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RECURSIONS ASSOCIATED TO TRAPEZOID, SYMMETRIC AND ROTATION SYMMETRIC FUNCTIONS OVER GALOIS FIELDS

Algebra · math.CO · math.RT · posed by Francis N. Castro, Robin Chapman, Luis A. Medina, L. Brehsner Sepúlveda · open

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Statement

Let (5.3)δo(j)={0if j is even1if j is odd.(5.3) \qquad \delta_o(j) = \begin{cases} 0 & \text{if } j \text{ is even} \\ 1 & \text{if } j \text{ is odd.} \end{cases} Define rk(n){r_{k}(n)} by (5.4) rk(0)=kr_{k}(0)\quad=\quad k rk(j)=2j−δo(j)⋅2,for 1≤j≤k−1r_{k}(j)\quad=\quad 2^{j}-\delta_{o}(j)\cdot 2,\quad for \ 1 \leq j \leq k-1 rk(n)=2∑l=0k−2rk(n−(l+2)),for n≥k.r_{k}(n)\quad=\quad 2 \sum_{l=0}^{k-2}r_{k}(n-(l+2)),\quad for \ n \geq k. Then, S(R2,3,...,k(n))=rk(n)S(R_{2,3,...,k}(n))=r_{k}(n) for all values of n ≥k.

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  • RECURSIONS ASSOCIATED TO TRAPEZOID, SYMMETRIC AND ROTATION SYMMETRIC FUNCTIONS OVER GALOIS FIELDS
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: For k≥2k\ge 2, work over F2\mathbb F_2. Define

    R2,3,…,k(n)=∑i=1nXiXi+1⋯Xi+k−1,R_{2,3,\ldots,k}(n)=\sum_{i=1}^n X_iX_{i+1}\cdots X_{i+k-1},

    with indices modulo nn, and

    S(F)=∑x∈F2n(−1)F(x).S(F)=\sum_{x\in\mathbb F_2^n}(-1)^{F(x)}.

    Let δo(j)=1\delta_o(j)=1 if jj is odd and 00 otherwise. Define rkr_k by

    rk(0)=k,rk(j)=2j−2δo(j)(1≤j≤k−1),r_k(0)=k,\qquad r_k(j)=2^j-2\delta_o(j)\quad(1\le j\le k-1),

    and

    rk(n)=2∑ℓ=0k−2rk(n−ℓ−2)(n≥k).r_k(n)=2\sum_{\ell=0}^{k-2}r_k(n-\ell-2)\quad(n\ge k).

    The conjecture asserts:

    S(R2,3,…,k(n))=rk(n)(n≥k).S(R_{2,3,\ldots,k}(n))=r_k(n)\qquad(n\ge k).

    Result: The conjecture is true.

    Let A=AkA=A_k be the k×kk\times k matrix indexed by 0,…,k−10,\ldots,k-1 with

    Ai0=1(0≤i≤k−1),Ai,i+1=1(0≤i≤k−2),Ak−1,k−1=−1,A_{i0}=1\quad(0\le i\le k-1),\qquad A_{i,i+1}=1\quad(0\le i\le k-2),\qquad A_{k-1,k-1}=-1,

    and all other entries 00.

    A closed walk in this transition graph records a cyclic binary word: state ii means the current terminal run of 11’s has length ii, capped at k−1k-1. The only negative transition is k−1→k−1k-1\to k-1, exactly when a cyclic block of kk consecutive 11’s occurs. Hence, for n≥kn\ge k,

    S(R2,3,…,k(n))=tr⁡(An).S(R_{2,3,\ldots,k}(n))=\operatorname{tr}(A^n).

    Now compute the characteristic polynomial. Since AA and ATA^T have the same characteristic polynomial, expanding XI−ATXI-A^T along its first row gives

    χA(X)=Xk−2(Xk−2+Xk−3+⋯+X+1).\chi_A(X)=X^k-2(X^{k-2}+X^{k-3}+\cdots+X+1).

