Odd Shifted Parking Functions

Jesse Kim; Zachary Hamaker

REVIEW 3 major objections 6 minor 16 references

Odd Shifted Parking Functions

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2505.10763 v1 pith:FTDX3PMQ submitted 2025-05-16 math.CO

Zachary Hamaker , Jesse Kim This is my paper · ORCID

classification math.CO MSC 05E0505E1020C30

keywords shiftedparkingfunctionsoddV-basisFrobeniuscharacterspinrepresentationshiftificationnoncrossingmatchingsSchröderpaths

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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 $R_{2k}$.

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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

Colleague,

Quick 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.

What'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.

The 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}.

This 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.

If 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.

Best.

Referee Report

3 major / 6 minor

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.

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 (3)

[§3.2, Definition 3.12] 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.
[§3.2, Proposition 3.14] 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.
[§3.2, Example 3.15] 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.

minor comments (6)

[§2.2, Theorem 2.2] 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.
[§3.2, Definition 3.10] 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.
[§4.1, Proposition 4.2] The displayed formula for OKrew(μ) contains garbled binomial factors; please correct the expression and check the algebra preceding it.
[§4.1, equation (14)] 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.
[Abstract and Corollary 3.16] 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.
[Figure 1 caption] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the V-basis expansion is derived from independently defined objects and external symmetric-function identities, not from the conclusion.

full rationale

The derivation chain is self-contained and non-circular. SH_n is defined externally as sh(PF_n) using Stanley's shifted parking function symmetric function. The paper realizes C[NShPf(n)] as C[Pf(n)] tensor the exterior algebra, whose Frobenius character is SH_n by Proposition 1.1, a proof resting on Observation 2.4 and Stembridge's Proposition 2.3. Grouping sorted naive shifted parking functions by garages (Lemmas 3.5, 3.6) and computing fiber characters (Proposition 3.7) gives SH_n = sum R_lambda(p) without fitting any parameter to the target V-basis expansion. Odd shifted parking functions are then introduced independently, with a map phi_o to garages; Proposition 3.14 claims the fiber over each garage has character R_lambda by a term-by-term bijection with the expansion R_{2k} = sum_i P_{2i-1} P_{2(k-i)+1}. If correct, Theorem 1.2 follows directly from Theorem 3.9. There is no self-citation chain and no uniqueness theorem imported from the authors; the cited inputs are Stanley [13] and Stembridge [14], which are external published results. The reviewer's objection that Proposition 3.14 is false (e.g., the garage ((1,1),(+1,0)) has empty fiber because condition 1 in Definition 3.10 forces the merged sign to be -1 for every matched pair) is a correctness flaw in the proof, not a circularity: an invalid lemma can break a derivation without making the conclusion an input. Accordingly, no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The central claim rests on standard symmetric function theory, Stanley's relation for the P basis, and Stembridge's spin character framework. No free parameters are fitted to data; the new objects are pure combinatorial definitions.

assumptions (4)

standard math Shiftification sh(p_{2k-1})=2p_{2k-1}, sh(p_{2k})=0 is a well-defined algebra map and self-adjoint with respect to the Hall inner product.
Section 2.2; standard definition used to construct SH_n.
domain assumption The relation B = (-1+sqrt(1+4A^2))/2 for A=sum P_odd and B=sum P_even, from Stanley [13, Lem 3.1].
Section 2.2, used to derive relation (4) and the R-basis expansion.
domain assumption Stembridge's spin characteristic map and Proposition 2.3 (f * 2P_n = sh f and f * 2P'_n = sh f').
Section 2.3, used for spin character interpretations.
standard math The parking function symmetric function PF_n is the Frobenius character of the S_n-action on parking functions.
Background, due to Haiman and Stanley.

invented entities (2)

Odd shifted parking functions (p,sigma,tau) independent evidence
purpose: Combinatorial objects indexing the V-basis expansion of SH_n; resolve Stanley's open problem.
Defined purely combinatorially; their counts match OKrew for small n and the main theorem gives a checkable identity.
Garages independent evidence
purpose: Intermediate combinatorial structure used to group naive shifted parking functions and prove the R-basis expansion.
Well-defined from lattice paths; counts match large Schroeder numbers.

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Cite this review

Pith. "Pith review of Odd Shifted Parking Functions." pith.science (2026). https://pith.science/paper/FTDX3PMQ

@misc{pith2026250510763,
  author       = {Pith},
  title        = {Pith review of: Odd Shifted Parking Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FTDX3PMQ}},
  note         = {Machine review of arXiv:2505.10763}
}

abstract

Stanley recently introduced the shifted parking function symmetric function $SH_n$, which is the shiftification of Haiman's parking function symmetric function $PF_n$. The function $SH_n$ lives in the subalgebra of symmetric functions generated by odd power sums. Stanley showed how to expand $SH_n$ into the $V-$basis of this algebra, which is indexed by partitions with all parts odd and is analogous to the complete homogeneous (or elementary) basis of symmetric functions. We introduce odd shifted parking functions to give combinatorial and representation-theoretic realizations of the $V-$expansion of $SH_n$, resolving the main open problem in his paper. Further, we present two representation-theoretic realizations of shiftification allowing us to interpret $SH_n$ as the spin character of a projective representation. We conclude with further directions, including a relationship between $SH_n$ and Haglund's $(q,t)-$Schr\"oder theorem.

Figures

Figures reproduced from arXiv: 2505.10763 by the authors.

**Figure 1.** The lattice path L associated to the garage in Example 3.8. Steps are labelled by υ with σ in the superscripts. The matching τ (L) is shown in red. Combining Lemma 3.6 and Proposition 3.7, we have: Theorem 3.9. For each n, we have SHn = X (p,σ)∈Gar(n) Rλ(p). 3.2 Odd shifted parking functions We now introduce odd shifted parking functions, which are the key objects in Theorem 1.2. Definition 3.10. An odd shifted park… view at source ↗

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Bergeron and A

F. Bergeron and A. M. Garsia. Science fiction and Macdonald’s polynomials. In Algebraic methods and q-special functions, pages 1–52. Providence, RI: American Mathematical Society, 1999

work page 1999

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The bosonic-fermionic diagonal coinvariant modules conjecture

Fran¸ cois Bergeron. The bosonic-fermionic diagonal coinvariant modules conjecture. arXiv preprint arXiv:2005.00924, 2020

arXiv 2005

[3] [3]

A proof of the shuffle conjecture

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work page 2018

[4] [4]

A Schr¨ oder generalization of Haglund’s statistic on Catalan paths

Eric S Egge, Jim Haglund, Kendra Killpatrick, and Darla Kremer. A Schr¨ oder generalization of Haglund’s statistic on Catalan paths. The Electronic Journal of Combinatorics , page R16, 2003

work page 2003

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A positivity result in the theory of Macdonald polynomials

AM Garsia and J Haglund. A positivity result in the theory of Macdonald polynomials. Proceedings of the National Academy of Sciences , 98(8):4313–4316, 2001

work page 2001

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Jim Haglund. A proof of the q,t -Schr¨ oder conjecture.International Mathematics Research Notices, 2004(11):525–560, 2004

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