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Schedules and the Delta Conjecture

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Pith's one-line read Marked parking functions of each ordered-set-partition type sum to a $t^{\operatorname{maj}}$ product of q-integers.

desk verdict Solid generalization of schedules to marked parking functions with a well-motivated candidate basis for SDR_n; the main theorem is likely correct, but Lemma 3.8 needs a real proof before publication. read the letter →

arxiv 1908.04732 v2 pith:22DY5I5N submitted 2019-08-13 math.CO math.RT

James Haglund , Emily Sergel This is my paper · ORCID
classification math.COmath.RT MSC 05E0505E1005A1505A30
keywords DeltaConjecturevalley-markedparkingfunctionsschedulesorderedsetpartitionsHilbertseriessuper-diagonalcoinvariantsdiagonalq-integers
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

This paper establishes a "schedule" formula for the valley-marked parking functions that appear in the $\Delta$ Conjecture. For every ordered set partition $\Pi$, the sum of $t^{\operatorname{area}(\mathrm{MPF})} q^{\operatorname{dinv}(\mathrm{MPF})}$ over marked parking functions of type $\tau^*(\Pi)$ equals $t^{\operatorname{maj}(\tau(\Pi))}$ times a product of q-integers $\prod_c [w_\Pi(c)]_q$, where $w_\Pi(c)$ is the schedule number of car $c$. This condenses a large enumeration into a compact product and specializes to the earlier schedule formula for unmarked parking functions when every block is a singleton. The authors use the formula to conjecture a Hilbert series for the super-diagonal coinvariant ring $\mathrm{SDR}_n$ and to build a candidate monomial basis whose monomials mirror the statistics area, dinv, and marking count. If the conjectures are right, the combinatorial side of the $\Delta$ Conjecture acquires an algebraic home and a concrete basis, not just an enumeration.

What carries the argument

The central object is the ordered set partition schedule. From an ordered set partition $\Pi$ one forms a marked word $\tau^*(\Pi)$ whose runs correspond to diagonals of a parking function, marking every car that is not the leftmost element of its block; unmarked elements are block beginnings, so $\Pi$ can be recovered from $\tau^*(\Pi)$. For each car $c$, the schedule number $w_\Pi(c)$ counts the unmarked cars that would serve as insertion sites for $c$ during the insertion process, and the theorem packages those counts as the degrees of q-integers. The machinery that carries the argument is the pair of insertion algorithms together with Lemma 3.8: inserting a marked car into the $k$-diagonal creates exactly one new diagonal inversion per insertion site passed and none with cars to the right, which makes the dinv values of the new parking functions consecutive and turns the weighted sum over a whole insertion tree into a product of q-integers. The type map $\tau^*$ is what connects this tree structure to the $\Delta$ Conjecture's ordered set partitions.

What would settle it

Enumerate all ordered set partitions of $[n]$ for $n=5$ or $6$, generate every marked parking function of each type $\tau^*(\Pi)$, and compare the two sides of equation (3.5); any mismatch would show where the next-encounter rule of Insertion Algorithm 3.7 fails to produce the promised diagonal-inversion count.

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Extended reading notes

Core claim

The central claim is Theorem 3.13: for every ordered set partition $\Pi$ of $[n]$, $$\sum_{\mathrm{MPF}\in \mathrm{MPF}(\Pi)} $t^{{\operatorname{area}}$(\mathrm{MPF})}$q^{{\operatorname{dinv}}$(\mathrm{MPF})}=$t^{{\operatorname{maj}}$(\tau(\Pi))}\prod_c [w_\Pi(c)]_q,$$ where the product runs over all cars in $\Pi$. The marked parking functions of a fixed type are generated by two insertion algorithms: unmarked cars are inserted by the classical schedule algorithm, and marked cars are inserted by a new rule that scans to the right until it meets a larger unmarked car in the next diagonal or a return of the path to the appropriate diagonal. A newly inserted marked car creates exactly one new diagonal inversion with each insertion site it passes, so its dinv values are consecutive integers and the entire insertion tree contributes one q-integer per car; the exponent of $t$ records the total area, $\operatorname{maj}(\tau(\Pi))$. The paper proves that the marked permutations admitting an insertion tree are exactly the type words $\tau^*(\Pi)$ of ordered set partitions, so the marked parking functions decompose disjointly by schedule. From this it derives a conjectural Hilbert series for $\mathrm{SDR}_n$ and a candidate monomial basis $B_n$ whose x-, y-, and $\theta$-degrees are dinv, area, and marking count respectively.

Load-bearing premise

The factorization stands or falls on Lemma 3.8, the assertion that when a marked car is inserted it creates exactly one new diagonal inversion for each insertion site it moves past and none with cars to its right, so the dinv values are consecutive integers.

