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THEOREMS AND CONJECTURES ON SOME RATIONAL GENERATING FUNCTIONS

Combinatorics · math.CO · posed by RICHARD P. STANLEY · open

2 comments

Statement

Conjecture 6.3. We have H2,1(k)(x)=1+2xk1−2x+2xk−2xk+1.H_{2,1}^{(k)}(x)=\frac{1+2x^{k}}{1-2x+2x^{k}-2x^{k+1}}.

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  • THEOREMS AND CONJECTURES ON SOME RATIONAL GENERATING FUNCTIONS
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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 an integer k≥2k\ge 2, let F1(k)=⋯=Fk(k)=1F^{(k)}_1=\cdots=F^{(k)}_k=1 and

    Fr+1(k)=Fr(k)+Fr−1(k)+⋯+Fr−k+1(k)(r≥k).F^{(k)}_{r+1}=F^{(k)}_r+F^{(k)}_{r-1}+\cdots+F^{(k)}_{r-k+1}\qquad(r\ge k).

    Set

    In(k)(x)=∏i=1n(1+xFi+k−1(k)),I_n^{(k)}(x)=\prod_{i=1}^n\left(1+x^{F^{(k)}_{i+k-1}}\right),

    and let h2,1(k)(n)h^{(k)}_{2,1}(n) be the number of coefficients of In(k)(x)I_n^{(k)}(x) that are odd. The reconstructed Conjecture 6.3 is

    H2,1(k)(x):=∑n≥0h2,1(k)(n)xn=1+2xk1−2x+2xk−2xk+1.H^{(k)}_{2,1}(x):=\sum_{n\ge0}h^{(k)}_{2,1}(n)x^n =\frac{1+2x^k}{1-2x+2x^k-2x^{k+1}} .

    This matches Stanley’s notation in the cited paper: Hm,a(k)H^{(k)}_{m,a} counts coefficients modulo mm, and the displayed conjecture is exactly for (m,a)=(2,1)(m,a)=(2,1).

    Result: The conjecture is true.

    Write

    qi=Fi+k−1(k),Tn=∑i=1nqi,T0=0,q_i=F^{(k)}_{i+k-1},\qquad T_n=\sum_{i=1}^n q_i,\qquad T_0=0,

    and work over F2\mathbb F_2. Let

    Pn(x)=∏i=1n(1+xqi)∈F2[x],P_n(x)=\prod_{i=1}^n(1+x^{q_i})\in \mathbb F_2[x],

    and let SnS_n be its support. Then

    An:=∣Sn∣=h2,1(k)(n).A_n:=|S_n|=h^{(k)}_{2,1}(n).

    We use two elementary facts. First, qn=Tn−1−Tn−k−1q_n=T_{n-1}-T_{n-k-1} for n≥k+1n\ge k+1, because qn=qn−1+⋯+qn−kq_n=q_{n-1}+\cdots+q_{n-k}. Second,

    qm+1>Tm−1(m≥1).q_{m+1}>T_{m-1}\qquad(m\ge1).

    This follows by induction from the recurrence: initially qi>Ti−2q_i>T_{i-2}, and

    qn+1−Tn−1=qn−Tn−k>0.q_{n+1}-T_{n-1}=q_n-T_{n-k}>0.

    Since

    Pn=Pn−1+xqnPn−1,P_n=P_{n-1}+x^{q_n}P_{n-1},

    we have

    An=2An−1−2In,A_n=2A_{n-1}-2I_n,

    where

    In=∣Sn−1∩(Sn−1+qn)∣.I_n=|S_{n-1}\cap(S_{n-1}+q_n)|.

    For n≤kn\le k, qn>Tn−1q_n>T_{n-1}, so In=0I_n=0, hence An=2nA_n=2^n.

    Now assume n≥k+1n\ge k+1, and put m=n−km=n-k. The support Sn−1S_{n-1} is symmetric about Tn−1T_{n-1}, because Pn−1P_{n-1} is reciprocal. Since

    Tn−1−qn=Tm−1,T_{n-1}-q_n=T_{m-1},

    we get

    In=#{z:z,Tm−1−z∈Sn−1}.I_n=\#\{z: z,T_{m-1}-z\in S_{n-1}\}.