    By Cayley-Hamilton,

    An=2(An−2+An−3+⋯+An−k)(n≥k),A^n=2(A^{n-2}+A^{n-3}+\cdots+A^{n-k})\qquad(n\ge k),

    so

    tr⁡(An)=2∑ℓ=0k−2tr⁡(An−ℓ−2).\operatorname{tr}(A^n)=2\sum_{\ell=0}^{k-2}\operatorname{tr}(A^{n-\ell-2}).

    It remains to check the initial values. Clearly

    tr⁡(A0)=k.\operatorname{tr}(A^0)=k.

    For 1≤j≤k−11\le j\le k-1, every cyclic binary word of length jj contributes +1+1, except the all-one word, which contributes (−1)j(-1)^j. Thus

    tr⁡(Aj)=2j−1+(−1)j=2j−2δo(j).\operatorname{tr}(A^j)=2^j-1+(-1)^j=2^j-2\delta_o(j).

    Therefore tr⁡(An)\operatorname{tr}(A^n) satisfies exactly the same recurrence and initial conditions as rk(n)r_k(n). Hence

    S(R2,3,…,k(n))=tr⁡(An)=rk(n)S(R_{2,3,\ldots,k}(n))=\operatorname{tr}(A^n)=r_k(n)

    for all n≥kn\ge k.

    Verification audit: the case k=2k=2 is included; the boundary case n=kn=k uses the paper’s convention that the nn cyclic shifts are all counted, so duplicated monomials cancel according to parity in F2\mathbb F_2. No extra hypotheses are introduced.

    Citation: The conjecture is Conjecture 5.2 in Castro, Chapman, Medina, and Sepúlveda, “Recursions associated to trapezoid, symmetric and rotation symmetric functions over Galois fields,” Discrete Mathematics 341 (2018), 1915–1931. The proof above is self-contained.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The proof attacks the correct R2,3,…,k(n)R_{2,3,\ldots,k}(n) statement. The weighted transfer matrix correctly counts cyclic binary words with sign (−1)# cyclic k-blocks of ones(-1)^{\#\text{ cyclic }k\text{-blocks of ones}}, hence equals S(R2,3,…,k(n))S(R_{2,3,\ldots,k}(n)) for n≥kn\ge k. The characteristic polynomial gives exactly the stated recurrence, and the trace initial values match rk(0)=kr_k(0)=k and rk(j)=2j−2δo(j)r_k(j)=2^j-2\delta_o(j). I found no fatal gap or mismatch.

    Novelty assessment

    TYPE1

    Classification rationale: The result appears genuinely new as an explicit resolution of Castro–Chapman–Medina–Sepúlveda Conjecture 5.2, but it is a very small contribution. The proof is a standard transfer-matrix/finite-state automaton argument for cyclic binary words with a weight on runs of kk ones, followed by an elementary characteristic-polynomial computation. It resolves a narrow one-parameter conjecture, but introduces no new method and is likely too short/routine for a standalone combinatorics paper except perhaps as a brief note or part of a broader work.

    Literature check: I found the conjecture in the original arXiv/journal paper, where only Conjecture 4.4 is noted as having been upgraded to a theorem; Conjecture 5.2 remains listed as an observation/conjecture. Searches for the exact conjecture label and notation, including “Conjecture 5.2” with “rotation symmetric Boolean functions”, R2,3,…,kR_{2,3,\ldots,k}, S(R2,3,…,k(n))S(R_{2,3,\ldots,k}(n)), and the displayed recurrence/characteristic polynomial, did not reveal an open-access paper, note, forum post, or repository proving the statement. I also checked the arXiv author page/feed for Luis A. Medina and related follow-up papers by the authors; later work concerns symmetric polynomials/value distributions, not this rotation-symmetric conjecture. General transfer-matrix and pattern-occurrence methods certainly contain the technique, but I found no explicit prior statement of this recurrence in the literature.

    Citation: Francis N. Castro, Robin Chapman, Luis A. Medina, and L. Brehsner Sepúlveda, “Recursions associated to trapezoid, symmetric and rotation symmetric functions over Galois fields,” Discrete Mathematics 341 (2018), 1915–1931; arXiv:1702.08038.

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