slug: chenkitaevzhang-2024-separable-record-peak-three bibkey: chenkitaevzhang2024separable doi: 10.1016/j.dam.2024.05.004 url: https://arxiv.org/abs/2404.18517v1 triage: theorem motivation_gids:

• D5/S1/Words/Patterns/Separable/RecordPeak.actual_record_rising

CKZ Conjecture 2: the four actual record distributions

Problem

Chen, Kitaev and Zhang, Distributions of statistics on separable permutations, arXiv:2404.18517v1, Section 3, Conjecture 2, page 17, asks:

The distribution of rmax or lmin (resp., lmax or rmin) on irreducible (resp., reducible) separable permutations is unimodal with the peak being at k=3 for n>=5 (see Table 4).

The journal reference is Discrete Applied Mathematics 355 (2024), 169–179. The carrier is the actual classical $2413/3142$-avoiding permutations, rather than a recursive replacement class. A proper direct cut separates a nonempty prefix whose values are all below those of the nonempty suffix. Irreducibility means positive length and no proper direct cut; reducibility means a proper direct cut. The empty permutation belongs to neither positive class. The singleton is irreducible and has one record of each type; the reducible singleton class is empty.

The four strict record statistics are right maxima, left minima, left maxima and right minima. A maximum exceeds every value on its indicated side, and a minimum is below every value on its indicated side. These counts are unshifted. For any of the pairs $(\mathrm{irreducible},\mathrm{rmax})$, $(\mathrm{irreducible},\mathrm{lmin})$, $(\mathrm{reducible},\mathrm{lmax})$ and $(\mathrm{reducible},\mathrm{rmin})$, let $a(n,k)$ be the corresponding actual record-fiber cardinality. The complete target is, for every $n\ge 5$ and every natural $k$,

$$\begin{array}{rl}k<3& ⟹a(n,k)\le a(n,k+1),\\ k\ge 3& ⟹a(n,k+1)\le a(n,k),\\ a(n,k)& \le a(n,3).\end{array}$$

A maximum at three is interpreted weakly. Strict inequalities on zero tails and uniqueness are not required. Issue 12664 registers this full target. A partial rising or transport result does not settle the conjecture.

Motivation

The actual rising side holds for every $n\ge 4$ and every $k<3$ in all four distributions. RecordPeak.actual_record_rising supplies the actual right-maximum comparison from two to three. The RecordTransport.actual_four_record_transports_and_rising theorem proves the all-length exact actual record transports, their singleton corrections, the low-record zero fibers and the complete four-distribution rising side. Writing $i_t,d_t$ for the positive irreducible and reducible record counts, these exact transports are

$$\begin{array}{rl}{i}_{\mathrm{lmin}}(n,k)& =i_{\mathrm{rmax}}(n,k),\\ d_{\mathrm{lmax}}(n,k)+\delta (n,k)& =i_{\mathrm{rmax}}(n,k),\\ d_{\mathrm{rmin}}(n,k)+\delta (n,k)& =i_{\mathrm{rmax}}(n,k),\end{array}$$

where $\delta (n,k)=1$ exactly when $n=k=1$. They reduce the remaining four declining laws at $n\ge 5$ to the actual irreducible right-maximum law. RecordFirstDecline.actual_four_record_first_decline proves the first declining comparison $a(n,4)\le a(n,3)$ at every natural length, and its strict form for every $n\ge 3$, on all four actual class/statistic pairs. The proof establishes its exact positive quotient and infinite tail inside the actual weighted inference.

Gap

The actual irreducible right-maximum declining comparisons for arbitrary $k\ge 4$ and the global maximum remain open. RecordNewtonPositivity.actual_q_record_newton_nonnegative proves the unbounded coefficient sign law $[t^n]G_r(q(t))\ge 0$ for every natural $r,n$. Its proof constructs the Motzkin series from Mathlib’s Catalan supplier, derives its differential recurrence at every index, and propagates two pairs of consecutive positive-composition certificates to the universal tail. This removes the universal kernel-sign obligation.

The exact remaining bridge is the normalized actual $J_k$ coefficient formula and its Newton transform. They must identify each actual decline with a nonnegative combination of the defined kernels; the sign theorem has no actual-record-fiber correspondence in its conclusion. No new actual declining range, full open-problem resolution or KPI increment is attributed to this partial unit.

Route

Use the actual record quadratic and scalar series from the right-maximum proof to derive the normalized Motzkin formula and Newton transform inside the full declining proof. Apply the proved all-index kernel sign law and extract the actual record coefficients. The exact transports give the remaining three declining laws at $n\ge 5$. Combine rising and declining inequalities to obtain the global maximum. A conditional generating-function theorem or finite row computation does not discharge this route.

Falsifier

An actual source-class permutation count at some $n\ge 5$ and natural $k$ violating a required adjacent inequality, or exceeding the count at three, refutes the corresponding conjectured law. A counterexample to an auxiliary coefficient estimate refutes that estimate; it does not automatically refute the actual permutation conjecture.

Evidence

The source v1 and the author and institutional manuscript bodies retain Conjecture 2. The bounded literature screen, detailed in the Library entry, reports no equivalent proof or refutation in the source manuscripts, five known citing bodies and the related papers inspected. The journal version of record body remains unverified. This is a non-discovery in the searched scope, not exhaustive global novelty, priority or exclusive ownership. Fu–Lin–Zeng’s descent real-rootedness problem and arXiv:2608.27583v1 Conjecture 15 are separate open problems.

Triage

Theorem: a Tier 2 published universal mathematical conjecture with exact source classes and record definitions. The partial actual-carrier theorems leave the complete conjecture open. None is a full external settlement or carries an OpenProblemResolutionClaim.

ASSUMED-UNVERIFIED

The journal version of record body, exhaustive citation coverage and global novelty are unverified. Uncompiled algebraic research does not establish the missing normalized actual coefficient formula or Newton transform. Reg enrollment is user-paused; no completed registration audit is claimed.