bibkey: chenkitaevzhang2024separable authors: Joanna N. Chen, Sergey Kitaev, and Philip B. Zhang year: 2024 title: “Distributions of statistics on separable permutations” doi: 10.1016/j.dam.2024.05.004 url: https://arxiv.org/abs/2404.18517v1 claim: “Strict record distributions on literal 2413/3142 avoiders; Section 3 Conjecture 2 asks for all-length class-conditioned unimodality with a maximum at record count three.” strata_touched:

• D5/S1/Words/Patterns/Separable/ActualCardinality
• D5/S1/Words/Patterns/Separable/MinimumCutKernel
• D5/S1/Words/Patterns/Separable/CutFactorization
• D5/S1/Words/Patterns/Separable/ProperCut
• D5/S1/Words/Patterns/Separable/RecordPeak
• D5/S1/Words/Patterns/Separable/RecordTransport
• D5/S1/Words/Patterns/Separable/RecordFirstDecline
• D5/S1/Words/Patterns/Separable/RecordNewtonPositivity license: citation-only triage: anchor

Record distributions on separable permutations

Verified locator

• Primary v1: https://arxiv.org/abs/2404.18517v1, 29 April 2024, 19 pages; Conjecture 2 and Table 4 are in Section 3, page 17. Definitions are in Sections 1 and 1.2, and functional equations are in Section 2.
• DOI: https://doi.org/10.1016/j.dam.2024.05.004

The journal reference is Discrete Applied Mathematics 355 (2024), 169–179. The institutional preprint and author manuscript also retain Conjecture 2. Neither manuscript establishes the content of the journal version of record; that body remains unverified.

The source class is the actual classical $2413/3142$-avoiding permutations. The four statistics count positions with strict comparisons: a right maximum exceeds all later values, a left minimum is below all earlier values, a left maximum exceeds all earlier values, and a right minimum is below all later values. Counts are unshifted. Direct sums put prefix values below suffix values; skew sums put them above. Irreducibility means positive length and no proper direct cut. The empty permutation belongs to neither the irreducible nor the reducible class; the singleton is irreducible and the reducible class at length one is empty.

For a class $X$ and statistic $t$, write $a_{X,t}(n,k)=|\{\pi \in X_n:t(\pi )=k\}|$. Conjecture 2 asks, for each of $(X,t)=(\mathrm{irreducible},\mathrm{rmax}), (\mathrm{irreducible},\mathrm{lmin}), (\mathrm{reducible},\mathrm{lmax}), (\mathrm{reducible},\mathrm{rmin})$,thatforeveryinteger$n\ge5$ and every integer $k\ge 0$,

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

A maximum at three is interpreted weakly; uniqueness is not needed for these inequalities. All four full assertions remain open in the project.

The strict-record mathematical source states the known block-sum identities for arbitrary actual permutations, including empty factors: a direct sum has only the suffix right maxima when the suffix is nonempty, retains the prefix count when the suffix is empty, and right maxima add in a skew sum. It also gives the record-weighted least-proper-cut convolution for both signs and all natural lengths and record indices on the actual avoiding class. The direct summand is $j(0,c)U(n-c,k)$ and the skew summand is ${\sum }_{b=0}^{k}J(1,c,b)U(n-c,k-b)$, summed over $0<c<n$, where $j$ counts the no-proper-cut prefix class, $J$ its exact record fiber, and $U$ the full avoiding record fiber. Both factor lengths are positive. At lengths zero and one there are no proper cuts. These helper formulas are known source and normalization reductions, obtained by strict block comparisons and weighting the existing minimum-cut reconstruction. They are used locally in the actual right-maximum proof below; no new standalone normalization helper theorems are retained.

ActualCardinality.actual_schroder_cardinality supplies the existing actual signed least-cut equivalence with exact dependent-length reconstruction. Its factorization uses MinimumCutKernel, CutFactorization, and ProperCut. Fu–Lin–Zeng Proposition 2.1 supplies the recursive direct/skew characterization of separable permutations; it does not itself supply a least-cut record equivalence. The least-cut reconstruction used by the mathematical source is the existing project supplier. Fu–Lin–Zeng’s later greatest-cut tree construction is a separate decomposition.

For every $n\ge 4$, the actual $2413/3142$ avoiders with no proper direct cut and exactly two strict right maxima are at most those with exactly three: $a_{\mathrm{irreducible},\mathrm{rmax}}(n,2)\le a_{\mathrm{irreducible},\mathrm{rmax}}(n,3)$. This is supplied by RecordPeak.actual_record_rising with no generating-function premise. Local finite-fiber and scalar generating-function derivations, together with a universal positive quotient and derivative composition argument, establish this adjacent comparison.

