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Enumeration of crossings in two-step puzzles

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The number of label-7 crossings and soft crossings in a two-step puzzle is fixed by the three boundary strings alone.

desk verdict New boundary-only formula for crossing counts in two-step puzzles, with a genuine but likely fixable gap in the p=1 reduction step. read the letter →

arxiv 2411.08412 v1 pith:RYAEIJLD submitted 2024-11-13 math.CO

Quentin François (CEREMADE , DMA) This is my paper · ORCID
classification math.CO MSC 05A1505E1414M15
keywords two-steppuzzlescolormapsSchubertcalculustriangularlatticetilingscrossingenumerationgashnumbersflagvarieties
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 that the number of crossings of either type inside a two-step puzzle is determined solely by the three 012 strings on the puzzle boundary. Two-step puzzles are tilings that compute Schubert structure constants of two-step flag varieties, and inside them the 0 and 1 boundary lines cross in two ways: hard crossings, recorded by the label 7, and soft crossings, the composed pieces containing a 3 label and any number of 2 labels. For boundary strings $u$, $v$, $w$ with $n_0$ zeros and $n_1$ ones each, the two counts are $G(u)+G(v)+G(w)-n_0n_1$ and $2n_0n_1-G(u)-G(v)-G(w)$, where $G(u)$ counts, for each zero position $i$ in $u$, the number of ones among positions $1$ through $i$. Because the formula never refers to a particular tiling, any two puzzles sharing a boundary have the same number of each crossing type. The proof translates puzzles into color maps and uses local moves called gashes to reduce the count to the boundary cases.

What carries the argument

The carrying object is a color map, a tiling of the triangular lattice by edge-labeled tiles whose allowed face patterns are $(0,0,0)$, $(1,1,1)$, $(1,0,3)$ and $(0,1,m)$ up to rotation; the 3 and m edges are the ones that record crossings. The proof is carried by the gash numbers $G(C,l)$ of the three boundary sides together with a local operation called a gash, a pair of same-type edges colored 0 and 1 that can be propagated through the map by six local rules and then removed at a terminal configuration. A propagation step lowers $G(C,2)$ by exactly one and changes $G(C,1)$, the m-count and the 3-count in a controlled way, so the main identity is proved by induction on $G(C,2)$. The base case reduces every color map to a canonical reduced form, where the remaining 3 and m edges organize into nonintersecting paths; counting horizontal and vertical steps of those paths yields the explicit formula.

What would settle it

Enumerate all color maps on a small triangular lattice, say $n=3$, for a fixed boundary with $n_0$ zeros and $n_1$ ones per side, and check whether every one satisfies $m(C)=G(C,0)+G(C,1)+G(C,2)-n_0n_1$ and $s(C)=2n_0n_1-G(C,0)-G(C,1)-G(C,2)$. Because the tiling set is finite, one boundary admitting two color maps with different $m$ or $s$ counts would disprove the theorem.

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Extended reading notes

Core claim

The central claim of the paper is Theorem 2.5: for any color map $C$ on the triangular lattice of size $n$ with $n_0$ edges of color 0 and $n_1$ edges of color 1 on each of the three boundary sides, the number $m(C)$ of m-colored edges is $G(C,0)+G(C,1)+G(C,2)-n_0n_1$ and the number $s(C)$ of 3-colored edges is $2n_0n_1-G(C,0)-G(C,1)-G(C,2)$. The statistic $G(C,l)$ is the gash number of side $l$: the sum over 0-colored boundary edges $e$ of the number of 1-colored boundary edges lying closer to the corner. Corollary 2.6 then transfers the statement to two-step puzzles through the bijection of [7]: m-colored edges become label-7 pieces, 3-colored edges become soft crossings, and the gash numbers become the string statistics $G(u)$, $G(v)$, $G(w)$. Thus the interior crossing counts are forced by the boundary, with no reference to the particular tiling.

Load-bearing premise

The proof's load-bearing premise is that the local reductions and arrow reversals in Section 4 always leave the boundary data that feed the formula unchanged, and that the final step collapsing the reduced map to one block is valid; this last step is asserted more briefly than the lemma it relies on.

Editorial extensions

If this is right

  • Every two-step puzzle with a given boundary has exactly the same number of label-7 crossings and exactly the same number of soft crossings; no information about the interior is needed.
  • The two counts can be computed in linear time by scanning the three boundary strings once, rather than enumerating tilings.
  • Because the two formulas sum to $n_0n_1$, the total number of crossings of both kinds is fixed once the number of zeros and ones on the boundary is known.
  • The companion Corollary 5.6 gives that every color map with $n_0$ and $n_1$ boundary colors has exactly $n_0(n_0+1)/2$ and $n_0(n_0-1)/2$ direct and reverse faces with all edges colored 0, and likewise for color 1, so those face counts are also boundary-determined.

