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arxiv: 2403.06318 · v4 · submitted 2024-03-10 · 🧮 math.CO

Lattice Points and Rational q-Catalan Numbers

Drew Armstrong This is my paper

Pith reviewed 2026-05-24 03:00 UTC · model grok-4.3

classification 🧮 math.CO
keywords rational q-Catalan numbersJohnson statisticsroot latticelattice pointstype Aq-analoguesBrion's theorem
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The pith

Johnson statistics on the root lattice of type A generate the coefficients of rational q-Catalan numbers.

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

The paper seeks a new explanation for the nonnegative integer coefficients of the rational q-Catalan number Cat(a,b)_q by working with lattice points in the root lattice rather than with Dyck paths. It conjectures that certain Johnson statistics on the root lattice of type A_{a-1} produce exactly these coefficients and proves the conjecture holds when a is at most 20. The new statistics are shown to obey a q-analogue of Brion's theorem for simplices and other properties that the area-dinv statistic on paths does not obviously explain. A reader would care because the lattice-point model offers an independent combinatorial source for the same polynomials and might settle positivity questions that remain open under the Dyck-path realization.

Core claim

For each pair of coprime integers a and b the rational q-Catalan number Cat(a,b)_q equals the sum over selected points x in the root lattice R of type A_{a-1} of q raised to the value of a Johnson statistic J(x). The paper conjectures that such statistics J exist and generate the same polynomial as the area-dinv statistic on rational Dyck paths but do so independently; the conjecture is established for all a at most 20. These statistics are further shown to satisfy a q-analogue of Brion's theorem for simplices together with several other remarkable properties.

What carries the argument

Johnson statistics: integer-valued functions J defined on the root lattice R of type A_{a-1} that enumerate lattice points to produce the coefficients of Cat(a,b)_q.

If this is right

  • The statistics satisfy a q-analogue of Brion's theorem for simplices.
  • The statistics generate Cat(a,b)_q independently of the area-dinv statistic on Dyck paths.
  • The statistics obey many additional remarkable properties beyond those already known from the Dyck-path model.
  • The conjecture that Cat(a,c)_q minus Cat(a,b)_q has nonnegative coefficients when b is less than c is compatible with this lattice-point realization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The lattice-point approach may eventually prove the difference positivity conjecture that the shuffle conjecture leaves open.
  • Similar Johnson statistics could be defined on root lattices of other types and produce analogous q-Catalan numbers.
  • The model supplies a direct way to study the polynomials by enumeration inside a fixed lattice rather than by path combinatorics that changes with b.

Load-bearing premise

Johnson statistics exist on the root lattice of type A_{a-1} that generate the coefficients of Cat(a,b)_q independently of the area-dinv statistic on Dyck paths.

What would settle it

An explicit computation for some a greater than 20 that exhibits no function J on the root lattice whose generating function equals Cat(a,b)_q for every coprime b, or that violates the q-analogue of Brion's theorem.

Figures

Figures reproduced from arXiv: 2403.06318 by Drew Armstrong.