Editorial extensions

If this is right

  • If Theorem 3.13 holds, the Hilbert series of $\mathrm{SDR}_n$ has the closed form of Conjecture 3.14: $\sum_\Pi z^{n-|\Pi|} t^{\operatorname{maj}(\tau(\Pi))}\prod_c [w_\Pi(c)]_q$ over ordered set partitions of $[n]$.
  • The candidate set $B_n$ contains exactly one monomial per marked parking function, with $x$-degree equal to dinv, $y$-degree equal to area, and $\theta$-degree equal to the number of markings, so Conjecture 4.3 would turn the schedule formula into a genuine basis.
  • Every marked parking function satisfies $\operatorname{area}+\operatorname{dinv}+\binom{k+1}{2}\le \binom{n}{2}$ (Theorem 4.7), yielding one direction of the nonempty-component inequality for $\mathrm{SDR}_n$ and, through the alternant bound, evidence for Conjecture 4.5.
  • At $\theta$-degree $0$ the candidate basis matches the known monomial basis for $\mathrm{DR}_n$, and at $\theta$-degree $n-1$ it collapses to a one-dimensional sign representation, matching the conjectured character of $\mathrm{SDR}_n$.
  • Computer checks show $B_n$ spans the homogeneous components for $n\le 4$ and for many components at $n=5$, so the basis conjecture is consistent in small cases.

Reading between the lines

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

  • The insertion-tree proof of Theorem 3.13 is a natural place to attack the full Delta Conjecture: if the same factorization can be refined to track the full fundamental-quasisymmetric-function expansion rather than only the $h_n^1$ specialization, it would prove the valley version directly.
  • The basis conjecture suggests a degeneration proof of $\mathrm{SDR}_n$'s freeness: ordering monomials by schedule numbers might show $B_n$ is a standard monomial basis, and the paper's note that no basis is currently known for the $t=0$ super-coinvariant ring makes this a concrete target.
  • The injective cell map in Theorem 4.7 may be a bijection exactly on the tight cases of the degree inequality; if so, the boundary of Conjecture 4.5 would have a Catalan-like enumerator that could be tested by computing the number of marked parking functions with $\operatorname{area}+\operatorname{dinv}+\binom{k+1}{2}=\binom{n}{2}$.
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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

3 major / 5 minor

Summary. The paper develops a marked-parking-function analogue of the Haglund–Loehr schedule formula. It introduces an insertion algorithm for marked cars (Algorithm 3.7), proves a factorization theorem (Theorem 3.13) for the weighted sum over marked parking functions of a given type τ*(Π), and uses this to conjecture a Hilbert series (Conjecture 3.14) and a monomial basis (Conjecture 4.3) for the super-diagonal coinvariant ring SDRn introduced by Zabrocki. The paper also proves several supporting results: a degree inequality for marked parking functions (Theorem 4.7), an existence result for all degrees satisfying the bound (Theorem 4.8), and a location theorem for the alternants of SDRn (Theorem 4.11), together with small-case computer checks.

Significance. If the main theorem is correct, it gives the first schedules formula for the Delta Conjecture side of the super-diagonal coinvariant story, and it provides a natural conjectural monomial basis for SDRn, extending the Carlsson–Oblomkov basis for DRn. The insertability characterization (Theorem 3.11), the degree inequalities, and the alternant computation are valuable contributions in their own right. The paper is honest about its conjectural steps and includes detailed proofs for Lemma 3.5, Theorem 3.11, and the degree inequalities, as well as explicit computer checks for n ≤ 4 and partial checks for n = 5. The connection to Schröder paths and Kronecker coefficients is elegant and adds independent interest.

