Show that the jump number of a random linear extension of a grid poset (i

Definitions and mathematical statement

For positive integers $m_1,\ldots,m_n$, let \[ P=[m_1]\times\cdots\times[m_n],\qquad [m_i]=\{1,\ldots,m_i\}, \] ordered coordinatewise, and put $N=\prod_i m_i$. A linear extension is an ordering $L=(x_1,\ldots,x_N)$ respecting this partial order. Its jump count is \[ J(L)=\#\{i\in\{1,\ldots,N-1\}:x_i \text{ and }x_{i+1}\text{ are incomparable in }P\}. \] Let $J_{\max}(P)=\max_LJ(L)$ and choose $L$ uniformly from all linear extensions. This distinguishes the jump count of an extension from the usual optimization parameter sometimes called the jump number of a poset, which uses a minimum.

The source asks to show that $J(L)$ is close to $J_{\max}(P)$ with high probability for grid posets. A precise theorem needs an asymptotic family of chain lengths/dimensions and a specified closeness scale, such as an additive error or a relative error. "With high probability" means probability tending to one in that family. No family, error bound, or rate is supplied, so one is not silently selected. Relative normalization also needs care when the maximum is zero, as for a single chain.

The source explicitly reports this behavior known for the symmetric grid $[m]^n$ in the regime \[ n=\exp(o(\log m)),\qquad\text{equivalently}\quad \frac{\log n}{\log m}\to0 \quad(m\to\infty). \] The precise closeness estimate for that reported result is not given. The requested extension to general products retains these missing quantitative choices for review.

Short English statement

Does a uniformly random linear extension of a grid poset have almost as many adjacent incomparable pairs as any linear extension can have? The source reports a symmetric-grid regime but does not define the quantitative error or general limiting regime.

The supplied summary mentions Cooper and Cooper--Reiss with an uncertain paper-title fragment and reports partial progress without exact results. The OPEN-TRIAGE label and October 2020 source snapshot are retained; provenance says later resolution was not checked item by item.

Sources

• Local UnsolvedMath registry, permanent ID 3100042, source code catalogue entry, random-linear-extension grid-poset question. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Cooper, Combinatorial Problems I Like, October 2020 snapshot, Matroids, Lattices, and Posets, source-order bullet 42, https://people.math.sc.edu/cooper/combprob.html; HTML extraction, accessed 2026-07-29.
• The supplied summary mentions Cooper--Reiss and an uncertain "THE MAXIMAL JUMP number" title fragment, without complete authorship, publication details, or theorem. No reference is completed by inference.
• Source provenance marks status evidence NEEDS_REVIEW and rights NEEDS_REVIEW: public access was verified, but redistribution permission was not established. This qualification is separate from the dataset licence.