Figure 1
Figure 1. Figure 1: A p4, 7q-Dyck path. Let Dycka,b be the set of pa, bq-Dyck paths. The case of coprime a and b is special. It has been known since Bizley [6] that # Dycka,b “ 1 a ` b ˆ a ` b a ˙ when gcdpa, bq “ 1. The proof interprets each lattice path in the rectangle as a word of length a ` b with a copies of the letter u (up) and b copies of the letter r (right). For example, the Dyck path in [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 2
Figure 2. Figure 2: The tilted partial order on the weight lattice for [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of “ 4 2 ‰ q “ 1 ` q ` 2q 2 ` q 3 ` q 4 . The F¨urlinger-Hofbauer q-Catalan numbers are defined as Catpnqq “ 1 rnsq „ 2n n ´ 1 ȷ q . It turns out that this is also a polynomial with non-negative integer coefficients. The most elementary proof uses the “major index of Dyck paths”, which according to F¨urlinger and Hofbauer [9] goes back to MacMahon and Aissen. Given an n ˆ n Dyck path P P Dyckn… view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of Catp3, 4qq “ 1 ` q 2 ` q 3 ` q 4 ` q 6 . More recently, these q-Catalan numbers have been generalized to the rational case. We saw in Section 4 that the binomial coefficient ` a´1`b a´1 ˘ is divisible by a whenever gcdpa, bq “ 1, and we called the quotient a rational Catalan number Catpa, bq “ ` a´1`b a´1 ˘ {a. It turns out that the q-binomial coefficient “ a´1`b a´1 ‰ q is also divisible b… view at source ↗
Figure 5
Figure 5. Figure 5: The sweep map. Each of the papers [3, 13, 21] conjectured the symmetry Catpa, bqq,t “ Catpa, bqt,q and the specialization Catpa, bqq “ q pa´1qpb´1q{2 Catpa, bqq,1{q 10It is certainly not obvious how to invert the sweep map. This was solved by Thomas and Williams [32], though the solution is quite complicated. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Tilted partial order on the set L X 5∆ when a “ 3. This fact is not difficult to prove. We give two different proofs for the sake of exposition. First we observe that the tilted partial order on L X b∆ is isomorphic to Stanley’s poset Lpa ´ 1, bq of Young diagrams that fit inside an pa ´ 1q ˆ b rectangle (see [27, Chapter 6]). A Young diagram (also called an integer partition) is a weakly-decreasing sequen… view at source ↗
Figure 7
Figure 7. Figure 7: A Young diagram. This bijection between Young diagrams and lattice paths shows that #Lpa ´ 1, bq “ ` a´1`b a´1 ˘ . The partial order on Lpa ´ 1, bq is defined as the componentwise order on the corresponding integer vectors y “ py1, . . . , ya´1q, which we can think of as “inclusion of diagrams”. Thus the rank of the diagram y in the poset Lpa ´ 1, bq is its area y1 ` ¨ ¨ ¨ ` ya´1, which shows that the rank… view at source ↗
Figure 8
Figure 8. Figure 8: The unique Johnson statistic when a “ 3. Theorem 7.3. Let J : R Ñ Z be a Johnson statistic and let gcdpa, bq “ 1. Then we have Catpa, bqq “ ÿ xPpRXBoxq q Jpxq „ a ´ 1 ` tpb ´ ř i xiq{au a ´ 1 ȷ q a , where the sum is taken over points x “ px1, .., xa´1q in R X Box. Proof. Since R is the disjoint union of cosets x`aL we note that RXb∆ is the disjoint union of the sets px ` aLq X b∆, where x runs over the po… view at source ↗
Figure 9
Figure 9. Figure 9: Two vertex contributions to Catp3, 7qq. We can use this extra level of generality to show that Johnson statistics satisfy a certain q-analogue of Brion’s theorem. For brevity we omit some computational details. First let K “ tpx1, . . . , xa´1q : 0 ď xi for all i u be the positive cone and consider the “rotated” cones Ki b :“ ϕ i b pKq “ tpx1, . . . , xa´1q : 0 ď xj for j ‰ i and aÿ´1 j“1 xj ď bu. 26 [PIT… view at source ↗
Figure 10
Figure 10. Figure 10: The unique ribbon partition when a “ 3. point in the ribbon (the rightmost point in the row). The value of J is shown in the rightmost column. For example, we have Jp1, 1, 3q :“ Tp1, 1, 2q “ 1 ¨ 1 ` 2 ¨ 1 ` 3 ¨ 2 “ 9. Horizontal lines in the table separate the points in the different slices of Box, corresponding to the q-Catalan germs. For example, the slice Boxr6, 7s contains 16 points of L and 4 points … view at source ↗
Figure 11
Figure 11. Figure 11: The remarkable ribbon partition when a “ 4. spaces of invariants. Stanley [25] generalized this idea to other posets arising from Lie groups and then Proctor [23] boiled down the argument to its bare essentials. We briefly describe Proctor’s approach, translated into our language. Let Lpa ´ 1, bq be the poset of Young diagrams fitting in an pa ´ 1q ˆ b rectangle, which is the same as our tilted partial or… view at source ↗
Figure 12
Figure 12. Figure 12: The poset L X Boxr2, 2s for a “ 5 has two ribbon partitions. We can imagine a similar method to produce standard partitions, and hence Johnson statistics. Given one of the posets LXBoxrc 1`1, cs we let V “ V0‘V1‘¨ ¨ ¨‘Va´1 be the cyclically-graded vector space where Vk is the formal sum of points x P pL X Boxrc 1 ` 1, csq satisfying Tpxq ” k mod a. One might might use the poset structure to construct an o… view at source ↗
Figure 13
Figure 13. Figure 13: Johnson statistic of type B2. In type G2 we have pd1, d2q “ p2, 6q, pc1, c2q “ p2, 3q, c “ 6 and f “ 1. Let ω1, ω2 be the fundamental basis of the weight lattice L so the vertices of the fundamental alcove ∆ are 0, ω1{2, ω2{3. The simple roots α1, α2 (expressed in weight coordinates) are the columns of the Cartan matrix C “ ˆ 2 ´1 ´3 2˙ , that is, α1 “ 2ω1 ´ 3ω2 and α2 “ ´3ω1 ` 2ω2. Let us define pw1, w2q… view at source ↗
Figure 14
Figure 14. Figure 14: Johnson statistic of type G2. Acknowledgements We thank the following people for helpful input: Francois Bergeron, Benjamin Braun, There￾sia Eisenk¨olbl, Eugene Gorsky, Christian Krattenthaler, Daniel Provder, Michael Schlosser, Nathan Williams, and the anonymous referees. We thank Heather Armstrong for help with Latex and ChatGPT for help with Sage coding. This project was partially supported by the SPP … view at source ↗
read the original abstract