major comments (3)
  1. [§3.2, Lemma 3.8 and Theorem 3.13] The proof of Lemma 3.8 is a sketch and does not establish the load-bearing geometric claim. The proof states that the new marked car c creates a diagonal inversion with the chosen insertion site and with all potential sites to its left, and none to its right, but this is asserted without proof. In particular, the algorithm's 'look to the right ... for the next time' rule could, in principle, send two different potential sites to the same final placement of c if no unmarked b' or path-return event occurs between them; the proof does not rule out such collisions. Likewise, the proof does not exclude an unmarked car s' < c in the k-diagonal lying between the chosen site and the final position of c, which would create an additional inversion not accounted for in the stated consecutive sequence dinv(MPF), dinv(MPF)+1, ..., dinv(MPF)+|Insert*(MPF,c,k)|-1. Since Theorem 3.13 multiplies the q-integer [w_Π(c)]_q over all cars and depends on these exact dinv increments, this gap is load-bearing. The authors should either provide a complete proof of Lemma 3.8 or replace it with a separately stated and fully proved geometric lemma.
  2. [§4.1, Theorem 4.8] Theorem 4.8 claims the existence of a marked parking function with arbitrary area a, dinv b, and c markings whenever a+b+binom(c+1,2) ≤ binom(n,2). The proof constructs MPF(a,c) with maximal dinv and then invokes Theorem 3.13 to obtain MPF(a,b,c) for every b in the feasible interval. This is conditional on the unproved part of Lemma 3.8; until Lemma 3.8 is repaired, Theorem 4.8 and Corollary 4.9 should be regarded as conditional. The paper should say so explicitly or prove Theorem 4.8 by a direct construction.
  3. [§3.3, Lemma 3.15 and Theorem 3.11] The sufficiency direction of Lemma 3.15 is not fully justified. In the final paragraph, the proof considers the leftmost car in the (k+1)-diagonal; if it is smaller than c, the proof says 'the car beneath it is also smaller than c, and again c has at least one insertion site.' But Insertion Algorithm 3.7 requires an unmarked car s < c in the k-diagonal as an insertion site, and the argument does not rule out the possibility that the car beneath the leftmost upper-diagonal car is marked. Similarly, the reverse direction of Theorem 3.11 ends with 'A moment’s thought shows...' rather than a precise argument. Because Theorem 3.11 is used to identify the domain MPF(Π) in Theorem 3.13, this proof should be completed.
minor comments (5)
  1. [§3.2, Definition 3.10 and the example after Definition 3.12] The placement of the marking stars in displayed marked permutations is inconsistent with the definition. For example, the text writes τ*(235|1679|48) = 2*3*56*7*9*14*8, but the definition 'mark every number which is not the left-most element of its block in τ(Π)' would give a different assignment of stars; the schedule-number computation that follows matches the definition, not the displayed string. Please correct the typesetting of the starred elements.
  2. [§3.2, after Definition 3.12] The sentence claiming 'we calculate wΠ(c) = 2 if c = 1, 7, 9 and wΠ(c) = 1 otherwise' is correct for the intended marked word, but only if the star positions are fixed. The example would be clearer with an explicit table showing the runs of τ*(Π) and the contribution to each schedule number.
  3. [§3.2, proof of Theorem 3.13] Theorem 3.13 is stated after 'All these observations lead us to the following theorem,' but no formal induction is written out. Since the proof depends on the precise iteration of Lemmas 3.5 and 3.8, including the role of the appended 0 and the schedule numbers, a short inductive argument would improve the paper's rigor.
  4. [§4.1, proof of Theorem 4.7] The proof that the maps φ and ψ are injective and have disjoint images is terse; the sentence 'no cell left of c1 and in the same diagonal can make a diagonal inversion with c2' is particularly compressed. A more formal argument, or a second illustrated example, would make the proof easier to verify.
  5. [References] The reference [HL05] is listed as 'Discete Math.'; it should be 'Discrete Math.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Delta schedule formula is derived from insertion algorithms and definitions, and the conjectural Hilbert-series and basis applications are explicitly labeled as conjectures.

full rationale

The central derivation (Theorem 3.13) does not assume the Delta Conjecture, the Shuffle Theorem, or any fitted data. It is proved by iterating Insertion Algorithms 3.3 and 3.7 through Lemmas 3.5 and 3.8, with schedule numbers wPi(c) defined independently in Definition 3.12 from the marked permutation tau*(Pi). The weighted sum over MPF(Pi) on the left of equation (3.5) and the q-integer product on the right are not equal by construction: the product is the bookkeeping consequence of the area and dinv changes asserted in the lemmas. The only load-bearing geometric content is Lemma 3.8's claim about consecutive dinv values; that claim is asserted rather than fully demonstrated, but an unproved or potentially flawed assertion is a correctness risk, not a circularity. Citations to [HRW18], [Zab19], [HL05], and [CO18] are motivational or contextual: the Delta Conjecture is not an input to Theorem 3.13, and the SDRn basis in Section 4 is presented as a conjecture, with supporting checks explicitly limited to small n. The paper is self-contained against external benchmarks and does not rename a known result as a new one.