Reading between the lines

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

  • A testable extension would be to check whether an equivariant or quantum version of two-step puzzles keeps this boundary-only structure; the proof here treats the unweighted count, and equivariant weights typically add geometric terms that could break the simple formula.
  • The fact that $m(C)+s(C)=n_0n_1$ means the two crossing types are complementary ways of reading the same 0/1 line arrangement; a geometric interpretation of the difference $n(P,7)-n(P,sc)$ on the two-step flag variety may be worth pursuing.
  • Because the proof reduces color maps to nonintersecting path configurations, the same reduction could give enumerative formulas for puzzles with fixed boundaries, and possibly new bijections between color maps and other path models in Schubert calculus.
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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

2 major / 4 minor

Summary. The paper studies color maps, tilings of the triangular lattice by edge-labeled tiles, obtained from two-step puzzles. The main result, Theorem 2.5, gives formulas for the numbers of m-colored and 3-colored edges in a color map in terms of three boundary statistics G(C,0), G(C,1), G(C,2), which depend only on the boundary colors. Corollary 2.6 translates this into formulas for the number of labels 7 and the number of soft crossings in a two-step puzzle. The proof strategy is to reduce the case G(C,2)=0 to reduced color maps using local replacements and arrow reversals, count explicitly in reduced maps via non-intersecting paths, and then handle the general case by induction using gash propagation.

Significance. If the proof is completed, the paper gives a striking boundary-only formula for interior crossing counts in two-step puzzles, with no fitted parameters or auxiliary data. The color-map theorem is self-contained except for the translation step in Corollary 2.6, and the explicit path-counting in Proposition 4.11 is a concrete, checkable combinatorial statement. The result is of clear interest to the Schubert calculus and puzzle combinatorics community. The main weakness is that several reduction steps are justified by pictures and local assertions rather than by complete formal arguments, and one of those steps, in Proposition 4.7, contains a genuine gap in the stated proof.

major comments (2)
  1. [4.1, Proposition 4.7] The reduction of a color map with G(C,2)=0 to a reduced color map is incomplete for the case p=1. Lemma 4.6 is stated under the hypothesis p(C) ≥ 2 and its proof merges the last two blocks, decreasing p by 1. Proposition 4.7 first applies Lemma 4.6 'until p=1' and then, if the resulting map has b1 ≥ 1, says to 'apply Lemma 4.6 a number of times equal to n0' so that b1 becomes 0. When p=1 there is no second block to merge, so the cited lemma cannot be applied even once. The claimed removal of b1 n0 edges of color 3 and the corresponding addition of m-colored edges is therefore not justified by any previously defined move. Since Proposition 4.7 is the bridge between the structural Lemma 4.5 and the explicit count in Proposition 4.11, the base case G(C,2)=0 of the induction in Theorem 2.5 is not established as written.
  2. [4.1, Lemmas 4.3 and 4.6] The global validity of the arrow-reversal moves is asserted rather than proved. In Lemma 4.3 the proof relies on the statement that 'reversing an arrow between endpoints x and x+ℓ+1 does not modify the colors of the edges e having origin y such that y0 ≥ x0', but no proof of this invariance is supplied. Lemma 4.6 similarly describes a sequence of replacements and arrow reversals that is only illustrated in Figures 8–12, without a formal verification that the resulting map is a color map and that the claimed equality p(φ(C)) = p or p−1 holds in all cases of the boundary data. These transformations are load-bearing for Proposition 4.7, so a complete proof should state the exact region affected by each move and prove that colors outside that region are unchanged.
minor comments (4)
  1. [4.1, Proposition 4.7] The expression 'x1(\tilde C)=0' appears to be a typo for 'b1(\tilde C)=0'; as written, x1 is not defined in that context.
  2. [2, Corollary 2.6] The translation from color maps to two-step puzzles depends on the bijection of the author's prior preprint [7]. Since that reference is not yet published and is a self-citation, the paper should state explicitly which results of [7] are used and confirm that they are proven there rather than assumed.
  3. [5, Corollary 5.6] In the statement of Corollary 5.6 the notation 'n(j), n(j)' repeats the same symbol; one of the two quantities should be distinguished, for instance by writing n_+(j) and n_-(j) for direct and reverse triangular faces.
  4. [4.1, Lemma 4.6] The proof of Lemma 4.6 is entirely visual. A short formal description of the sequence of replacements, including the coordinates of the vertices involved and the colors of all modified edges, would substantially improve verifiability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the color-map theorem is derived from scratch, and the puzzle corollary's reliance on the prior bijection [7] is self-citation but does not reduce the central result to its inputs.