For each pair of coprime integers $a$ and $b$ we have a rational $q$-Catalan number $\operatorname{Cat}(a,b)_q=\binom{a+b}{a}_q/[a+b]_q$. It is known that this is a polynomial in $q$ with nonnegative integer coefficients, but the nature of these coefficients is still mysterious. Our current understanding is based on the rational shuffle conjecture that was conjectured by Bergeron, Garsia, Leven and Xin in 2014 and proved by Mellit in 2016, based on earlier work with Carlsson. This theorem realizes $\operatorname{Cat}(a,b)_q$ as the generating function for the statistic "area $-\ \mathrm{dinv}+\frac{(a-1)(b-1)}{2}$" defined on rational Dyck paths. However, this statistic is difficult to work with and leaves some phenomena unexplained. For example, it does not prove the conjecture that the difference $\operatorname{Cat}(a,c)_q-\operatorname{Cat}(a,b)_q$ has nonnegative coefficients whenever $\gcd(a,b)=\gcd(a,c)=1$ and $b<c$. The current paper proposes to look at lattice points instead of Dyck paths. Our idea is to fix $a$ and express everything in terms of the weight lattice $\mathrm{L}$ and root lattice $\mathrm{R}$ of type $A_{a-1}$. Based on ideas of Paul Johnson, we conjecture the existence of certain "Johnson statistics" $J:\mathrm{R}\to\mathbb{Z}$ and we prove this conjecture for $a\le 20$. We show that these statistics satisfy many remarkable properties including a $q$-analogue of Brion's theorem for simplices.

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.

Referee Report

1 major / 2 minor

Summary. The paper proposes an alternative lattice-point interpretation of the rational q-Catalan numbers Cat(a,b)_q for coprime a,b, based on the weight lattice L and root lattice R of type A_{a-1}. It conjectures the existence of 'Johnson statistics' J:R→Z that generate the coefficients of Cat(a,b)_q, proves this conjecture computationally for a≤20, and shows that the statistics satisfy several properties, including a q-analogue of Brion's theorem for simplices. This is positioned as a potential improvement over the area-dinv statistic on rational Dyck paths from the rational shuffle conjecture.

Significance. If the Johnson statistics exist in general and are independent of the Dyck-path model, the approach could yield more tractable proofs for open positivity conjectures, such as the nonnegativity of Cat(a,c)_q - Cat(a,b)_q when b<c. The manuscript explicitly credits the computational verification for a≤20 and the establishment of the q-Brion analogue as concrete advances; the former supplies machine-checkable evidence within a bounded range, which is a clear strength of the work.

major comments (1)
  1. [Abstract and computational verification section] Abstract (conjecture statement and verification claim): the assertion that the conjecture is proved for a≤20 is load-bearing for the paper's main results, yet the abstract (and presumably the corresponding section) supplies no description of the algorithm, input representation of the root lattice, error-handling protocol, or method used to verify the q-Brion property. Without these details the computational evidence cannot be independently assessed.
minor comments (2)
  1. [Introduction] The relationship between the new Johnson statistics and the existing area-dinv statistic is stated to be independent, but a brief explicit comparison (even for a single small a) would help readers see the distinction.
  2. [Section 2 (lattice definitions)] Notation for the lattices L and R is introduced without an accompanying figure or explicit basis description; adding one would improve readability for readers outside type-A combinatorics.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need for greater transparency in our computational claims. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and computational verification section] Abstract (conjecture statement and verification claim): the assertion that the conjecture is proved for a≤20 is load-bearing for the paper's main results, yet the abstract (and presumably the corresponding section) supplies no description of the algorithm, input representation of the root lattice, error-handling protocol, or method used to verify the q-Brion property. Without these details the computational evidence cannot be independently assessed.

    Authors: We agree that the current manuscript does not supply sufficient detail on the verification procedure. In the revised version we will add an explicit subsection describing (i) the algorithm used to enumerate lattice points in the relevant fundamental domain of the root lattice R, (ii) the integer-coordinate representation of those points, (iii) the error-handling and exact-arithmetic protocol employed, and (iv) the direct computational check of the q-Brion identity on the simplices. These additions will be placed in the computational-verification section and referenced from the abstract, thereby making the evidence independently reproducible while leaving the statement of the conjecture and the range a≤20 unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper explicitly frames its central claim as a conjecture on the existence of Johnson statistics J:R→Z that generate the coefficients of Cat(a,b)_q, with a proof supplied only for a≤20 together with additional properties such as a q-analogue of Brion's theorem. No step reduces by definition or construction to its own inputs, no fitted parameter is relabeled as a prediction, and the argument does not depend on load-bearing self-citations or imported uniqueness theorems; the conjecture is stated independently and verified against external q-Catalan data within the limited range.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the conjectural existence of Johnson statistics, which are not derived from prior results but proposed here and verified only computationally for small a; standard q-analog and lattice axioms are invoked without new free parameters.

axioms (2)
  • standard math q-binomial coefficients and q-integers are the standard q-analogues used to define Cat(a,b)_q.
    Definition of the rational q-Catalan number in the abstract.
  • standard math Weight lattice L and root lattice R of type A_{a-1} carry the usual inner product and root system structure from Lie theory.
    The paper expresses all constructions inside these lattices.
invented entities (1)
  • Johnson statistics J: R → Z no independent evidence
    purpose: Assign integer weights to root lattice points so that the generating function equals Cat(a,b)_q.
    Newly conjectured object whose existence is asserted and verified only for a≤20.

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