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

No free parameters are fitted; the paper's central theorem is a combinatorial identity proven from definitions. The axioms listed are external theorems and conjectures invoked for motivation or standard background; the main proof of Theorem 3.13 uses only the insertion algorithms and elementary combinatorial facts.

assumptions (5)
  • standard math Shuffle Theorem (Carlsson-Mellit): ∇e_n equals the weighted parking function enumeration.
    Used as a proved external theorem for Corollary 3.1 and as the z=0 special case of Delta.
  • standard math Haiman's theorem: Frob(DR_n;q,t) = ∇e_n.
    Connects the Hilbert series of diagonal coinvariants to parking function counts.
  • domain assumption Zabrocki's Conjecture 2.7: Frob(SDR_n;q,t,z) = sum_k z^k Δ'_{e_{n-k-1}} e_n.
    Conjectural module interpretation; motivates Conjecture 3.14 and the candidate basis, but is not needed for Theorem 3.13.
  • domain assumption Delta Conjecture valley version (HRW18), Equation (2.7).
    The combinatorial side that the new schedule formula reorganizes; still open and not assumed true for Theorem 3.13.
  • standard math Standard facts on Schur functions, the dual Cauchy identity, and Frobenius characteristics.
    Used in Section 4.2 to reduce alternant coefficients to hook Schur function coefficients.

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Pith. "Pith review of Schedules and the Delta Conjecture." pith.science (2026). https://pith.science/paper/22DY5I5N

@misc{pith2026190804732,
  author       = {Pith},
  title        = {Pith review of: Schedules and the Delta Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/22DY5I5N}},
  note         = {Machine review of arXiv:1908.04732}
}
abstract

In a recent preprint, Carlsson and Oblomkov (2018) obtain a long sought after monomial basis for the ring $\operatorname{DR}_n$ of diagonal coinvariants. Their basis is closely related to the "schedules" formula for the Hilbert series of $\operatorname{DR}_n$ which was conjectured by the first author and Loehr (2005) and first proved by Carlsson and Mellit (2018), as a consequence of their proof of the famous Shuffle Conjecture. In this article we obtain a schedules formula for the combinatorial side of the Delta Conjecture, a conjecture introduced by the first author, Remmel and Wilson (2018) which contains the Shuffle Conjecture as a special case. Motivated by the Carlsson-Oblomkov basis for $\operatorname{DR}_n$ and our Delta schedules formula, we introduce a (conjectural) basis for the module $\operatorname{SDR}_n$ of super-diagonal coinvariants, an $S_n$ module generalizing $\operatorname{DR}_n$ introduced recently by Zabrocki (2019) which conjecturally corresponds to the Delta Conjecture.

Figures

Figures reproduced from arXiv: 1908.04732 by the authors.

Figure 2.1
Figure 2.1. A parking function of size 8. Conjecture 4.5. Let a, b, c be non-negative integers. Then the homogeneous component of SDRn of order n with x-degree a, y-degree b, and z-degree c is non-empty if and only if a + b +  c + 1 2  ≤  n 2  We prove one direction of this conjecture for the alternants of SDRn; see Theorem 4.11. 2 The Delta Conjecture In this section we present the important combinatorial objects and resul… view at source ↗
Figure 2
Figure 2. shows a parking function of size 8. However it will be convenient here to allow a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 2.2
Figure 2.2. A valley-marked parking function of size 8 with 2 marked valleys. [PITH_FULL_IMAGE:figures/full_fig_p007_2_2.png] view at source ↗
Figures from the paper (9 more)
Figure 3.1
Figure 3.1. Figure 3.1: A parking function together with those obtained by inserting a 5 into the 2- [PITH_FULL_IMAGE:figures/full_fig_p009_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: The tree of parking functions built from the schedule [PITH_FULL_IMAGE:figures/full_fig_p011_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: A marked parking function together with those obtained by inserting a marked [PITH_FULL_IMAGE:figures/full_fig_p013_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: Building Figure 2.2 using Insertion Algorithms 3.3 and 3.7. [PITH_FULL_IMAGE:figures/full_fig_p015_3_4.png]
Figure 3.4
Figure 3.4. Figure 3.4: The full tree would have 144 leaves with weights summing to [PITH_FULL_IMAGE:figures/full_fig_p016_3_4.png]
Figure 4.1
Figure 4.1. Figure 4.1: A marked parking function of size 9 with the images of [PITH_FULL_IMAGE:figures/full_fig_p022_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: The marked parking function MPF(16, 3) of size n = 8 4.2 The location of the alternants Let Gn be the polynomial ring C[θ1, θ2, . . . , θn] in the Grassmannian θ variables modulo the ideal generated by θ1 + θ2 + · · · + θn. Sn acts on Gn by permuting the θ variables,…
Figure 4.3
Figure 4.3. Figure 4.3: A Schr¨oder path of size 9. Remark 4.12. Finding a combinatorial interpretation for the Kronecker coefficients is an important unsolved problem, but some special cases are solved. In particular, when either λ or µ is a hook, Blasiak [Bla16] gives a combinatorial inte…
Figure 4
Figure 4. Figure 4: we have marked cells in the image of [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]

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