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Odd diagrams, Bruhat order, and pattern avoidance

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Pith's one-line read Odd diagram classes are shown to be Bruhat intervals, with the 213-avoiding and 312-avoiding elements as unique maximum and minimum.

arxiv 2009.08865 v2 pith:3MUGYEMB submitted 2020-09-18 math.CO

Francesco Brenti , Angela Carnevale , Bridget Eileen Tenner This is my paper · ORCID
classification math.CO
keywords diagrambruhatclassespermutationsclassadditionalavoidanceavoiding
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The pith

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The reading

Permutations can be drawn as dots on a grid, one dot in each row and column. The classical diagram of a permutation is the set of empty boxes left when you draw vertical and horizontal lines away from every dot. The odd diagram keeps only those boxes whose row and column indices have opposite parity. Two permutations with the same odd diagram belong to the same odd diagram class.

The paper's first main result says that a class can contain at most one permutation that avoids the pattern 213 and at most one that avoids 312; if such permutations exist, they are, respectively, the largest and smallest elements in the class under Bruhat order. Bruhat order is the partial order obtained by swapping entries of permutations: one permutation is smaller than another if you can reach it by a sequence of swaps that each reduce the number of inversions. The second main result is stronger: the whole class is exactly the set of permutations lying between that smallest and largest element, called a Bruhat interval.

To prove this, the authors characterize, in Theorem 4.3, exactly when a swap of two entries preserves the odd diagram. These 'legal swaps' must obey three conditions about the parity of the swapped positions and the values surrounding them. The remainder of the paper uses these swaps to show that the class is connected, that one can walk monotonically from the minimum to the maximum through legal swaps, and that every rank gap in such a walk can be filled by another legal swap. The proof technique, based on filling squares and flipping chains, is self-contained and case-based.

Extended reading notes

Core claim

Theorem 6.1: Let D be an odd diagram. Then Perm_n(D) = [u, v] for some u, v in S_n. In words, every odd diagram class is a Bruhat interval. If correct, the set of permutations with a fixed odd diagram is exactly the interval between its unique Bruhat-minimal and Bruhat-maximal elements.

Load-bearing premise

The proof of Theorem 6.1 relies on Proposition 6.9, which asserts that any saturated 3-chain x < y < z inside an odd diagram class completes to a square [x,z] also inside the class. The proof's final paragraph omits three of the seven configurations (a < c < b = d, c < a < b = d, c < a = d < b) as 'analogous' without demonstration. If any of these configurations is not actually analogous, the flip argument that all maximal chains in [u,v] lie in the class would not go through, and the class might not equal the interval. Also, the proof of Theorem 6.5 is sketched only.

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Formalized claims in Lean

  1. Claim #1: Theorem 6.1: Let D be an odd diagram. Then Perm_n(D) = [u, v] for some u, v in S_n. In words, every odd diagram class is a Bruhat interval. If correct, the set of permutations with a fixed odd diagram is exactly the interval between its unique Bruhat-minimal and Bruhat-maximal elements.

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

No free parameters are introduced; the results are theorem-proving with no data fitting or undetermined constants. The axioms are standard results in Bruhat order theory and permutation combinatorics. No new particles, forces, or unobservable entities are introduced; the objects (odd diagrams, legal moves, class graphs) are mathematical definitions within the existing framework.

assumptions (4)
  • standard math Bruhat covering in S_n is given by transpositions with no intermediate values (Proposition 2.1).
    Used to identify legal covers and to verify that transpositions used in constructions are covers.
  • standard math For a saturated 3-chain x < y < z in a Bruhat interval, [x,z] is a Boolean algebra of rank 2 ([2, Lemma 2.7.3]).
    Used in Proposition 6.9 to ensure a unique fourth element completing the square.
  • standard math Every Bruhat interval has a unique increasing maximal chain, lexicographically first ([9, Proposition 4.3]).
    Used in Proposition 6.10 to show any two maximal chains are flip-connected.
  • standard math The lexicographic order on reflections is a reflection order.
    Used to order chains and define flips in Proposition 6.10.

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Pith. "Pith review of Odd diagrams, Bruhat order, and pattern avoidance." pith.science (2026). https://pith.science/paper/3MUGYEMB

@misc{pith2026200908865,
  author       = {Pith},
  title        = {Pith review of: Odd diagrams, Bruhat order, and pattern avoidance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MUGYEMB}},
  note         = {Machine review of arXiv:2009.08865}
}
read the original abstract

The odd diagram of a permutation is a subset of the classical diagram with additional parity conditions. In this paper, we study classes of permutations with the same odd diagram, which we call odd diagram classes. First, we prove a conjecture relating odd diagram classes and 213- and 312-avoiding permutations. Secondly, we show that each odd diagram class is a Bruhat interval. Instrumental to our proofs is an explicit description of the Bruhat edges that link permutations in a class.

Figures

Figures reproduced from arXiv: 2009.08865 by the authors.

Figure 1
Figure 1. G(41325). The diagram D(w) of a permutation w is D(w) := {(i, j) ∈ [n] 2 : j < w(i), w−1 (j) > i}. The diagram can be seen by drawing lines to the south (legs) and to the east (arms) of each point (i, w(i)) ∈ G(w), and keeping the empty boxes that remain (see [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. D(41325) consists of the four empty boxes. Definition 2.3. The odd diagram of a permutation w, as defined in [6], is the subset of D(w) defined by Do(w) := {(i, j) ∈ D(w) : i 6≡ w −1 (j) (mod 2)}. We will often mark the elements of Do(w) by stars ∗, and refer to them as such. The odd diagram of 41325 ∈ S5 is depicted in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Do(41325) consists of the three boxes that are marked by ∗s [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The graphs of the permutations v and w as described in the proof of Theorem 3.2, showing a 213-pattern in v and a 312-pattern in w. The graphs are identical to the left of the dashed line. Corollary 3.4. (a) The map w 7→ Do(w) is injective on Avn(213). That is, if v 6=…
Figure 5
Figure 5. Figure 5: In a Bruhat edge v ↔ v, the black arms and legs will arise for both permutations, whereas the red points and segments appear for only one permutation, and the blue points and segments only appear for the other. The following result is fundamental in all that follows. T…
Figure 6
Figure 6. Figure 6: Building the minimal and maximal elements of Perm7(Do(7461325)). minimal (5431627, whose graph is represented by ◦) and maximal (7461523, whose graph is represented by ) elements in the class. Perm7(D) is a Bruhat interval of size 18 and rank 5. Theorems 6.4 and 6.5 im…
Figure 7
Figure 7. Figure 7: The partition of the symmetric group S4 into odd diagram classes. Solid edges connect permutations within an odd diagram class. Each class in S4 is either a singleton or a rank 1 Bruhat interval. Proof. It is well known (see, e.g., [9, Proposition 4.3]) that there is a…

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