{"id":"a8d74c0e-ba46-4770-ab34-1b1533ffb0b2","arxiv_id":"1908.04732","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A new schedules formula for marked parking functions matches the Delta Conjecture's combinatorial side and motivates a conjectural monomial basis for super-diagonal coinvariants.","lead":"Parking functions with marked valleys are grouped by a new schedule rule so their weighted sums factor into simple q-products, giving a schedule formula for the Delta Conjecture side. This leads the authors to conjecture a monomial basis for the super-diagonal coinvariant ring, with small-case checks and degree bounds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.8's next-encounter claim is asserted rather than proved; a failure there would change the q-integer product in Theorem 3.13.","rationale":"The reader's weakest-assumption analysis correctly isolates Lemma 3.8 as the hinge of Theorem 3.13. I read the paper in good faith: the surrounding construction is coherent, the unmarked schedule formula is a known theorem, and the marked analogue is a natural extension with a plausible geometric mechanism. I could not construct a counterexample to Lemma 3.8, and the special cases and MAPLE checks for n = 3, 4, and part of n = 5 provide supporting evidence that the formula is likely correct. However, the written proof of Lemma 3.8 is not detailed enough to be fully convincing: the claim that the new marked car creates exactly one new diagonal inversion per potential site passed, and none with anything to the right, depends on an unproved statement about when the Dyck path returns to y = x + k relative to other cars in the same and adjacent diagonals. Because this lemma is the only step that distinguishes the marked schedule formula from the unmarked one, it is the single most load-bearing assumption in the central theorem. I do not think this concern overturns the reader's ACCEPT: the claimed theorem is explicit, finite, and directly checkable, and the proposed exhaustive test would settle whether the geometric assertion actually holds. If the test passes, the proof gap is a presentation issue rather than a substantive error; if it fails, the central theorem would need revision.","tokens_in":22079,"tokens_out":44074,"duration_ms":436695,"concrete_test":"Exhaustively implement Insertion Algorithms 3.3 and 3.7 together with Definition 2.4, and for every ordered set partition Pi of [n] with n <= 6 generate the full insertion tree of type tau*(Pi). Compute the bivariate sum of t^{area} q^{dinv} over all leaves and compare it with t^{maj(tau(Pi))} * prod_c [w_Pi(c)]_q. A match for every Pi through n = 6 would confirm the consecutive-dinv assertion of Lemma 3.8 in all small cases; any mismatch would identify the exact configuration where the next-encounter rule fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.13 is proved by induction using Lemmas 3.5 and 3.8, and Lemma 3.8 carries the genuinely new content of the marked insertion argument. Its proof asserts that Insertion Algorithm 3.7 places the new marked car c so that the only new diagonal inversions are with the chosen potential insertion site and with every potential site to its left, yielding dinv values dinv, dinv+1, ..., dinv+|Insert*(MPF,c,k)|-1. This requires a precise geometric fact about the next-encounter rule: between the chosen same-diagonal site s (or upper-diagonal site b) and the cell where c is placed, there can be no unmarked same-diagonal car s'<c and no unmarked upper-diagonal car b'>c; otherwise a potential site to the left of the placement would be missed or the dinv sequence would skip a value. The one-paragraph proof does not establish this fact, and the sentence 'c is inserted so that it does not create a diagonal inversion with anything to the right of s or b' is ambiguous, especially in the case where c is placed directly under an encountered upper car b'. Since Theorem 3.13 multiplies [w_Pi(c)]_q over all cars, any violation of the claimed consecutive-dinv sequence would change the product and invalidate the schedule formula. No counterexample is produced here; the concern is that the load-bearing geometric lemma is asserted rather than rigorously demonstrated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22344,"tokens_out":26242,"duration_ms":229070,"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":[{"comment":"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.","section":"§3.2, Lemma 3.8 and Theorem 3.13"},{"comment":"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.","section":"§4.1, Theorem 4.8"},{"comment":"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.","section":"§3.3, Lemma 3.15 and Theorem 3.11"}],"minor_comments":[{"comment":"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.","section":"§3.2, Definition 3.10 and the example after Definition 3.12"},{"comment":"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.","section":"§3.2, after Definition 3.12"},{"comment":"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.","section":"§3.2, proof of Theorem 3.13"},{"comment":"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.","section":"§4.1, proof of Theorem 4.7"},{"comment":"The reference [HL05] is listed as 'Discete Math.'; it should be 'Discrete Math.'","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses an interesting and timely problem, and the main theorem is plausible. However, the published version must contain a complete proof of Lemma 3.8; in its current form the proof is a sketch of the central geometric step. I also recommend that the authors clarify the sufficiency part of the insertability proof and fix the star-placement typography in the examples. These are within the scope of a revision and do not require a fundamentally new approach."