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Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations

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Pith's one-line read The paper proves nine conjectured enumeration formulas for pattern-avoiding uniquely sorted permutations of odd length, using bijections to Dyck, S-Motzkin, and Schröder paths.

desk verdict Nine conjectures settled by explicit bijections, but Theorem 7.1 rests on an under-derived functional equation that needs referee attention. read the letter →

arxiv 1908.04025 v4 pith:DVM3FHJF submitted 2019-08-12 math.CO

Hanna Mularczyk This is my paper · ORCID
classification math.CO MSC 05A0505A1505A19
keywords uniquelysortedpermutationsstack-sortingmappatternavoidancecanonicalhookconfigurationDyckpathsS-MotzkinSchröder3-Catalannumbers
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

This paper proves nine conjectured enumeration formulas for classes of uniquely sorted permutations that avoid one length-three pattern and one length-four pattern. Each class is shown to be equinumerous with a family of lattice paths, giving explicit counts: two classes are counted by the binomial coefficient $\binom{2k-1}{k}$, five by the 3-Catalan numbers $\frac{1}{2k+1}\binom{3k}{k}$, one by the little Schröder numbers, and one by the coefficients of $C(xC(x))$. These results settle nine previously open conjectures and show that the descent structure of uniquely sorted permutations mirrors the prefix conditions of Dyck, S-Motzkin, and Schröder paths.

What carries the argument

The central mechanism is the canonical hook configuration (CHC), a tuple of hooks in the permutation plot that records, for each descent top, the leftmost available northeast endpoint. A known theorem says a permutation is sorted exactly when it has a CHC; together with the characterization that uniquely sorted permutations are exactly sorted permutations with $k$ descents in length $2k+1$, this turns unique sortedness into a concrete geometric condition. For each pattern pair, the paper uses pattern avoidance to force the permutation into a restricted shape (vee, svee, layered, stair-svee, stair-layered, modsvee, vee-step), then reads ascent tops and descent bottoms in a prescribed order to produce a lattice path—a Dyck path, an S-Motzkin path (a Motzkin path with $k$ up, $k$ down, and $k$ east steps, starting with an east step and with exactly one up step between consecutive east steps), or a Schröder path without horizontal steps on the axis. The path's prefix conditions are exactly the CHC conditions, so the map is reversible and gives the stated count.

What would settle it

Enumerate the permutations in $U_{2k+1}(231,4312)$ by brute force for $k=1,2,3,4$ and compare the counts with the coefficients of $C(xC(x))$; a single mismatch at small $k$ would disprove Theorem 7.1. Alternatively, deriving the functional equation $\tilde{B}(x)=x+xC(x^2)\tilde{B}(x)^2$ directly from the Section 7 decomposition would remove the only unsupported step.

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Extended reading notes

Core claim

The central claim is that for each of the nine pattern pairs marked with an asterisk in Table 1, the set $U_{2k+1}(\tau^{(1)},\tau^{(2)})$ of uniquely sorted permutations of length $2k+1$ avoiding one length-three and one length-four pattern is enumerated exactly by the stated sequence. The proofs are constructive: for each class, a canonical decomposition of the permutation plot is described, and the decomposition is matched invertibly to a Dyck path, an S-Motzkin path, a Schröder path without horizontal steps on the axis, or to a recursive decomposition governed by a functional equation. In particular, the paper proves $|U_{2k+1}(132,4312)|=|U_{2k+1}(132,3421)|=\binom{2k-1}{k}$, five classes have the 3-Catalan count $\frac{1}{2k+1}\binom{3k}{k}$, the class $U_{2k+1}(231,1432)$ is counted by the little Schröder numbers, and $\sum_{k\ge 0}|U_{2k+1}(231,4312)|x^k=C(xC(x))$, where $C(x)$ is the Catalan generating function.

Load-bearing premise

The count for the class $U_{2k+1}(231,4312)$ rests on a generating-function equation $\tilde{B}(x)=x+xC(x^2)\tilde{B}(x)^2$ that the paper quotes from a previous proof rather than deriving, and if that equation or the copied argument does not carry over, the enumeration in Theorem 7.1 is unsupported.

Editorial extensions

If this is right

  • The nine pattern pairs marked with an asterisk in Table 1 now have proven enumeration formulas, so those rows of the conjecture table are settled.
  • Five classes are placed in bijection with S-Motzkin paths, so their 3-Catalan counts identify them with ternary trees through a single path model.
  • Inversion gives an explicit bijection between $U_{2k+1}(132,3421)$ and $U_{2k+1}(132,4312)$, showing both are counted by $\binom{2k-1}{k}$.
  • The bijection between $U_{2k+1}(231,1432)$ and Schröder paths without horizontal steps on the axis proves the little Schröder count for that class.
  • The generating function $B(x)=C(xC(x))$ for $U_{2k+1}(231,4312)$ places the class in the same enumerative family as previously counted uniquely sorted classes with the same generating function.

