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Enumerating several statistics of r-Colored Dyck paths with no dd-steps having the same colors

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Pith's one-line read The paper proves explicit closed-form formulas for four statistics on r-colored Dyck paths with no two consecutive down-steps sharing a color, with the first three encoded as Riordan arrays and the fourth as a product formula.

desk verdict Solid, competent Riordan-array enumeration of four statistics on a natural new class of colored Dyck paths; the formulas look right and the last-return decomposition is sound, but several proof details are skipped. read the letter →

arxiv 2506.08407 v1 pith:FNGTJ2GM submitted 2025-06-10 math.CO

Yidong Sun , Jinyi Wang , Xinyu Wang This is my paper
classification math.CO MSC 05A1505A1005A19
keywords r-coloredDyckpathsnoequal-colordd-stepsRiordanarray(ab)-SchröderLagrangeinversionlatticepathstatisticsudu-stepsclosed-formenumeration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper works with the set $\mathcal{A}_{n,0}^{(r)}$ of $r$-colored Dyck paths of length $2n$ in which consecutive down-steps never have the same color, and it derives explicit counting formulas for four statistics: the number of points at level $\ell$, the number of up-steps at level $\ell+1$, the number of peaks at level $\ell+1$, and the number of $udu$-steps. For the first three statistics the counts are given as finite binomial sums, and the triangles of counts are shown to be entries of Riordan arrays whose second component is always $x(r-1)S_r(x)^2$, where $S_r(x)=S(1,r-1;x)$ is the generating function of weighted $(1,r-1)$-Schröder paths. For the fourth statistic the paper proves the product formula $T_{n,\ell}^{(r)}=\binom{n-1}{\ell}T_{n-\ell,0}^{(r)}r^{\ell}$ and gives an explicit sum for the $\ell=0$ case. A closing section transfers the three Riordan-array results to weighted $(a,b)$-Dyck paths. A sympathetic reader would care because the formulas turn a neighborhood-restricted path model into concrete, checkable numbers rather than a bare recurrence.

What carries the argument

The machinery is the last-return decomposition combined with a Riordan-array representation. A Riordan array is an infinite lower-triangular matrix whose $k$-th column has generating function $d(x)h(x)^k$; here the second component is always $h(x)=x(r-1)S_r(x)^2$, and the first component changes by statistic. The function $S_r(x)=S(1,r-1;x)$ is the weighted generating function of $(1,r-1)$-Schröder paths and satisfies $S_r(x)=1+xS_r(x)+x(r-1)S_r(x)^2$, which is the quadratic that Lagrange inversion turns into binomial sums. The auxiliary Lemma 2.2 packages a nested composition $Z_m(a,b;x)=S\left(-a,\frac{b^{2m}}{a^{2m-1}};x\right)$ and yields the identity $S_r(x)S_r\left(-x(r-1)S_r(x)^2\right)=S(1,-(r-1)^2;x)$ used for the corollaries. For the $udu$ statistic the load-bearing object is the generating function $T_0(x)$ of $udu$-avoiding paths, whose quadratic functional equation encodes the three forms $\varepsilon$, $ud_j$, and $uD_1d_jD_2$.

What would settle it

Enumerate all 3-colored Dyck paths of length 8 (that is, $n=4$) with no two consecutive down-steps of the same color, and count the total number of points at level 2. The formula in Theorem 3.1 evaluates to $P_{4,2}^{(3)}=1128$; if a complete direct enumeration gives any other number, the last-return decomposition used in the proof is not a bijection and the Riordan-array formulas fail.