full rationale

The main color-map identity (Theorem 2.5) is derived directly from the structure of color maps: Section 4 reduces to reduced color maps and counts m- and 3-edges explicitly (Propositions 4.7 and 4.11), while Section 5 propagates gashes to induct on G(C,2). There are no fitted parameters, and the target counts m(C), s(C) are not defined in terms of the gash numbers G(C,l); the proof genuinely connects two distinct quantities. Corollary 2.6 translates the color-map theorem to puzzle labels using the bijection from the author's prior preprint [7, Theorem 5.3]. This is a self-citation, and it is load-bearing for the puzzle formulation, but it is not circular: the bijection is an independent prior construction whose content is not the formula being proved, and Theorem 2.5 stands independently of it. One non-circular proof gap should be flagged: in Proposition 4.7, after reducing until p=1, the text says, 'If the resulting color map C̃ has b1(C̃) ≥ 1, apply Lemma 4.6 a number of times equal to n0', but Lemma 4.6 is stated only under the hypothesis p(C) ≥ 2. The final exchange of b1(C̃)n0 3-edges for m-edges is therefore not explicitly justified by that lemma. This is an omitted justification in the reduction argument, not a circular step, and should be weighed as a correctness risk. Overall, the derivation does not reduce to its own inputs, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

There are no fitted parameters in the paper. The central counting theorem for color maps is self-contained except for standard lattice geometry. The puzzle translation relies on the bijection from [7], and the proof relies on several diagrammatically justified local moves whose full validity is not formally established.

assumptions (3)
  • domain assumption Bijection from [7] between two-step puzzles and dual two-colored hives/color maps, including the identification of 7-labels with m-edges and soft crossings with 3-edges.
    Corollary 2.6 depends on this external result. It is proved in the author's prior preprint and is not reproved or machine-checked in this paper.
  • domain assumption Arrow reversal and gash propagation/removal operations produce valid color maps and change the counts of m and 3 edges exactly as stated.
    Lemmas 4.3, 4.6 and Section 5 use these operations as local valid moves. The paper illustrates them with figures but does not give a complete formal proof of global validity.
  • standard math Standard geometry of the triangular lattice and the Lindstrom-Gessel-Viennot path-counting formula.
    Used in Remark 4.12 to count reduced color maps as nonintersecting paths; this is a standard background result, not central to the main identity.
invented entities (2)
  • Color maps with edge colors 0, 1, 3, m independent evidence
    purpose: Tiling model dual to two-step puzzles in which the main counting theorem is proved.
    Color maps are defined by explicit local tile rules and are bijectively related to two-step puzzles through prior work [7]; they are constructive rather than unexplained postulates.
  • Gashes and gash propagation independent evidence
    purpose: Induction tool that moves a local imbalance across the board and reduces G(C,2) by one.
    Gashes have a precise local definition and their propagation is claimed to terminate in Proposition 5.4; they are used as a constructive proof device.

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Pith. "Pith review of Enumeration of crossings in two-step puzzles." pith.science (2026). https://pith.science/paper/RYAEIJLD

@misc{pith2026241108412,
  author       = {Pith},
  title        = {Pith review of: Enumeration of crossings in two-step puzzles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RYAEIJLD}},
  note         = {Machine review of arXiv:2411.08412}
}
read the original abstract

We prove a formula which gives the number of occurrences of certain labels and local configurations inside two-step puzzles introduced by Buch, Kresch, Purbhoo and Tamvakis from the work of Knutson. Puzzles are tilings of the triangular lattice by edge labeled tiles and are known to compute the Schubert structure constants of the cohomology of two-step flag varieties. The formula that we obtain depends only on the boundary conditions of the puzzle. The proof is based on the study of color maps which are tilings of the triangular lattice by edge labeled tiles obtained from puzzles.

Figures

Figures reproduced from arXiv: 2411.08412 by the authors.