Reading between the lines

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

  • For the nine still-open conjectures, the paper's own remarks suggest standard lattice-path bijections may fail because the relevant Motzkin model has the wrong length, so a generating-function approach may be more productive.
  • The Section 4 inverse bijection hints that inversion symmetry between pattern pairs could yield further equinumerosities among uniquely sorted classes; testing this on other inverse pairs is a natural next step.
  • The repeated 'descents become up steps, ascent tops become down steps' reading suggests a general dictionary between fertility conditions and path prefix conditions that could generate new path families for the remaining classes.
  • Deriving the quoted functional equation for $U_{2k+1}(231,4312)$ from the decomposition would not only complete Theorem 7.1 but may provide a template for enumerating other classes of the form $U_{2k+1}(231,4xxx)$.
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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

1 major / 5 minor

Summary. The paper studies uniquely sorted permutations (permutations with fertility 1 under West's stack-sorting map) that avoid one pattern of length 3 and one of length 4. It proves nine of Defant's conjectures by establishing bijections between these classes and lattice paths (Dyck paths, S-Motzkin paths, and a subclass of Schröder paths) and by a generating function argument. The enumerations include binomial coefficients, 3-Catalan numbers, little Schröder numbers, and coefficients of C(xC(x)).

Significance. If the gap in Section 7 is repaired, the paper settles nine conjectures in the enumeration of pattern-avoiding uniquely sorted permutations. The bijections in Sections 4-6 are largely explicit and reversible, and they introduce useful structural notions (modsvee, modvee, stair-svee, stair-layered, svee-increasing, vee-layered, vee-step). The generating-function route in Section 7 is the only part that is not fully carried out in the manuscript.

major comments (1)
  1. [7, proof of Theorem 7.1] The proof of Theorem 7.1 stops at the generating function for nice permutations, x^2 C(x^2) \tilde B(x), and then asserts the functional equation \tilde B(x) = x + x C(x^2) \tilde B(x)^2, saying that the rest of the proof is identical to that of Theorem 8.1 in Defant's article [8]. This equation is not derived for the class U_{2k+1}(231,4312), and Theorem 8.1 of [8] concerns the different class U_{2k+1}(231,4132). The author must show how non-nice permutations are recursively decomposed (for example, that a non-nice element can be split into a nice element and an arbitrary element) and must verify that the decomposition preserves 231 and 4312 avoidance and unique sortedness. As written, the enumeration of this class is unsupported, and Theorem 7.1 is one of the paper's nine central claims.
minor comments (5)
  1. [Lemma 3.3] The proof begins "Given π ∈ U_{2k+1}(132, 312)", but the statement is about U_{2k+1}(231,312); the class name should be corrected.
  2. [Theorem 5.4] The proof says "Consider some π ∈ U(312, 2431)", but the theorem is about U_{2k+1}(312,3421); also the final equality omits the subscript 2k+1 and should read |U_{2k+1}(312,3421)|.
  3. [Theorem 5.3] The inverse of the described bijection is only implicit in the sentence about recovering the type of ascent; a precise reconstruction of a stair-svee permutation from an S-Motzkin path would make the bijection easier to verify.
  4. [Section 7] The notation for the generating function \tilde B(x) alternates between "B~(x)" and "~B(x)", and the equality \tilde B(x) = x + x C(x^2) \tilde B(x)^2 is introduced without derivation; at minimum the notation should be made consistent.
  5. [Theorem 4.2] The preservation of the canonical hook configuration under the modvee/mod-svee inversion is argued in one sentence ("by the logic in the lemmas"); a few more details would make this central step easier to check.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the enumeration theorems are derived from explicit bijections and external lemmas, not from the target formulas.

full rationale

The paper's central claims are the nine asterisked enumeration theorems. Each is proven either by a direct decomposition-and-counting argument (Theorem 4.1), by an explicitly defined bijection with S-Motzkin paths or Schröder paths whose cardinalities are cited from independent sources (Theorems 5.3–5.6, 6.1), or by composing previously established bijections (Theorem 5.7). No parameter is fitted to data and then renamed a prediction; no class is defined in terms of its own enumeration; and no load-bearing step reduces, by the paper's own equations, to its target formula. The only potentially weak point, flagged by the skeptic, is Theorem 7.1: the paper derives the generating function for nice permutations as x^2 C(x^2) B~(x) and then states that 'the rest of the proof is identical to that of Theorem 8.1 in Defant's article [8]', asserting the functional equation B~(x) = x + x C(x^2) B~(x)^2 without deriving it in the present text. This is an omitted derivation and a dependence on an external proof, not a circular reduction: the asserted equation is not shown to be an input of the argument, nor is Defant's theorem a self-citation by the present author. Accordingly, the circularity burden is zero, and the appropriate finding is 'no significant circularity' with any concern about Theorem 7.1 treated as a correctness or completeness issue rather than circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

