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arxiv: 2605.10113 · v1 · submitted 2026-05-11 · 🧮 math.CO

Recognition: no theorem link

Motzkin paths with two variants of level steps on odd levels -- a kernel method approach

Helmut Prodinger
Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:50 UTC · model grok-4.3

classification 🧮 math.CO MSC 05A1505A16
keywords Motzkin pathskernel methodholonomic recurrencegenerating functionslattice pathsodd levelsfunctional equations
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The pith

Motzkin paths allowing two level-step variants only on odd levels satisfy a holonomic recurrence derived automatically from functional equations via the kernel method.

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

The paper sets up generating functions for Motzkin paths that admit two kinds of horizontal steps but restrict them to odd levels, then uses the kernel method to solve the resulting functional equations. This produces both the enumeration of complete paths and the counts for partial paths that end at a fixed height or at any height. A reader would care because the same manipulation yields a recurrence relation for the counting sequence in an almost automatic way, without needing to guess the form of the relation by hand. The work also treats the symmetric case where the restriction applies to even levels instead.

Core claim

Functional equations are written from a first-step decomposition that distinguishes the two types of level steps and enforces their occurrence only on odd levels; the kernel method then cancels the kernel polynomial to isolate an equation whose coefficients satisfy the holonomic recurrence previously observed for sequence A176677.

What carries the argument

Kernel method applied to functional equations from first-step decomposition, which encode the parity restriction on the two level-step variants.

If this is right

  • The sequence A176677 obeys a specific holonomic recurrence that follows directly from the solved functional equation.
  • Generating functions exist for the number of partial paths that reach a prescribed level or any level.
  • Switching the parity restriction from odd to even levels produces an analogous set of functional equations and the same style of recurrence.
  • The same kernel-method steps extend immediately to three or more variants of level steps under the same parity rule.

Where Pith is reading between the lines

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

  • The technique may apply to other lattice-path families whose step rules depend on the current height parity.
  • Automatic extraction of recurrences could shorten the work of finding closed forms or asymptotics for similar restricted paths.
  • Partial-path counts to arbitrary height may satisfy simpler recurrences or admit combinatorial interpretations in terms of trees or matchings.

Load-bearing premise

The first-step decomposition of the paths produces functional equations that correctly capture the rule allowing the two level-step variants only on odd levels.

What would settle it

Generate the initial terms of the sequence from the derived holonomic recurrence and compare them term-by-term with the published terms of A176677; any mismatch disproves the encoding of the restriction.

Figures

Figures reproduced from arXiv: 2605.10113 by Helmut Prodinger.

Figure 1
Figure 1. Figure 1: Distinguishing even and odd levels (even levels on top). On odd levels, two variants of level steps are possible (depicted in brown). Odd numbered states on top and even numbered states at the bottom cannot be reached. . . [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simplified version Then we need bivariate generating functions as well: F(z, u) = X i≥0 u i fi(z), G(z, u) = X i≥0 u i gi(z); by summing up the recursions one finds F(z, u) = X i≥0 u i fi = 1 + zf0 + zg1 + X i≥1 u i [zfi + zgi+1 + zgi−1] = 1 +X i≥0 u i zfi + X i≥0 u i zgi+1 + X i≥1 u i zgi−1 = 1 + zF(z, u) + z u [G(z, u) − g0] + zuG(z, u). Notice that g0 = 0 for combinatorial reasons; G(z, u) = X i≥0 u i g… view at source ↗
Figure 3
Figure 3. Figure 3: Brown loops refer to two variants of level steps, black loops just to one variant. F(z, u) = 1 + 2zF(z, u) + z u G(z, u) + zuG(z, u), G(z, u) = zG(z, u) + z u [F(z, u) − f0] + zuF(z, u) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

The sequence A176677 in the Encyclopedia of Integer Sequences enumerates Motzkin paths where two types of horizontal steps may occur, but only on odd indexed levels. We show how to perform the enumeration, also dealing with partial such Motzkin paths leading to a particular level or to any level (open paths). The method is the kernel method where functional equations are manipulated in a suitable way. The coefficients of sequence A176677 satisfy a holonomic recursion that was recently discussed on the arxiv. We show how this can be established in an (almost) automatic fashion. Eventually we switch the roles of `odd' and `even'. One could also allow more versions of horizontal steps but we leave this to the interested readers.

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

2 major / 2 minor

Summary. The paper enumerates Motzkin paths in which two distinct types of level steps are permitted only on odd-indexed levels (sequence A176677), together with their partial/open variants. It applies the kernel method to first-step decompositions to obtain bivariate generating functions, extracts the univariate generating function via kernel cancellation, and derives the holonomic recurrence satisfied by the coefficients in an essentially automatic manner. The same approach is applied after swapping the roles of odd and even levels.