Watch

Extended reading notes

Core claim

The central claim is that the generating function $S_r(x)=S(1,r-1;x)$ carries the entire enumeration. After the last-return decomposition $D=D_1uD_2d_j$ with $D_1,D_2\in\mathcal{A}_{\cdot,0}^{(r)}$, the color rule forces exactly $r-1$ choices for the final down-step $d_j$ when $D_2$ is nonempty and $r$ choices when $D_2$ is empty; summing over this decomposition gives recurrences that solve to the Riordan-array identities: the point counts satisfy $P_{n,\ell}^{(r)}+\frac{1}{r-1}\delta_{n,0}$ as the $(n,\ell)$-entry of $\left(S_r(x)^2+\frac{1}{r-1},\,x(r-1)S_r(x)^2\right)$, the up-step counts are the entries of $\left(S_r(x)^2(1+(r-1)S_r(x)),\,x(r-1)S_r(x)^2\right)$, and the peak counts are the entries of $\left(rS_r(x)^2,\,x(r-1)S_r(x)^2\right)$. Lagrange inversion applied to $S_r(x)=1+xS_r(x)+x(r-1)S_r(x)^2$ converts each Riordan entry into the displayed binomial sum. For $udu$-steps the paper proves a bijective deletion argument: deleting the $d_ju$ steps of each $udu$ occurrence sends a path with $\ell$ $udu$'s to a $udu$-avoiding path, each choice of an endpoint contributing a factor $r$, giving $T_{n,\ell}^{(r)}=\binom{n-1}{\ell}T_{n-\ell,0}^{(r)}r^{\ell}$, with $T_{n,0}^{(r)}$ itself extracted by Lagrange inversion from the quadratic $T_0(x)=1+rx-(r-1)xT_0(x)+(r-1)xT_0(x)^2$.

Load-bearing premise

The recurrences in Sections 3-5 all rest on the assertion that in the decomposition $D=D_1uD_2d_j$ of a path in $\mathcal{A}_{n+1,0}^{(r)}$, the color of the final down-step $d_j$ is independent of everything else: it has $r-1$ choices when $D_2$ is nonempty and $r$ choices when $D_2$ is empty, with no additional restriction coming from the last step of $D_1$ or the first step of $D_2$.

Editorial extensions

If this is right

  • The point statistic $P_{n,\ell}^{(r)}$ can be read directly from the Riordan array $\left(S_r(x)^2+\frac{1}{r-1},\,x(r-1)S_r(x)^2\right)$, so row sums, alternating sums, and generating functions of level totals are available without further recurrences.
  • The specialization $r=2$ recovers statements about large Schröder numbers: for example, the alternating row-sum identity $\sum_{\ell=0}^n(-1)^\ell P_{n,\ell}^{(2)}=S_n$ and analogous identities for up-steps and peaks.
  • For peaks, the parameter-weighted identity of Corollary 5.3 with $m=1$ gives the clean linear relation $\sum_{\ell=0}^n(-1)^\ell(\ell+1)p_{n,\ell}^{(r)}=r(n+1)$, showing that an alternating moment of the peak distribution is exactly linear in $n$.
  • The $udu$ statistics satisfy $T_{n,\ell}^{(r)}=\binom{n-1}{\ell}T_{n-\ell,0}^{(r)}r^{\ell}$ and the moment identity $\sum_{\ell=1}^n\ell T_{n+1,\ell}^{(r)}=rnS_n^{(r)}$, linking total $udu$-occurrences to the Schröder-like numbers $S_n^{(r)}$.
  • The same Riordan-array derivation goes through for weighted $(a,b)$-Dyck paths, giving parameterized analogues (Theorems 7.1-7.3) that specialize to the colored results when $a=r$, $b=r-1$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the only input to the recurrences is the last-return decomposition and the choice counts $r$/$r-1$, the same proof should work for the variant that forbids equal colors on runs of down-steps of length at most $k$, yielding a family of Riordan arrays parameterized by $k$.
  • The product formula for $udu$-steps suggests a direct bijective proof of the whole distribution: choose $\ell$ of the $n-1$ up-step endpoints, insert colored $d_ju$ pairs, and keep the $udu$-avoiding core; this could be exported to Motzkin or Schröder analogues.
  • The Riordan-array formulation raises natural positivity questions: the tables in Sections 3-5 are likely totally positive, which would imply log-concavity or unimodality of each row as $\ell$ varies, a property not stated in the paper.
  • The functional identity $S_r(x)S_r\left(-x(r-1)S_r(x)^2\right)=S(1,-(r-1)^2;x)$ is the kind of self-composition relation that often signals an underlying orthogonal-polynomial or continued-fraction structure; testing small $r$ for such a representation is a natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. This paper studies the set A_{n,0}^{(r)} of r-colored Dyck paths of semilength n with no two consecutive down-steps sharing a color. The authors derive explicit enumeration formulas for four statistics: the total number of points at level ℓ (Theorem 3.1), the number of up-steps at level ℓ+1 (Theorem 4.1), the number of peaks at level ℓ+1 (Theorem 5.1), and the number of udu-steps (Theorem 6.1). The first three statistics are expressed as entries of Riordan arrays built from S_r(x), the generating function for these paths, and are converted into closed binomial sums via Lagrange inversion. Theorem 6.1 gives the product formula T_{n,ℓ}^{(r)} = binom(n-1,ℓ) T_{n-ℓ,0}^{(r)} r^ℓ, with T_{n,0}^{(r)} given as a binomial sum. Several corollaries record alternating-sum identities, and Section 7 states generalizations to (a,b)-Dyck paths without proofs.