All load-bearing premises are external theorems from the stack-sorting and permutation-pattern literature that the paper cites without reproof, chiefly from Defant [8], [12], [13], [14] and Prodinger-Selkirk [23]. No free parameters or invented entities are introduced. The most fragile premise is the assertion in Section 7 that Defant's generating-function derivation transfers verbatim.

assumptions (7)
  • domain assumption A permutation π ∈ S_n is uniquely sorted if and only if it is sorted and has exactly (n−1)/2 descents.
    Theorem 1.1, Section 1, cited from [14]. This characterization anchors every counting argument; the paper does not reprove it.
  • domain assumption A permutation is sorted if and only if it has a canonical hook configuration.
    Proposition 2.1, Section 2.2, cited from [13]. Used to invoke CHC conditions throughout.
  • domain assumption In any π ∈ U_{2k+1}, the set of descent bottoms and the set of NE endpoints of the CHC partition the points at positions 2 through 2k+1.
    Lemma 2.2, Section 2.2, cited from [8]. The proof of every bijection uses this to equate ascent tops with NE endpoints.
  • standard math |U_{2k+1}(231,312)| = C_k, the k-th Catalan number.
    Lemma 3.3 in Section 3; the paper references previous enumeration. Used as the building block in Theorem 7.1.
  • standard math The number of S-Motzkin paths of length 3k is (1/(2k+1)) binom(3k,k).
    Theorem 5.2, cited from [23]. Needed to convert the five bijections in Section 5 into closed formulas.
  • domain assumption There is a bijection swu : U_{2k+1}(231) → U_{2k+1}(132) whose restriction maps Av(231,1423) onto Av(132,3412).
    Theorem 5.7, citing [8, Thm 6.1] and [12, Thm 5.1]. This is the whole proof of Theorem 5.7; no independent verification is given.
  • domain assumption The functional equation \tilde{B}(x) = x + xC(x^2)\tilde{B}(x)^2 for the generating function of nice permutations in the class (231,4312) follows by the same derivation as Defant's Theorem 8.1 in [8].
    Section 7, final paragraph. The paper asserts that 'the rest of the proof is identical' and refers the reader to [8], leaving the derivation unstated.

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Pith. "Pith review of Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations." pith.science (2026). https://pith.science/paper/DVM3FHJF

@misc{pith2026190804025,
  author       = {Pith},
  title        = {Pith review of: Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DVM3FHJF}},
  note         = {Machine review of arXiv:1908.04025}
}
read the original abstract

Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.

Figures

Figures reproduced from arXiv: 1908.04025 by the authors.