Figure 1
Figure 1. Possible tiles for color maps As there is an equal number of both 0 and 1 labels on each side of two-step puzzles, we will consider boundary conditions ∂C ∈ {0, 1} 3n having an equal number of 0 and 1 colored edges respectively denoted by n0 and n1 so that n0 + n1 = n, see [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. A color map on E5 with boundary condition ∂C = ((1, 0, 1, 1, 0),(1, 1, 0, 0, 1),(0, 1, 0, 1, 1)). Definition 2.4 (Gash numbers). Let C : En → {0, 1, 3, m} be a color map. For any l ∈ {0, 1, 2} and edge e ∈ ∂ (n) l denote by n(C, e) = |{e ′ ∈ ∂ (n) l : h(e ′ ) < h(e) and C(e ′ ) = 1}| the number of 1 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. A soft crossing in two step puzzles. Recall that the clockwise labels on boundaries of two-step puzzles are 012 strings. Let u be a 012 string of length n: u = u1 . . . un where ui ∈ {0, 1, 2}, 1 ≤ i ≤ n. In accordance with Definition 2.4, define G(u) = X 1≤i≤n:ui=0 |{j ≤ i : uj = 1}|. Theorem 2.5 yields a direct computation of the number of labels 7 and soft crossings in any two-step puzzle given in Corollary 2.6. … view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: First row from left to right : an opening a with color 0 and type 0 at x, the case C(e ′′) = c, the case C(e ′′) ̸= c and an arrow of length r = 4. The second row is the analog for color 1. Let A be an arrow of length r ≥ 1 at an opening with center x. The reversal of …
Figure 5
Figure 5. Figure 5: Reversal of an arrow of length 4 at x. 4. The case G(C, 2) = 0 In this section, we prove (2.2) for color maps C such that G(C, 2) = 0. We first reduce the color map C to C ′ so that all the 0 colored edges on ∂0C ′ are consecutive starting from the bottom left coner of…
Figure 6
Figure 6. Figure 6: Region at the bottom of C. Lozenge regions are filled with 3 lozenges and trapeze regions are filled with 0 colored edges. □ Recall that for 1 ≤ i ≤ p, ri(C) = yi − xi denotes the number of edges of ∂0C which are in the i-th trapeze region of [PITH_FULL_IMAGE:figures/…
Figure 7
Figure 7. Figure 7: Replacement at vertex v. Consider the lozenge region L[bp + 2, rp,(yp−1 − 1, 0)] where bp = xp − yp−1 and rp = yp − xp ≥ 1, see [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: Initial color map C. 0 0 0 0 0 0 0 0 1 1 3 L(bp, rp,(yp−1, 0)) yp−1 xp 0 0 0 0 m 1 1 0 0 0 0 0 0 [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 11
Figure 11. Figure 11: By reverting arrows with openings at (yp−1, 0) + qξ, 1 ≤ q ≤ rp each with length bp, the resulting color map φ(C) satisfies (4.6) p(φ(C)) = p and rp(φ(C)) = rp − 1 if rp ≥ 2, p(φ(C)) = p − 1 and rp(φ(C)) (4.7) (φ(C)) = rp−1 + 1 if rp = 1, see [PITH_FULL_IMAGE:figures…
Figure 13
Figure 13. Figure 13: The region R in a reduced color map. Edges outside R have their color fixed. The next Lemma shows that edges of color 0 in R can only be of type 1. Lemma 4.9 (Edges of color 0 in R). Let R be the region above associated a to reduced color map C. Then, every edge of co…
Figure 14
Figure 14. Figure 14: The two possible orientations of 3 and m lozenges in the region R. We know from Lemma 4.9 that the color map C in R consists in triangular faces f such that C(f) = (1, 1, 1) together with either 3 or m lozenges oriented as in [PITH_FULL_IMAGE:figures/full_fig_p009_14.png]
Figure 15
Figure 15. Figure 15: Possible adjacent configuration to a gash of type 2 in dashed edges. Only the original colors of gashes are represented. Definition 5.2 (Gash propagation). Let g be gash of type 2 with center x adjacent to a configuration (i),(ii) or (iii). We define the propagation o…
Figure 16
Figure 16. Figure 16: Propagation of a gash through configurations (i), (ii) and (iii). New colors of gashes are written in parenthesis. If g is adjacent to a configuration (iv), notice that there is a 0 opening at its center x and thus an arrow of color 0 at x with type 0. Reverting this …
Figure 17
Figure 17. Figure 17: Gash removal in configurations (v) and (vi). 5.2. Proof of Theorem 2.5. We are now in position to prove Theorem 2.5 by induction on G(C, 2). The case G(C, 2) = 0 has been treated in Section 4.3. Assume that the identities (2.2) and (2.3) hold for color maps C such tha…

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