A general obstruction to the pinnacle middle-order lattice

J. Liang and B. E. Sagan, Log-concavity and log-convexity via distributive lattices, Order 43 (2026), Article 21, DOI 10.1007/s11083-026-09735-2, arXiv:2408.02782v2, Section 8, printed pages 34–35, Question 8.6 and its preceding definition:

Let σ be a permutation of distinct positive integers. Let Pin_n(σ) = {π ∈ S_n | the pinnacles of π are the elements of σ in that order}. … Computation of examples raise the following questions. Question 8.6. For all n and σ: (a) Is the restriction of the middle order to Pin_n(σ) a distributive lattice? (b) Is the sequence (pin_n(σ)) log-concave?

Section 6, “Permutations”, printed pages 17–18, defines inversions and inversion tops:

Given π = π1 π2 . . . πn ∈ Sn in one-line notation, its set of inversion (values) is Inv π = {(πi, πj) | i < j and πi > πj}.

We also say that πi is an inversion top if (πi, πj) ∈ Inv π for some πj.

On printed page 18 the inversion coordinate and middle order are specified separately:

where ιi = # of inversions in which i is an inversion top.

We can now use the bijection just given to define the middle order (Sn, ⪯) by π ⪯ σ if and only if ι(π) ≤ ι(σ) where the order on inversion tables is component-wise.

Pinnacles are the values π_j with π_{j−1}<π_j>π_{j+1}.

Formal reading

Lean names have prefix D5.S0.Certificates.Combinatorics.LiangSaganPinnacle.Result.

| Source object | Lean name | Correspondence | | --- | --- | --- | | Permutation word | Equiv.Perm (Fin n) | Increasing positions; Fin value k represents k+1. | | Ordered pinnacle sequence | pinnacleWord | Values at internal strict local maxima, from left to right. | | Inversion-top table | iota | Number of smaller values to the right of a value. | | Middle order | middleLe | Componentwise comparison of iota. | | Ordered pinnacle class | Pin | Fibre of pinnacleWord. | | Universal distributive-lattice assertion | claim | A distributive-lattice structure with exactly this order on every fibre. | | Refutation | result | Exactly ¬ claim. |

The source uses distinct positive integers. Repeated values cannot occur in a pinnacle word. A source sequence containing a value outside [n] has empty fibre; the counterexample uses only distinct in-range values.

The settling declaration is D5/S0/Certificates/Combinatorics/LiangSaganPinnacle/Result.result. The source note is D5/L/PermutationPatterns/liang2026distributivelattices. The ordered pinnacle condition constrains positions, while the middle order constrains inversion counts. Their interaction can prevent joins.

Even existence of a common upper bound fails in the fibre with σ=(n−2,n) for every n≥6. Thus this fibre is not a lattice, and the universal distributive-lattice assertion is false.

In paper values let

$$A_n=(1,n-2,2,n,3,4,\ldots,n-3,n-1),$$ $$B_n=(n-1,1,n-2,2,n,3,4,\ldots,n-3).$$

Both have ordered pinnacle word (n−2,n). Their relevant inversion counts are ι_n(A_n)=n−4 and ι_{n−1}(B_n)=n−2. An upper bound U of B_n must place n−1 before every smaller value. The left neighbour of the first pinnacle n−2 is smaller than n−2, so n−1 precedes that neighbour. The first pinnacle therefore has zero-based position at least 2. The second pinnacle n comes later and cannot be adjacent, so its position is at least 4. For a maximal value, ι_n(U)=n−1−pos(n)≤n−5<n−4. This contradicts U being above A_n. A lattice join would be a common upper bound.

Two permutations in the same ordered pinnacle fibre with no common upper bound contradict the universal claim. A mismatch in the word order, value shift or inversion-top convention would invalidate the argument.

PROVED in Lean. no_common_upper_bound establishes the obstruction for every n≥6. result : ¬ claim refutes the universal assertion. Both statements are checked by the Lean kernel.

COMPUTED. The immutable experiment directory contains ls-pinnacle.py, SHA-256 4343fcc9ffc8cfd37742ca99fe76bc80567cf060d26fd934568621c3473fbb25. python3 ls-pinnacle.py, exit 0, checks the family for n=6,…,9 and all nonempty fibres for n=1,…,6. The command python3 ls-pinnacle.py 7, exit 0, additionally gives n=7 classes=50 nonlattice=14. Both commands use the same pinned script and SHA-256 above.

The supplied literature scope covers arXiv v2 and the Order publication, Vatter arXiv:2602.16355, and Middle orders: all distributive lattices between weak and Bruhat, arXiv:2606.12129. No settlement was found in that scope. This does not establish exhaustive coverage or priority.

What the settlement shows

• PROVED in Lean. no_common_upper_bound: for every n≥6, the class with σ=(n−2,n) contains two elements with no common upper bound.
• PROVED in prose. A large non-pinnacle value forced before every smaller value must lie at the front: it precedes the first pinnacle, which precedes the maximum, so it precedes every other value. This pushes the first pinnacle past a smaller left neighbour, then pushes the maximum two positions farther. The resulting cap on its inversion count excludes a common upper bound.
• COMPUTED. Every nonempty class is a lattice for n≤5. At n=6 the non-lattice classes are σ∈{(3,6),(4,6),(6,3),(6,4)}. The enumeration tests lattice existence, not distributivity of the remaining classes.
• COMPUTED. At n=7, 14 of 50 nonempty classes are non-lattices. The pinned script command python3 ls-pinnacle.py 7, exit 0, gives n=7 classes=50 nonlattice=14; its immutable directory and SHA-256 are stated in Evidence. These are finite readings, not a general classification theorem.
• OPEN. Question 8.6(b), log-concavity of the class-size sequence, is untouched. A classification of all lattice fibres is also open here.

Google Scholar's full cited-by list was unavailable in the supplied literature check. Quotations and printed page locators use the supplied pdftotext of arXiv:2408.02782v2; the PDF typography was not independently checked. External literature checks are the supplied bounded scope; no exhaustive novelty claim is made.