Significance. If the functional equations correctly encode the parity restriction, the work supplies a reproducible algebraic derivation of both the generating function and the holonomic recurrence for A176677, illustrating the kernel method's utility for automatically producing recurrences from combinatorial decompositions rather than data-fitting. This is a modest but concrete contribution to the enumeration of restricted lattice paths.

major comments (2)
  1. [§2] §2 (first-step decomposition) and the functional equations (presumably (1)–(3)): the manuscript must explicitly show how the odd-level restriction on the two level-step variants is enforced. Standard first-step decompositions track total height but do not automatically distinguish step types by parity; an auxiliary variable or separate even/odd generating functions appear necessary. Without a clear verification that the kernel solution excludes paths violating the parity condition, the claimed enumeration may count a superset of the intended objects.
  2. [§3] The extraction of the holonomic recurrence from the kernel solution (around the discussion of A176677): the paper should display the explicit functional equation after kernel cancellation and the subsequent coefficient extraction step that yields the recurrence. If this step relies on an unstated assumption about the form of the kernel or the series expansion, the automatic character of the derivation is weakened.
minor comments (2)
  1. Notation for the two level-step variants should be introduced once and used consistently; the current text occasionally switches between descriptive phrases and symbols without a clear table or definition.
  2. The abstract states that the recurrence 'was recently discussed on the arxiv'; a precise citation to that preprint should be added for completeness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [§2] §2 (first-step decomposition) and the functional equations (presumably (1)–(3)): the manuscript must explicitly show how the odd-level restriction on the two level-step variants is enforced. Standard first-step decompositions track total height but do not automatically distinguish step types by parity; an auxiliary variable or separate even/odd generating functions appear necessary. Without a clear verification that the kernel solution excludes paths violating the parity condition, the claimed enumeration may count a superset of the intended objects.

    Authors: We appreciate this observation. Our first-step decomposition employs separate generating functions for paths ending at even and odd heights; this auxiliary tracking enforces the parity restriction on the two level-step variants when they are introduced only at odd levels. The functional equations (1)–(3) are written with this distinction in mind. Nevertheless, we agree that the enforcement was not stated with sufficient explicitness. In the revised manuscript we will add a dedicated paragraph in §2 that (i) recalls the even/odd generating functions, (ii) shows how the parity condition is inserted into the decomposition, and (iii) verifies that any path violating the restriction would be excluded by the resulting kernel equation. This will confirm that the solution enumerates precisely the intended objects. revision: yes

  2. Referee: [§3] The extraction of the holonomic recurrence from the kernel solution (around the discussion of A176677): the paper should display the explicit functional equation after kernel cancellation and the subsequent coefficient extraction step that yields the recurrence. If this step relies on an unstated assumption about the form of the kernel or the series expansion, the automatic character of the derivation is weakened.

    Authors: We agree that greater transparency is desirable. In the revised version we will insert, immediately after the kernel cancellation, the explicit functional equation that remains and then detail the coefficient-extraction steps (including the series expansion and any algebraic manipulations) that produce the holonomic recurrence for A176677. This will make visible any assumptions on the kernel or the generating-function ansatz and will strengthen the claim that the recurrence is obtained in an essentially automatic manner. revision: yes

Circularity Check

0 steps flagged

No circularity: kernel method derives recursion from combinatorial functional equations

full rationale

The paper sets up functional equations via first-step decompositions that incorporate the odd-level restriction on the two level-step variants, then applies algebraic kernel cancellation to obtain the generating function and the holonomic recursion for A176677. No parameter is fitted to a subset of data and renamed as a prediction; no self-citation supplies a uniqueness theorem or ansatz; the recursion is not presupposed but extracted from the solved equations. The derivation remains self-contained against the combinatorial model.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard combinatorial first-step decompositions for Motzkin paths together with the algebraic validity of the kernel method; no new free parameters, invented entities, or non-standard axioms are introduced.

axioms (1)
  • domain assumption The first-step decomposition of a Motzkin path with the stated level-step restriction produces a functional equation whose kernel can be solved by the standard kernel-method substitution.
    This is the usual modeling assumption in kernel-method papers on lattice paths; it is domain-specific rather than ad-hoc.

pith-pipeline@v0.9.0 · 5415 in / 1218 out tokens · 50434 ms · 2026-05-12T02:50:34.261002+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    Prodinger

    H. Prodinger. The kernel method: a collection of examples.S´ em. Lothar. Combin., 50:Art. B50f, 19, 2003/04

    work page 2003

  2. [2]

    Salvy and P

    B. Salvy and P. Zimmermann, GFUN: a maple package for the manipulation of generating and holo- nomic functions in one variable. Transactions on Mathematical Software 20 (1994), no. 2, 163–177

    work page 1994

  3. [3]

    Neil J. A. Sloane and The OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, 2023

    work page 2023

  4. [4]

    A short proof of Mathar's 2016 recurrence conjecture for OEIS A176677

    Tong Niu, A short proof of Mathar’s 2016 recurrence conjecture for OEIS A176677. ArXiv 2605.04369, (2026). Department of Mathematics, University of Stellenbosch 7602, Stellenbosch, South Africa and NITheCS (National Institute for Theoretical and Computational Sciences), South Africa. Email address:warrenham33@gmail.com

    work page internal anchor Pith review Pith/arXiv arXiv 2016