Significance. The main formulas are new and the Riordan-array framework is elegant, connecting the statistics to (a,b)-Schröder numbers. The proofs are based on a sound last-return decomposition and standard Riordan-array and Lagrange-inversion techniques; the recurrences are stated in enough detail that the small table values can be checked by hand. The paper is honest in scope: no parameters are fitted, and the results are derived from the recursive structure of the class. The explicit formulas and tables will be useful to researchers in enumerative combinatorics. The main proofs are essentially complete; the gaps are in supporting material rather than in the central claims.

minor comments (5)
  1. [Lemma 2.2] The proof of Lemma 2.2 establishes (2.2) for Z_1 but then says that (2.3) follows 'by induction on m' with the detail left to the reader; since (2.5) and Corollaries 4.2, 4.3, and 5.2 depend on this lemma, please include the induction step or at least a full outline.
  2. [Section 7] Theorems 7.1-7.3 are stated without proofs, with the note that 'the detailed proofs are omitted'; please either provide proofs or sketches, or clearly re-label these as remarks rather than theorems.
  3. [Theorems 3.1, 4.1, 5.1] In the Lagrange inversion steps, the transition from the coefficient-of-S_r expression to the final binomial sum is not shown, and in Theorem 5.1 even the intermediate Lagrange expression is omitted; please add at least one example of the binomial simplification, and for Theorem 5.1 include the intermediate coefficient expression.
  4. [Throughout] The notation A_{n,k}^{(r)} is used both for the set of paths and for its cardinality, for instance in Section 2; please introduce separate notation for the set and the number, or state explicitly that the meaning is clear from context.
  5. [General] There are several typographical errors: 'well-kown' on page 3, 'determinated' in Theorem 6.1, 'Enumberative Combinatorics' in reference [49], and 'nodd-steps' in the sentence near (2.5).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the statistics formulas are derived from path decompositions and Lagrange inversion, not from self-citation or fitted inputs.

