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Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons

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extensions-of-partial-cyclic-orders-euler-numbers-and-multidimensionalSpectral Theorymath.COmath.SPposed by Sanjay Ramassamyrecorded: open · 1 machine check, unexamined

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Statement

For any 1≤α≤61 \le \alpha \le 6, p∞(α):=lim⁡n→∞pn(α)p_{\infty}^{(\alpha)} := \lim_{n \to \infty} p_n^{(\alpha)} exists and is given by: p∞(1)=1π(6.1)p_{\infty}^{(1)} = \frac{1}{\pi} \qquad (6.1) p∞(2)=p∞(5)=12−1π(6.2)p_{\infty}^{(2)} = p_{\infty}^{(5)} = \frac{1}{2} - \frac{1}{\pi} \qquad (6.2) p∞(3)=p∞(4)=2π−12(6.3)p_{\infty}^{(3)} = p_{\infty}^{(4)} = \frac{2}{\pi} - \frac{1}{2} \qquad (6.3) p∞(6)=1−3π(6.4)p_{\infty}^{(6)} = 1 - \frac{3}{\pi} \qquad (6.4)

Context

Candidate 1 of the open problems stated in "Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

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    Problem: The reconstructed conjecture is Conjecture 6.1 of Ramassamy. For n≥2n\ge2, let P+n\mathcal P_{+^n} be the set of total cyclic orders ZZ on [n+2][n+2] such that

    (i,i+1,i+2)∈Z(1≤i≤n).(i,i+1,i+2)\in Z\qquad(1\le i\le n).

    Partition P+n\mathcal P_{+^n} into six classes according to the cyclic order of

    1, 2, n+1, n+2:1,\ 2,\ n+1,\ n+2: R(1):(1,2,n+1,n+2),R(2):(1,n+1,2,n+2),R(3):(1,n+1,n+2,2),R(4):(1,2,n+2,n+1),R(5):(1,n+2,2,n+1),R(6):(1,n+2,n+1,2).\begin{aligned} \mathcal R^{(1)} &: (1,2,n+1,n+2),& \mathcal R^{(2)} &: (1,n+1,2,n+2),\\ \mathcal R^{(3)} &: (1,n+1,n+2,2),& \mathcal R^{(4)} &: (1,2,n+2,n+1),\\ \mathcal R^{(5)} &: (1,n+2,2,n+1),& \mathcal R^{(6)} &: (1,n+2,n+1,2). \end{aligned}

    Here (a,b,c,d)(a,b,c,d) means that these four points appear in that cyclic order. Define

    pn(α)=#R+n(α)#P+n.p_n^{(\alpha)}=\frac{\#\mathcal R_{+^n}^{(\alpha)}}{\#\mathcal P_{+^n}}.

    The conjecture asserts that pn(α)p_n^{(\alpha)} has a limit and gives the six stated values.

    This is supported directly by the paper’s definitions of Pw\mathcal P_w, Rw(α)\mathcal R_w^{(\alpha)}, and the displayed Conjecture 6.1.

    Result: The conjecture is true.

    Represent a random cyclic order on [M][M], M=n+2M=n+2, by choosing independent uniform points q2,…,qM∈[0,1)q_2,\dots,q_M\in[0,1), fixing q1=0q_1=0, and reading clockwise order. Put L=M−1=n+1L=M-1=n+1 and

    Xi=(qi+1−qi) mod 1,1≤i≤L,X_i=(q_{i+1}-q_i)\bmod 1,\qquad 1\le i\le L,

    and

    U=(−X1−⋯−XL) mod 1=(q1−qM) mod 1.U=(-X_1-\cdots-X_L)\bmod 1=(q_1-q_M)\bmod 1.

    The map (q2,…,qM)↦(X1,…,XL)(q_2,\dots,q_M)\mapsto (X_1,\dots,X_L) is measure-preserving on the torus.

    For 1≤i≤L−11\le i\le L-1,

    (i,i+1,i+2)∈Z  ⟺  Xi+Xi+1<1.(i,i+1,i+2)\in Z \iff X_i+X_{i+1}<1.

    Hence P+n\mathcal P_{+^n} corresponds to the density

    ∏i=1L−11{Xi+Xi+1<1}.\prod_{i=1}^{L-1}\mathbf 1_{\{X_i+X_{i+1}<1\}}.

    Let X=X1X=X_1, Y=XLY=X_L. In coordinates,

    q2=X,qM=1−U,qM−1={1−U−Y,Y+U<1,2−U−Y,Y+U>1.q_2=X,\qquad q_M=1-U,\qquad q_{M-1}= \begin{cases} 1-U-Y,&Y+U<1,\\ 2-U-Y,&Y+U>1. \end{cases}

    Thus the six limiting events are determined by (X,Y,U)(X,Y,U).

    Define the compact self-adjoint operator

    (Tf)(x)=∫01−xf(y) dy.(Tf)(x)=\int_0^{1-x} f(y)\,dy.

    Its top eigenvalue is λ0=2/π\lambda_0=2/\pi, with positive eigenfunction

    ϕ(x)=cos⁡(πx/2).\phi(x)=\cos(\pi x/2).

