The height of Dyck paths and checkerboard labellings

Abstract: Dyck paths and certain black/white labelling of nodes leads to the \emph{white-height}. Using generating functions and tricks of the trade, we establish that the average white-height among Dyck paths of half length $n$ is asymptotic to $\frac12\sqrt{πn}$ for two different models. These are appealing results that could be presented to students to learn the trade.

• The paper derives exact binomial-sum enumerations for Dyck paths exceeding a given white-height in two parity-based checkerboard labelling models.
• Using kernel-method substitutions, Mellin transforms, and singularity analysis, it proves that the average white-height grows as one-half sqrt(pi n), matching half the usual Dyck-path height.
• The two models differ in lower-order behavior: average white-height is one-half sqrt(pi n) plus 1 in the ordinate-parity model and one-half sqrt(pi n) minus one-half in the checkerboard model.

Overview

The paper under review, "The height of Dyck paths and checkerboard labellings" (2606.10035), studies a colouring statistic on Dyck paths. Nodes lying between a Dyck path and the $x$-axis are coloured black or white according to a parity rule, and the white-height of a path is the maximum, over all abscissae, of the number of white nodes at that abscissa. The author derives exact enumerations of Dyck paths by white-height and computes the average white-height over all Dyck paths of semilength $n$ for two distinct labelling models. The main results are that the average white-height is asymptotic to $\frac{1}{2}\sqrt{\pi n}$ in both models, differing only in lower-order terms. The paper is explicitly framed as a pedagogical exposition of the de Bruijn–Knuth–Rice methodology [deBrKnRi], combining the kernel-method substitution $z = u/(1+u)^2$ with Mellin transforms and singularity analysis.

Model I: labelling by parity of ordinates

In the first model, the colour of a node is determined by the parity of its ordinate, which produces a checkerboard pattern after rotating the path by $45^\circ$. A first-return decomposition creates a technical complication: chopping off the first and last steps of a sojourn swaps the initial colour from white to black. The author therefore introduces two interlocking families of generating functions,

$$A_h = \frac{1}{1 - zB_{h-1}}, \quad A_0 = 0, \qquad B_h = \frac{1}{1 - zA_h}, \quad B_0 = 1,$$

where $A_h$ counts paths beginning white-black-… with white-height $\le h$ and $B_h$ counts paths beginning black-white-…. Eliminating $B_h$ yields a continued fraction for $n$0, and the substitution $n$1 — the standard device from de Bruijn, Knuth, and Rice — linearizes the recursion. The closed forms

$n$2

are found by guessing second-order recursions for numerators and denominators with the computer algebra package gfun, then verified by induction. The tail $n$3 counts paths of white-height exceeding $n$4.

Coefficient extraction via Cauchy's integral formula and the change of variables produces an exact binomial sum: the number of paths of semilength $n$5 with white-height $n$6 equals

$n$7

For the asymptotics, the author sums over $n$8, sets $n$9, and evaluates the Mellin transform

$\frac{1}{2}\sqrt{\pi n}$0

Shifting the contour left and collecting residues at $\frac{1}{2}\sqrt{\pi n}$1 gives the expansion $\frac{1}{2}\sqrt{\pi n}$2 in the $\frac{1}{2}\sqrt{\pi n}$3-plane. Singularity analysis then yields a coefficient asymptotic of $\frac{1}{2}\sqrt{\pi n}$4; dividing by the Catalan number asymptotic $\frac{1}{2}\sqrt{\pi n}$5 produces the main result of this section:

$\frac{1}{2}\sqrt{\pi n}$6

The leading constant $\frac{1}{2}\sqrt{\pi n}$7 matches the heuristic that white-height is essentially half the ordinary Dyck path height, whose average is asymptotically $\frac{1}{2}\sqrt{\pi n}$8; the additive constant $\frac{1}{2}\sqrt{\pi n}$9 is a genuine correction beyond that heuristic.

Model II: the checkerboard model

In the second model the colour is determined by the parity of abscissa plus ordinate, giving a checkerboard pattern without rotation. Here no auxiliary sequence is needed, but the recursion is subtler because white-height interacts with ordinary height: a sojourn of white-height $z = u/(1+u)^2$0 has ordinary height $z = u/(1+u)^2$1 or $z = u/(1+u)^2$2, so paths of white-height $z = u/(1+u)^2$3 decompose through paths of ordinary height $z = u/(1+u)^2$4. Using the classical height generating function $z = u/(1+u)^2$5, the author obtains

$z = u/(1+u)^2$6

with tail $z = u/(1+u)^2$7. The same coefficient extraction gives a binomial sum with step $z = u/(1+u)^2$8 instead of $z = u/(1+u)^2$9, and the Mellin transform simplifies to $45^\circ$0. Residue collection yields the $45^\circ$1-expansion $45^\circ$2, and division by the Catalan asymptotic gives

$45^\circ$3

The structural parallel between the two models is notable: the closed forms for $45^\circ$4 in Model I and $45^\circ$5 in Model II are identical up to an index shift, which explains why both models share the same leading asymptotics $45^\circ$6 and differ only in the additive constant ($45^\circ$7 versus $45^\circ$8).

Methodological remarks and limitations

The analytic pipeline — kernel substitution, Lagrange-style contour extraction of binomial sums, Mellin transform of the resulting lattice-sum generating function, residue collection, and singularity analysis — is the one developed in [deBrKnRi] and revisited by the author in earlier work [prodinger-ars], with technical details of the residue computations discussed at greater length in [HPW]. The paper presents the residue calculations as performed "with a computer" and does not display full error-term analysis; the "smaller order terms" in both main results are not made explicit, so the stated asymptotic expansions are leading-order statements rather than complete asymptotic expansions with controlled remainders. The paper also does not establish distributional results (e.g., a limiting law for white-height) or variance estimates, which the classical ordinary-height analysis does provide; these remain open questions raised by the present treatment. The author notes the presentation is intended to have pedagogical value, and the derivations are correspondingly self-contained.

Conclusion

The paper extends the de Bruijn–Knuth–Rice height analysis to Dyck paths with checkerboard-style black/white labellings, giving exact binomial-sum formulas for the number of paths exceeding a given white-height in two labelling models and proving that the average white-height among Dyck paths of semilength $45^\circ$9 is $$A_h = \frac{1}{1 - zB_{h-1}}, \quad A_0 = 0, \qquad B_h = \frac{1}{1 - zA_h}, \quad B_0 = 1,$$0 in the parity-of-ordinate model and $$A_h = \frac{1}{1 - zB_{h-1}}, \quad A_0 = 0, \qquad B_h = \frac{1}{1 - zA_h}, \quad B_0 = 1,$$1 in the checkerboard model. The agreement of the leading terms across models, with a shift of $$A_h = \frac{1}{1 - zB_{h-1}}, \quad A_0 = 0, \qquad B_h = \frac{1}{1 - zA_h}, \quad B_0 = 1,$$2 in the constants, is the paper's cleanest quantitative observation. Natural open problems include the full asymptotic expansion with explicit error terms, the limiting distribution of white-height, and the extension to further colouring rules of the kind studied in [Baril] and [china].