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. This paper does two things: it proves a schedule formula for valley-marked parking functions (Theorem 3.13), generalizing the HL05 schedules to the Delta Conjecture setting, and it uses that formula to propose a monomial basis for the super-diagonal coinvariant ring SDR_n (Conjecture 4.3). The marked insertion algorithm is a real new idea, and the schedule numbers attached to ordered set partitions are natural. The main theorem is proved by induction using the insertion lemma plus an insertability result; the latter is handled properly. I also like Theorem 4.7, where the authors extend the classical area+dinv bound to marked parking functions with an explicit injection; that is a clean piece of combinatorics on its own.\n\nThe soft spot is Lemma 3.8. The proof is a sketch: it asserts that a newly inserted marked car creates diagonal inversions exactly with the chosen insertion site and all sites to its left, and none to the right, giving consecutive dinv values. That geometric fact is load-bearing for the q-integer product in Theorem 3.13, but the one-paragraph argument doesn't actually prove it. The sentence about not creating inversions to the right of s or b is ambiguous, especially when c ends up directly under an upper-diagonal car. I don't have a counterexample, and I expect the lemma is true, but a referee should require a full argument before publication.\n\nTwo smaller things: the n=4 and n=5 computer checks are mentioned but no code or data is included, so they're not reproducible; and Theorem 4.8's claim that every cell above D(a) is in the image of φ or ψ is asserted a bit quickly. Neither is fatal, but both should be tightened.\n\nThe paper is honest: all conjectures are labeled, the reliance on Zabrocki's conjecture is motivational rather than circular, and the Shuffle Theorem is used only as a proved benchmark. The citations look right for this area. This is a solid contribution for anyone working on parking functions, diagonal coinvariants, or the Delta Conjecture. It deserves a serious referee. I'd recommend sending it to peer review, with the expectation that Lemma 3.8's proof be expanded and the small computations made reproducible.","headline":"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.","tokens_in":22883,"tokens_out":4862,"would_cite":true,"duration_ms":45011,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","05A15","05A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Marked parking functions of each ordered-set-partition type sum to a $t^{\\operatorname{maj}}$ product of q-integers.","keywords":["Delta Conjecture","valley-marked parking functions","schedules","ordered set partitions","Hilbert series","super-diagonal coinvariants","diagonal coinvariants","q-integers"],"falsifier":"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.","tokens_in":21847,"feed_emoji":"🚗","tokens_out":12011,"duration_ms":104042,"temperature":0.7,"pith_summary":"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.","feed_headline":"Schedules shrink Delta Conjecture sums to a single product","feed_subtitle":"Grouping marked parking functions by ordered set partitions yields a product formula and a candidate basis for super-diagonal coinvariants.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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 $\\operatorname{area}+\\operatorname{dinv}+\\binom{k+1}{2}=\\binom{n}{2}$."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the original schedule formula for unmarked parking functions that Theorem 3.13 generalizes.","marker":"[HL05]"},{"why":"defines valley-marked parking functions, their dinv statistic, and the Delta Conjecture whose combinatorial side is being enumerated.","marker":"[HRW18]"},{"why":"introduces SDR_n and conjectures its Frobenius characteristic equals the z-graded Delta Conjecture sum, the target of Conjectures 3.14 and 4.3.","marker":"[Zab19]"},{"why":"gives the monomial basis for DR_n built from schedule numbers that motivates the candidate basis B_n.","marker":"[CO18]"},{"why":"proves the Shuffle Theorem, providing the unmarked t^{area}q^{dinv} expansion that the marked formula extends.","marker":"[CM18]"},{"why":"proves the q,t-Schröder formula for hook coefficients of ∇e_n used in the alternant degree bound.","marker":"[Hag04]"},{"why":"identifies Frob(DR_n) with ∇e_n, connecting the parking-function Hilbert series to the coinvariant ring.","marker":"[Hai01]"},{"why":"proves the b=0 case of the degree inequality used to identify the top θ-degree component.","marker":"[Wal19]"}],"fun_headline_variants":["Schedules turn Delta Conjecture sums into product formulas","Insertion algorithm yields product formula for Delta Conjecture","Schedule-based basis for super-diagonal coinvariants proposed","Delta Conjecture sums reduced to single product via schedules","New proof: Delta Conjecture schedules formula via insertion trees"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Schedules turn Delta Conjecture sums into product formulas","Insertion algorithm yields product formula for Delta Conjecture","Schedule-based basis for super-diagonal coinvariants proposed","Delta Conjecture sums reduced to single product via schedules","New proof: Delta Conjecture schedules formula via insertion trees"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000241,"raw_usage":{"total_tokens":1578,"prompt_tokens":1060,"completion_tokens":518,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":676,"completion_tokens_details":{"reasoning_tokens":436}},"tokens_in":676,"tokens_out":518,"duration_ms":4841,"temperature":1.0,"reasoning_tokens":436,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:33:25.828812+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}