Dyck paths on black-and-white lattices

Sela Fried

arxiv: 2606.18754 · v1 · pith:TCKLITXJnew · submitted 2026-06-17 · 🧮 math.CO

Dyck paths on black-and-white lattices

Sela Fried This is my paper

Pith reviewed 2026-06-26 20:40 UTC · model grok-4.3

classification 🧮 math.CO

keywords Dyck pathsCatalan numberslattice pathschessboard coloringcolumn-alternating coloringenumerationblack and white cells

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The pith

Dyck paths with equal black and white cells beneath them are enumerated for chessboard and column-alternating grid colorings.

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

The paper sets out to enumerate Dyck paths of semilength n from (0,0) to (n,n) that never rise above the diagonal y=x and that have equal numbers of black and white cells under them. It applies this balance condition under two explicit grid colorings: the chessboard pattern and the column-alternating pattern. A sympathetic reader would care because the standard count of all Dyck paths is the Catalan numbers, so the color balance supplies a natural refinement that may produce new sequences or generating functions. If the enumerations succeed, they give concrete counts for each n and each coloring.

Core claim

We color the cells of the integer grid in black and white according to two natural patterns, namely chessboard and column-alternating, and enumerate the Dyck paths having equal numbers of black and white cells beneath them.

What carries the argument

The chessboard coloring and the column-alternating coloring of the integer grid, which define the numbers of black and white cells beneath a Dyck path and allow the equality condition to be imposed.

If this is right

The enumeration supplies explicit sequences counting the balanced paths for each of the two colorings.
These sequences refine the Catalan numbers by the additional color-balance constraint.
The balance condition partitions the full set of Dyck paths into those with equal black and white cells and those without.

Where Pith is reading between the lines

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

Similar balance conditions could be imposed on other families of lattice paths such as Motzkin or Schröder paths.
The column-alternating coloring introduces a periodic flavor that might link to generating functions with roots of unity or periodic coefficients.
Bijective proofs connecting the balanced paths to other Catalan objects counted by the same numbers would strengthen the results.

Load-bearing premise

The two colorings are well-defined on the integer grid such that the number of cells of each color beneath any given Dyck path is unambiguously determined.

What would settle it

For small fixed n, compute by hand or program the exact number of Dyck paths satisfying the equal-color condition under each coloring and check whether the numbers match the paper's enumeration formula.

Figures

Figures reproduced from arXiv: 2606.18754 by Sela Fried.

**Figure 2.** Figure 2: The two colorings studied in this work. Given a Dyck path p, let black(p) and white(p) denote the numbers of black and white cells that lie below p. We say that p is black-white balanced (with respect to a given coloring, which should be clear from context) if black(p) = white(p). Our goal is to determine, for each of the two colorings, how many Dyck paths of semilength n are black-white balanced. For ex… view at source ↗

**Figure 3.** Figure 3: The 14 Dyck paths of semilength 4 with chessboard coloring. The [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: The distribution of the right-steps in black-white balanced Dyck [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: An illustration of the bijection described in the proof of Theorem [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: The 14 Dyck paths of semilength 4 with column-alternating col [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

read the original abstract

A Dyck path of semilength $n$ is a lattice path from $(0,0)$ to $(n,n)$ consisting of $n$ right-steps $(1,0)$ and $n$ up-steps $(0,1)$ that never rises above the line $y=x$. These paths are enumerated by the Catalan numbers and play a central role in enumerative combinatorics. We color the cells of the integer grid in black and white according to two natural patterns, namely chessboard and column-alternating, and enumerate the Dyck paths having equal numbers of black and white cells beneath them.

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.

Desk Editor's Note private letter to a colleague

The paper counts Dyck paths with equal black and white cells under two explicit grid colorings, but supplies no formulas or small-n data in the abstract so the actual results stay opaque.

read the letter

The core contribution is a pair of colorings on the grid—chessboard by (i+j) mod 2 and column-alternating by x-coordinate parity—plus the count of Dyck paths of semilength n whose area below the path has equal numbers of each color. The definitions line up with standard lattice-path conventions and the balance condition is unambiguous, so the setup itself is clean.

What the work does is take the usual Catalan setting and impose this extra parity constraint on the colored cells. That is a modest but honest refinement; similar colored or weighted Dyck-path problems appear in the literature, and this version does not obviously collapse to a prior result.