full rationale

The derivation chain is self-contained. The main recurrences rest on the last-return decomposition D = D_1 u D_2 d_j (Theorem 3.1 proof), in which the constraint 'no two consecutive d-steps having the same colors' is handled locally: d_j has r-1 choices when D_2 is nonempty and r choices when D_2 is empty. The same decomposition is reused for Theorems 4.1 and 5.1. These recurrences produce generating-function equations that are solved using only S_r(x) = 1 + x S_r(x) + x(r-1) S_r(x)^2, which is itself derived from the same path class in Section 2, not from the statistics being counted. The final closed forms are obtained by Lagrange inversion on α = S_r(x)-1 = x(α+1)((r-1)(α+1)+1), so the binomial-sum formulas are coefficient extractions, not fitted quantities. Theorem 6.1 is also non-circular: the factor binom(n-1,ℓ) r^ℓ comes from an insertion bijection that inserts ℓ d_j u steps into ℓ chosen u-step endpoints, with the no-equal-color rule automatically preserved because each inserted d_j is flanked by u-steps. The cited works [13], [51], [53], [54] supply the Riordan-array framework and template identities, but none of the target entries or the recurrences are imported from them. Some proofs are abbreviated ('the detail is omitted', Section 7 'detailed proofs are omitted'), which is a completeness concern, not circularity: the omitted arguments are deferred analogues, not restatements of the conclusions. No parameter is fitted to data and renamed as a prediction, and no statistic is defined in terms of the quantity it is claimed to enumerate.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper uses standard tools (Lagrange inversion, Riordan arrays, weighted Schröder generating functions) and one domain assumption about the first-return decomposition. No free numerical parameters are fitted and no ad hoc entities are introduced beyond the path class under study.

assumptions (4)
  • standard math Lagrange inversion formula
    Used throughout, e.g., before Lemma 2.2 and in the coefficient extraction for Theorems 3.1, 4.1, 5.1, 6.1.
  • standard math Riordan array multiplication rule B(x) = d(x) A(h(x))
    Used to interpret P_{n,ell}, U_{n,ell}, p_{n,ell} as entries of Riordan arrays in Theorems 3.1, 4.1, 5.1.
  • standard math Weighted Schröder generating function S(a,b;x) satisfies S = 1 + a x S + b x S^2 (equation (1.4))
    Stated as known from Chen and Pan [12]; used in Lemma 2.2 and in reducing A_r(x,y) to S(1, r-1; x).
  • domain assumption Every non-empty Dyck path has a unique first-return decomposition
    Invoked for the colored class in Section 2 and used to build recurrences in Sections 3, 4, 5.
invented entities (1)
  • A_{n,0}^{(r)}: the set of r-colored Dyck paths with no two consecutive down-steps sharing a color independent evidence
    purpose: The object of study for all four statistics in the paper.
    The counts S_n(1, r-1) match OEIS sequences (Table 2.1), and small values in Tables 3.1 to 6.2 are independently checkable, giving an outside handle.

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Pith. "Pith review of Enumerating several statistics of r-Colored Dyck paths with no dd-steps having the same colors." pith.science (2026). https://pith.science/paper/FNGTJ2GM

@misc{pith2026250608407,
  author       = {Pith},
  title        = {Pith review of: Enumerating several statistics of r-Colored Dyck paths with no dd-steps having the same colors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FNGTJ2GM}},
  note         = {Machine review of arXiv:2506.08407}
}
abstract

An $r$-colored Dyck path is a Dyck path with all $\mathbf{d}$-steps having one of $r$ colors in $[r]=\{1, 2, \dots, r\}$. In this paper, we consider several statistics on the set $\mathcal{A}_{n,0}^{(r)}$ of $r$-colored Dyck paths of length $2n$ with no two consecutive $\mathbf{d}$-steps having the same colors. Precisely, the paper studies the statistics ``number of points" at level $\ell$, ``number of $\mathbf{u}$-steps" at level $\ell+1$, ``number of peaks" at level $\ell+1$ and ``number of $\mathbf{udu}$-steps" on the set $\mathcal{A}_{n,0}^{(r)}$. The counting formulas of the first three statistics are established by Riordan arrays related to $S(a,b; x)$, the weighted generating function of $(a,b)$-Schr\"{o}der paths. By a useful and surprising relations satisfied by $S(a,b; x)$, several identities related to these counting formulas are also described.

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Reviewed August 7, 2026 · model on record in the stance chip above.