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arxiv: 2605.13003 · v1 · submitted 2026-05-13 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

Dyck Symmetric Functions and Applications to \(q,t\)-Catalan Polynomials

Graham Hawkes
Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:51 UTC · model grok-4.3

classification 🧮 math.CO
keywords Dyck sequencesDyck tableauxq,t-Catalan polynomialsSchur positivitysymmetric functionsrow insertionbijectiondeficit formulas
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The pith

Row-insertion on dual Dyck sequences produces a weight-preserving bijection to Dyck tableaux paired with semistandard Young tableaux of the same shape.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a row-insertion algorithm that acts on dual Dyck sequences and extends it to Dyck tableaux. This map sends dual Dyck factorizations to pairs consisting of one Dyck tableau and one semistandard Young tableau of identical shape, while keeping all weights intact. The bijection immediately shows that the dual Dyck symmetric functions expand positively in the Schur basis and that the associated affine versions expand in conjugate Schur functions. The same objects are then used to rewrite the q,t-Catalan polynomial as a sum over two-column Dyck tableaux indexed by Dyck m-skeletons, and to give an explicit interval formula for the low-deficit range defc ≤ 2n-8.

Core claim

The central construction is a row-insertion procedure on dual Dyck sequences that extends to Dyck tableaux and yields a weight-preserving bijection between dual Dyck factorizations and pairs (Dyck tableau, semistandard Young tableau of the same shape). This bijection proves that dual Dyck symmetric functions are Schur-positive and that affine Dyck symmetric functions have conjugate-shape Schur expansions. The same machinery supplies a two-column tableau formula for C_n(q,t) and organizes the low-area half of each low-deficit slice of C_n(q,t) via Dyck skeletons together with local East, West, up, and down moves.

What carries the argument

The row-insertion algorithm on dual Dyck sequences, extended to Dyck tableaux, which produces the weight-preserving bijection to pairs of a Dyck tableau and a semistandard Young tableau.

Load-bearing premise

The row-insertion algorithm extends consistently to Dyck tableaux while preserving the required weights and satisfying the commutation or jeu-de-taquin relations needed for the Schur expansion.

What would settle it

A concrete dual Dyck sequence of length five whose row-insertion produces a pair whose total weight differs from the original factorization weight, or a small dual Dyck symmetric function whose Schur expansion contains a negative coefficient.

read the original abstract

This paper develops three related combinatorial results for Dyck-type sequences. First, it constructs a row-insertion algorithm for dual Dyck sequences and extends it to Dyck tableaux. This construction gives a weight-preserving bijection between dual Dyck factorizations and pairs consisting of a Dyck tableau and a semistandard Young tableau of the same shape. As a consequence, the associated dual Dyck symmetric functions are Schur-positive, and the corresponding affine Dyck symmetric functions have the conjugate-shape Schur expansion. Second, it applies these Dyck symmetric functions to the \(q,t\)-Catalan polynomial. It gives a two-column tableau formula for \(C_n(q,t)\), expressing it as a sum over Dyck \(m\)-skeletons and at-most-two-column Dyck tableaux with summands involving two-variable Schur functions. Third, it develops a Dyck-skeleton formula for the deficit range \(\defc\le 2n-8\). Full and special Dyck skeletons, together with local \(\mathrm{East}\), \(\mathrm{West}\), \(\mathrm{up}\), and \(\mathrm{down}\) moves, organize the low-area half of each low-deficit slice into skeleton-indexed strings. The \(q,t\)-symmetry of \(C_n(q,t)\) supplies the complementary high-area half in the resulting interval formula.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Circularity Check

0 steps flagged

No circularity: derivation rests on explicit combinatorial construction

full rationale

The paper constructs a row-insertion algorithm for dual Dyck sequences, extends it to Dyck tableaux, and uses the resulting weight-preserving bijection to establish Schur-positivity of the dual Dyck symmetric functions. This is a direct, self-contained combinatorial argument with no reduction of the target quantities to fitted parameters, self-definitional loops, or load-bearing self-citations. The abstract and derivation chain describe an algorithmic extension and bijection without invoking prior results by the same author as an unverified uniqueness theorem or ansatz. The q,t-Catalan applications follow from the same explicit objects. No step equates a prediction to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions of Dyck paths, tableaux, and symmetric functions; no new free parameters, invented entities, or non-standard axioms are introduced in the abstract.

axioms (2)
  • domain assumption Dyck sequences and their duals are well-defined combinatorial objects with standard weight functions (area, bounce, etc.).
    Invoked throughout the construction of the insertion algorithm and the formulas for C_n(q,t).
  • domain assumption The row-insertion procedure commutes appropriately with jeu-de-taquin or rectification to produce a valid bijection.
    Required for the weight-preserving property and the Schur expansion to hold.

pith-pipeline@v0.9.0 · 5541 in / 1464 out tokens · 36229 ms · 2026-05-14T18:51:09.354953+00:00 · methodology

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Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages

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    work page doi:10.1007/s00026-023-00662-2 2024