\lambda$, so the odd Kreweras numbers count explicit combinatorial objects.","$\\operatorname{OShPf}(n)$ carries a projective $S_n^-$-action whose spin character is $2^{n/2}SH_n$, giving the spin analogue that the paper that introduced $SH_n$ suspected might be too much to hope for.","At $q=1$, $SH_n(1,t)=\\sum t^{\\operatorname{area}_o(p,\\sigma,\\tau)}V_{\\operatorname{shape}(p)}$, extending the expansion to a graded identity with an area statistic.","The bijection between naive and odd shifted parking functions gives a parity-flavoured class of objects enumerated by large Schr\\\"oder numbers.","Pairing the $q,t$-analogue with the $q,t$-Schr\\\"oder theorem gives $\\langle SH_n(q,t),h_n\\rangle=\\sum_{S\\in\\mathcal S(n)} q^{\\operatorname{area}(S)}t^{\\operatorname{bounce}(S)}$, linking odd shifted parking functions to Schr\\\"oder paths."],"supporting_citations":[{"why":"introduces $SH_n$, the $V$-basis expansion, and the open problem that odd shifted parking functions resolve.","marker":"[13]"},{"why":"supplies the projective representation framework, the spin characteristic map, and the Clifford-algebra construction used to interpret $SH_n$ as a spin character.","marker":"[14]"},{"why":"provides the diagonal-harmonics context and the $q,t$-analogue of $PF_n$ that motivates $SH_n(q,t)$.","marker":"[7]"},{"why":"gives the background quotient-ring conjectures for diagonal harmonics used in the motivating discussion.","marker":"[8]"},{"why":"proves the $q,t$-Schr\\\"oder theorem that yields the Schr\\\"oder-path identity for $\\langle SH_n(q,t),h_n\\rangle$.","marker":"[6]"},{"why":"proves the shuffle conjecture, making $PF_n(q,t)=\\nabla e_n$ available for the shiftification argument.","marker":"[3]"}],"fun_headline_variants":["Odd shifted parking functions resolve open problem","Combinatorial model for odd Kreweras numbers","New parking functions realize SH_n expansion","Shiftification meets parking functions","Odd parking functions give V-basis meaning"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Proposition 3.14, that every garage's preimage under $\\varphi_o$ has Frobenius character $R_{\\lambda(p)}$, which requires the parity constraints on odd shifted parking functions to allow exactly the right number of sign and split choices for each matched pair; if this fails for even one small garage, the grouping proof of Theorem 1.2 needs repair.","fun_headline_variants_meta":{"raw":{"variants":["Odd shifted parking functions resolve open problem","Combinatorial model for odd Kreweras numbers","New parking functions realize SH_n expansion","Shiftification meets parking functions","Odd parking functions give V-basis meaning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000659,"raw_usage":{"total_tokens":3042,"prompt_tokens":1001,"completion_tokens":2041,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":1978}},"tokens_in":617,"tokens_out":2041,"duration_ms":13762,"temperature":1.0,"reasoning_tokens":1978,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:09:02.846579+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate the $\\varphi_o$-fiber of the size-2 garage $(p,\\sigma)=((1,1),(+1,0))$: the parity condition $\\sigma_a=-\\sigma_b$ on the only matched pair forces $\\sigma_a=-1$, so no odd shifted parking function maps to this garage, while $R_{(2)}=P_2$ is nonzero. This directly tests Proposition 3.14, the load-bearing step of the proof of Theorem 1.2.","supporting_citations":[{"cited_title":"A Shifted Parking Function Symmetric Function","cited_arxiv_id":"2405.02164","evidence_quote":"introduces $SH_n$, the $V$-basis expansion, and the open problem that odd shifted parking functions resolve."},{"cited_title":"Shifted tableaux and the projective representations of symmetric groups","cited_arxiv_id":null,"evidence_quote":"supplies the projective representation framework, the spin characteristic map, and the Clifford-algebra construction used to interpret $SH_n$ as a spin character."},{"cited_title":"Hilbert schemes, polygraphs and the Macdonald positivity conjecture","cited_arxiv_id":null,"evidence_quote":"provides the diagonal-harmonics context and the $q,t$-analogue of $PF_n$ that motivates $SH_n(q,t)$."},{"cited_title":"Conjectures on the quotient ring by diagonal invariants","cited_arxiv_id":null,"evidence_quote":"gives the background quotient-ring conjectures for diagonal harmonics used in the motivating discussion."},{"cited_title":"A proof of the q,t -Schr¨ oder conjecture.International Mathematics Research Notices, 2004(11):525–560, 2004","cited_arxiv_id":null,"evidence_quote":"proves the $q,t$-Schr\\\"oder theorem that yields the Schr\\\"oder-path identity for $\\langle SH_n(q,t),h_n\\rangle$."},{"cited_title":"A proof of the shuffle conjecture","cited_arxiv_id":null,"evidence_quote":"proves the shuffle conjecture, making $PF_n(q,t)=\\nabla e_n$ available for the shiftification argument."}],"review_version":1}