    Standard spectral decomposition gives, under the above conditioned measure,

    (X1,XL)⟹(X,Y),(X_1,X_L)\Longrightarrow (X,Y),

    where X,YX,Y are independent with density

    h(x)=π2cos⁡(πx/2),0≤x≤1.h(x)=\frac{\pi}{2}\cos(\pi x/2),\qquad 0\le x\le1.

    Moreover, using the twisted operators

    (Tθf)(x)=∫01−xeiθyf(y) dy,(T_\theta f)(x)=\int_0^{1-x} e^{i\theta y}f(y)\,dy,

    one has r(T2πm)<2/πr(T_{2\pi m})<2/\pi for every nonzero integer mm. Therefore the fractional closing gap

    U=(−X1−⋯−XL) mod 1U=(-X_1-\cdots-X_L)\bmod1

    converges to an independent uniform [0,1][0,1] variable.

    So

    p∞(α)=∫01∫01∫011α(x,y,u) h(x)h(y) du dy dx.p_\infty^{(\alpha)} =\int_0^1\int_0^1\int_0^1 \mathbf 1_{\alpha}(x,y,u)\,h(x)h(y)\,du\,dy\,dx.

    The relevant interval lengths in uu yield:

    p∞(1)=E(1−X−Y)+=1π.p_\infty^{(1)}=\mathbb E(1-X-Y)_+=\frac1\pi.

    Also,

    p∞(1)+p∞(2)=E[1−max⁡(X,Y)]=∫01sin⁡2(πt/2) dt=12,p_\infty^{(1)}+p_\infty^{(2)} =\mathbb E[1-\max(X,Y)] =\int_0^1 \sin^2(\pi t/2)\,dt =\frac12,

    and

    p∞(1)+p∞(2)+p∞(3)=E(1−Y)=2π.p_\infty^{(1)}+p_\infty^{(2)}+p_\infty^{(3)} =\mathbb E(1-Y)=\frac2\pi.

    Thus

    p∞(2)=12−1π,p∞(3)=2π−12.p_\infty^{(2)}=\frac12-\frac1\pi,\qquad p_\infty^{(3)}=\frac2\pi-\frac12.

    By symmetry,

    p∞(4)=p∞(3),p∞(5)=p∞(2).p_\infty^{(4)}=p_\infty^{(3)},\qquad p_\infty^{(5)}=p_\infty^{(2)}.

    Finally,

    p∞(6)=E(X+Y−1)+=2EX−1+p∞(1)=1−3π.p_\infty^{(6)}=\mathbb E(X+Y-1)_+ =2\mathbb E X-1+p_\infty^{(1)} =1-\frac3\pi.

    Therefore

    p∞(1)=1π,p∞(2)=p∞(5)=12−1π,p∞(3)=p∞(4)=2π−12,p∞(6)=1−3π.p_{\infty}^{(1)}=\frac1\pi,\quad p_{\infty}^{(2)}=p_{\infty}^{(5)}=\frac12-\frac1\pi,\quad p_{\infty}^{(3)}=p_{\infty}^{(4)}=\frac2\pi-\frac12,\quad p_{\infty}^{(6)}=1-\frac3\pi.

    Citation: Problem source: Sanjay Ramassamy, “Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons,” Electron. J. Combin. 25(1), Paper 1.66, 2018; arXiv:1706.03386. No prior resolution of the conjecture is used here.

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE2

      PASS

      The argument attacks the correct six-class refinement of P+n\mathcal P_{+^n}. The random-points model gives uniform cyclic orders, the gap change of variables is measure-preserving, and the condition (i,i+1,i+2)∈Z(i,i+1,i+2)\in Z is correctly converted to Xi+Xi+1<1X_i+X_{i+1}<1. The transfer-operator spectral argument yields the stated endpoint density, and the nonzero Fourier-mode spectral gap for the twisted operators gives an independent uniform closing gap. The subsequent interval-length integrals give exactly the conjectured constants. I found only the original paper and related consecutive-coordinate-polytope work, not a prior resolution of these asymptotic densities.

      Novelty assessment

      TYPE2

      Classification rationale: The result appears genuinely new and resolves an explicit conjecture from a published EJC paper. It is narrow and not a major advance, but the proof uses a nontrivial transfer-operator/asymptotic argument and would plausibly support a short standalone note in a standard combinatorics journal. It is not TYPE3-level because the conjecture is specialized and of limited broader impact.

      Literature check: I found no prior resolution of Ramassamy’s Conjecture 6.1/Conjecture 17. The closest related works are Ayyer–Josuat-Vergès–Ramassamy on consecutive-coordinate polytopes, which develops the polytope/transfer-map framework, and Diaconis–Wood on related adjacent-sum polytopes; neither states or proves the six limiting endpoint/cyclic-order densities. Citation searches show only a few citing works, none addressing this asymptotic density conjecture. Searches for the exact constants and for “asymptotic densities” with “partial cyclic orders”/“Ramassamy” did not reveal an existing proof.

      Citation: Sanjay Ramassamy, “Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons,” Electron. J. Combin. 25(1), P1.66, 2018; arXiv:1706.03386. Closely related: Ayyer, Josuat-Vergès, Ramassamy, “Extensions of partial cyclic orders and consecutive coordinate polytopes,” Ann. Henri Lebesgue 3 (2020), 275–297.

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