RecordTransport.actual_four_record_transports_and_rising gives the exact all-length actual transports. If $i_t$ and $d_t$ are the positive irreducible and proper-direct-cut record counts, then $i_{\mathrm{lmin}}=i_{\mathrm{rmax}}$ and $d_{\mathrm{lmax}}+\delta =d_{\mathrm{rmin}}+\delta =i_{\mathrm{rmax}}$, where $\delta (n,k)=1$ exactly at $n=k=1$. The theorem constructs actual reverse, complement and reverse-complement avoidance and cut transports, and restricts their record-position bijections to the exact record fibers. It excludes the empty permutation from irreducibility and treats the singleton separately. It also proves, for all four class/statistic pairs, zero counts at record zero for positive lengths, zero counts at record one for $n\ge 2$, and every adjacent rising comparison $k<3$ for every $n\ge 4$. The two-to-three right-maximum comparison uses the existing RecordPeak theorem directly.

RecordFirstDecline.actual_four_record_first_decline gives the actual comparison $a(n,4)\le a(n,3)$ at every natural length, and the strict comparison $a(n,4)<a(n,3)$ for every $n\ge 3$, for all four class/statistic pairs. Its actual weighted difference is $t^{3}F(q(t))$, where $F(x)=1+4x+2x^2-8x^3-6x^4+11x^5+15x^6+5x^7$. The exact quotient $(1+x{)}^2{F}^{\prime }(x)/(1-2x-x^2)$ has coefficients $4,20,32,16,47,286,889,2224,5372$ through index eight and an unbounded nonnegative tail satisfying $v_m=2v_{m-1}+v_{m-2}$. Its positive-index composition coefficients dominate those of $20q(t)$; the scalar Schröder recurrence gives positivity of the actual $q$ at every positive index. The derivative identity and $F(0)=1$ therefore establish strict positivity of every coefficient of $F(q(t))$. The theorem applies the existing record transports directly. It has no conditional generating-function premise and uses no finite length cutoff.

RecordNewtonPositivity.actual_q_record_newton_nonnegative proves $[t^n]G_r(q(t))\ge 0$ for all natural kernel indices $r$ and length indices $n$, using the actual scalar $q=tL(t)$. In the independent variable $u$, let $a=u(3+2u)$, $b=u(1+u{)}^3$, $c=u(4+3u)$ and $M(z)=(1-az{)}^{-1}C(b{z}^2/(1-az{)}^2)$, where $C$ is Mathlib’s Catalan series. With $T_r=[z^r]M$, the kernel is

$$G_r=(1+u{)}^2(T_r+T_{r+1})-u(1-u)(1+u)(T_r+2T_{r+1}+T_{r+2}).$$

The proof derives the universal recurrence $(r+4)T_{r+2}=(2r+5)a{T}_{r+1}+(r+1)c{T}_r$ inside the live theorem. Twelve exact derivative-quotient certificates give the initial kernels. Two consecutive certificates for each of $E_1{T}_r$ and $E_2{T}_r$, with $E_1=(1+u{)}^2(1-u-2u^2+3u^3)$ and $E_2=(1+u{)}^2(1-2u-2u^2+4u^3)$, propagate by that recurrence to every remaining kernel. The reduction uses the actual scalar identity $t(1+q)=q(1-q)$; there is no parameter cutoff or assumed sign premise.

The declining inequalities at arbitrary $k\ge 4$ and the global maximum at three remain open. The missing inference is the normalized actual record-coefficient formula and its Newton transform connecting each actual difference to these nonnegative kernels. The unbounded sign law alone supplies neither correspondence. The problem dossier keeps the complete target distinct from them. Full CKZ Conjecture 2 and C15 remain OPEN; no partial theorem is a full external resolution, novelty or priority claim. The relevant scalar cut and cardinality results are in ActualCardinality, MinimumCutKernel, CutFactorization and ProperCut; descent statistics in Fu–Lin–Zeng are a different question.

The supplied bounded literature screen reports examination of the primary and manuscript bodies, five known citing bodies, and related record, descent, and Schröder-class papers. The five citing bodies are Li–Kitaev, King permutations and partially ordered patterns, Integers 26 (2026), A59, DOI 10.5281/zenodo.19949822; Han, Joint distributions of 10 classical statistics over permutations avoiding two patterns of length 3, Integers 26 (2026), A18, DOI 10.5281/zenodo.18305021; Liao–Yang–Yu, arXiv:2510.12046v1; Gao–Kitaev–Li–Ruan, arXiv:2509.14341v1 and the journal-formatted manuscript for DOI 10.1016/j.dam.2025.09.017; and Han–Kitaev–Zhang, arXiv:2408.12865v1. The screen found no equivalent proof or refutation of Conjecture 2 within that corpus. This scope does not establish global priority, exhaustive citation coverage, exclusive ownership, or the unavailable journal body’s contents. No originality or open-problem resolution is attributed to these partial conclusions.