Figure 1
Figure 1. The CHC of 2 7 3 5 9 4 8 1 6 10 11 12. The following proposition allows us to determine whether a permutation is sorted using its CHC. Proposition 2.1 ([13]). A permutation π is sorted if and only if it has a canonical hook con￾figuration. This proposition, along with Theorem 1.1, gives us that π ∈ Sn is uniquely sorted if and only if it has a CHC and exactly n−1 2 descents. The permutation in [PITH_FULL_IMAGE:figu… view at source ↗
Figure 2
Figure 2. The Dyck path UDUUUDDUDD of semilength 5. 3.2. Three Classes of Permutations. We now highlight the structures of three particular classes of pattern-avoiding permutations: • A permutation π avoids 132 and 231 if and only if π = L1R, where L is decreasing and R is increasing. We call this type of permutation vee after its resemblance to the letter V. We call the imaginary vertical line at 1, separating L and R, the v… view at source ↗
Figure 3
Figure 3. The vee permutation 10 6 5 3 2 1 4 7 8 9 11 and its image under the bijection in Lemma 3.1, the Dyck path UDUUUDDUDD. This simple bijection touches upon an intriguing connection between Dyck paths and uniquely sorted permutations. While Defant uses a different map for proving that: |U2k+1(132, 312)| = |U2k+1(231, 312)| = Ck, we note here that a method similar to that in the proof of Lemma 3.1 can be used to make bij… view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: The modsvee permutation 643527819. The dashed boxes illustrate which pieces of the permutation are decomposed after each step in the decompo￾sition process. Following Defant in [8], we call a uniquely sorted permutation π ∈ U2k+1 nice if the SW endpoint of the hook in …
Figure 5
Figure 5. Figure 5: The decomposition of the modsvee permutation 643527819 into a nice modsvee uniquely sorted permutation (on the left) and a svee uniquely sorted permutation (on the right). Theorem 4.2. We have |U2k+1(132, 3421)| = [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: The modvee permutation 853241679, which is the inverse of the per￾mutation in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: The S-Motzkin path EUEUDEDUEUDD of length 12. Let MS k denote the set of S-Motzkin paths of length 3k. Theorem 5.2 ([23]). We have |MS k | = 1 2k+1 3k k  . Prodinger and Selkirk prove this with a bijection between S-Motzkin paths of length 3k and ternary trees on k no…
Figure 8
Figure 8. Figure 8: The structure of a stair-svee permutation. Now, we define a bijection from U2k+1(312, 2431) to MS k via the following rule. Let Λ be the path EUEU . . . EU containing alternating k E’s and k U’s. Given π ∈ U2k+1(312, 2431), let a1, . . . , ak be the ascent tops of π, o…
Figure 9
Figure 9. Figure 9: The stair-svee permutation 3 2 4 1 9 8 7 10 11 6 5 12 13 and its image under the bijection in Theorem 5.3, the S-Motzkin path EUDEUEDUEUDDEUEUDD. Theorem 5.4. We have |U2k+1(312, 3421)| = 1 2k+1 3k k  . Proof. Consider the plot of some π ∈ U(312, 3421) and some consec…
Figure 10
Figure 10. Figure 10: The structure of a stair-layered permutation. We define a map from U2k+1(312, 3421) to MS k using similar rules as we did for the previous class. Let Λ be the path EUEU · · · EU containing an alternating k E’s and k U’s. Consider some π ∈ U2k+1(312, 2431), with tl(π) …
Figure 11
Figure 11. Figure 11: The stair-layered permutation 3 2 4 1 8 7 6 9 11 10 5 12 13 and its im￾age under the bijection in Theorem 5.4, the S-Motzkin path EUDEUEDUEUDDEUEUDD. The rightmost block is the tail. Theorem 5.5. We have |U2k+1(312, 1432)| = 1 2k+1 3k k  . Proof. Consider π ∈ U2k+1(3…
Figure 12
Figure 12. Figure 12: The general structure of a svee-increasing permutation. Each of the blocks above the horizontal and the points below the horizontal can be deleted [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: The svee-increasing permutation 4 3 2 5 7 6 1 9 10 8 11 and its image under the bijection in Theorem 5.5, the S-Motzkin path EUEUDEDUEUEDDUD. Theorem 5.6. We have |U2k+1(231, 1423)| = 1 2k+1 3k k  . Proof. Consider π ∈ U2k+1(231, 1423) and some consecutive subsequenc…
Figure 14
Figure 14. Figure 14: The general structure of a vee-layered permutation. Each of the blocks to the right of the vertical and the points to the left of the vertical can be deleted. We define a map from U2k+1(231, 1423) to MS k using rules similar to those in the previous bijection: Given t…
Figure 15
Figure 15. Figure 15: The vee-layered permutation 9 2 1 5 4 3 6 8 7 10 11 and its image un￾der the bijection in Theorem 5.6, the S-Motzkin path EUEUDEDUEUEDDUD. Theorem 5.7. We have |U2k+1(132, 3412)| = 1 2k+1 3k k  . Proof. The result follows from Theorem 5.6 and some previous work. Theo…
Figure 16
Figure 16. Figure 16: The Schröder path UHUUUDDUHDDD of semilength 7. It is well-known that the Schröder paths can be divided into two equinumerous subclasses: paths that have a horizontal step on the horizontal axis and paths that do not have a horizontal step on the horizontal axis. Let …
Figure 17
Figure 17. Figure 17: The vee-step permutation 732154689 and its image under the bijec￾tion in Theorem 6.1, the Schröder path UUHDUDD. 7. Counting U2k+1(231, 4312) with a Generating Function In [8], Defant showed how to decompose permutations in U2k+1(231, 4132) to obtain an identity provi…
Figure 18
Figure 18. Figure 18: Decomposing the nice permutation π into τ and π ′ . 8. Conclusion The theorems in this paper prove nine out of the eighteen conjectures in [8], which enumerate classes of the form U2k+1(τ (1), τ (2)), where τ (1) has length 3 and τ (2) has length 4. The nine remaining…

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