The main limitation is that the abstract states the paths are enumerated without giving a closed form, generating function, recurrence, or even the sequence for n=1 to 5. Without those, it is impossible to judge whether the numbers are interesting, whether they match known sequences, or whether the proofs are routine or require new ideas. The full text presumably contains the details, but on the visible evidence the claim rests on an unshown enumeration.

This is niche enumerative combinatorics aimed at readers already working on refined Catalan objects or colored lattice paths. A specialist might extract a usable sequence or generating function from it; a general reader will not. The problem is well-posed and the colorings are standard, so it clears the bar for peer review even if the eventual formulas turn out to be unsurprising.

Referee Report

0 major / 2 minor

Summary. The manuscript defines Dyck paths of semilength n (lattice paths from (0,0) to (n,n) with right and up steps never rising above y=x) and two grid colorings (chessboard via parity of i+j, column-alternating via parity within each fixed-x column). It enumerates those paths for which the number of black and white unit squares in the region beneath the path are equal.

Significance. If explicit formulas or generating functions are derived and verified, the results would refine the Catalan numbers by a color-balance statistic under two natural colorings, adding to the body of work on weighted or colored Dyck paths and potentially yielding new bijections or q-analogues.

minor comments (2)

[Abstract] Abstract: the claim of enumeration is stated without any explicit formula, generating function, or proof outline, which slows assessment of the central contribution even though the full text presumably supplies these.
The region 'beneath' the path is described as standard, but a brief sentence confirming it consists of unit squares strictly below the path (with no boundary ambiguity for the equality condition) would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their concise summary of the manuscript and for recommending minor revision. No major comments were listed in the report, so there are no specific points requiring point-by-point response or revision.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a direct combinatorial enumeration of Dyck paths (semilength n) satisfying an equality condition on the number of black and white unit squares beneath the path, under two explicitly defined grid colorings (chessboard by (i+j) parity; column-alternating by column parity). Both colorings and the 'beneath' region are standard, unambiguous lattice definitions independent of the enumeration result itself. No equations, fitted parameters, self-citations, or ansatzes are present in the provided text that would reduce the claimed counts to the inputs by construction. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.1-grok · 5612 in / 1076 out tokens · 25966 ms · 2026-06-26T20:40:05.367068+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 2 canonical work pages

[1]

Cheng, S.-P

S.-E. Cheng, S.-P. Eu, and T.-S. Fu, Area of Catalan paths on a checkerboard,European J. Combin.28(2007), no. 4, 1331–1344. https://doi.org/10.1016/j.ejc.2006.01.006

work page doi:10.1016/j.ejc.2006.01.006 2007
[2]

Fried, Black-white cell capacity ink-ary words and permutations, arXiv preprint arXiv:2509.07533 [math.CO], 2025.https://arxiv

S. Fried, Black-white cell capacity ink-ary words and permutations, arXiv preprint arXiv:2509.07533 [math.CO], 2025.https://arxiv. org/abs/2509.07533

arXiv 2025
[3]

R. P. Stanley,Catalan Numbers, Cambridge University Press, 2015

2015
[4]

N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, OEIS Foundation Inc.,https://oeis.org
[5]

R. A. Sulanke, Moments, Narayana numbers, and the cut and paste for lattice paths,J. Statist. Plann. Inference135(2005), no. 1, 229–244. https://doi.org/10.1016/j.jspi.2005.02.016 8

work page doi:10.1016/j.jspi.2005.02.016 2005

[1] [1]

Cheng, S.-P

S.-E. Cheng, S.-P. Eu, and T.-S. Fu, Area of Catalan paths on a checkerboard,European J. Combin.28(2007), no. 4, 1331–1344. https://doi.org/10.1016/j.ejc.2006.01.006

work page doi:10.1016/j.ejc.2006.01.006 2007

[2] [2]

Fried, Black-white cell capacity ink-ary words and permutations, arXiv preprint arXiv:2509.07533 [math.CO], 2025.https://arxiv

S. Fried, Black-white cell capacity ink-ary words and permutations, arXiv preprint arXiv:2509.07533 [math.CO], 2025.https://arxiv. org/abs/2509.07533

arXiv 2025

[3] [3]

R. P. Stanley,Catalan Numbers, Cambridge University Press, 2015

2015

[4] [4]

N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, OEIS Foundation Inc.,https://oeis.org

[5] [5]

R. A. Sulanke, Moments, Narayana numbers, and the cut and paste for lattice paths,J. Statist. Plann. Inference135(2005), no. 1, 229–244. https://doi.org/10.1016/j.jspi.2005.02.016 8

work page doi:10.1016/j.jspi.2005.02.016 2005