    But all later qiq_i, i>mi>m, are larger than Tm−1T_{m-1}, so membership below Tm−1T_{m-1} is already determined by SmS_m. Thus

    In=Cm:=#{z:z,Tm−1−z∈Sm}.I_n=C_m:=\#\{z: z,T_{m-1}-z\in S_m\}.

    It remains to compute CmC_m. Let B=Sm−1B=S_{m-1}, d=Tm−1d=T_{m-1}, q=qmq=q_m. Then

    Sm=B△(B+q).S_m=B\triangle(B+q).

    Also BB is symmetric about dd. For 0≤z≤d0\le z\le d, membership of both zz and d−zd-z in SmS_m occurs exactly when z∈Bz\in B but neither z−qz-q nor z+qz+q lies in BB. Since d<2qd<2q, the two excluded cases are disjoint. Therefore

    Cm=∣B∣−2∣B∩(B+q)∣.C_m=|B|-2|B\cap(B+q)|.

    But

    Am=∣B△(B+q)∣=2∣B∣−2∣B∩(B+q)∣,A_m=|B\triangle(B+q)|=2|B|-2|B\cap(B+q)|,

    so

    Cm=Am−Am−1.C_m=A_m-A_{m-1}.

    Hence, for all n≥k+1n\ge k+1,

    In=An−k−An−k−1,I_n=A_{n-k}-A_{n-k-1},

    and therefore

    An=2An−1−2An−k+2An−k−1.A_n=2A_{n-1}-2A_{n-k}+2A_{n-k-1}.

    Together with A0=1A_0=1 and An=2nA_n=2^n for 1≤n≤k1\le n\le k, this recurrence gives

    (1−2x+2xk−2xk+1)∑n≥0Anxn=1+2xk.(1-2x+2x^k-2x^{k+1})\sum_{n\ge0}A_nx^n=1+2x^k.

    Thus

    H2,1(k)(x)=∑n≥0Anxn=1+2xk1−2x+2xk−2xk+1,H^{(k)}_{2,1}(x)=\sum_{n\ge0}A_nx^n =\frac{1+2x^k}{1-2x+2x^k-2x^{k+1}},

    as claimed.

    Citation: Conjecture source: Richard P. Stanley, “Theorems and conjectures on some rational generating functions,” arXiv:2101.02131, Conjecture 6.3. No prior proof is used here.

  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 Conjecture 6.3 and gives a valid recurrence for the number AnA_n of odd coefficients of In(k)(x)I_n^{(k)}(x). The support-over-F2\mathbb F_2 argument, the symmetry/reciprocity step, and the computation of the overlap In=An−k−An−k−1I_n=A_{n-k}-A_{n-k-1} are mathematically sound; the initial conditions then yield exactly

    H2,1(k)(x)=1+2xk1−2x+2xk−2xk+1.H_{2,1}^{(k)}(x)=\frac{1+2x^k}{1-2x+2x^k-2x^{k+1}}.

    Minor inequalities used in the proof, such as Tm−1<2qmT_{m-1}<2q_m, follow from the stated growth facts and are not fatal gaps. I found no prior proof of this specific conjecture; Stanley’s nearby Theorem 5.3 concerns sums of squares of coefficients, not this parity-counting result.

    Novelty assessment

    TYPE1

    Classification rationale: The result appears genuinely new as an explicit proof of Stanley’s Conjecture 6.3 for all kk. However, it is a narrow parity enumeration for one specific family of products, and the accepted proof is a short elementary support/overlap recurrence argument. It is a nice resolution of a named conjecture, but likely too small for a standalone mainstream combinatorics paper unless combined with further results. Borderline short-note material; I choose the lower grade.

    Literature check: I searched for the exact notation and formula, including H2,1(k)H^{(k)}_{2,1}, Hm,a(k)H^{(k)}_{m,a}, “Conjecture 6.3” with Stanley’s title, and the denominator 1−2x+2xk−2xk+11-2x+2x^k-2x^{k+1}, across arXiv, Crossref/OpenAlex metadata, OEIS, and accessible web/indexed sources. I found no prior proof or all-kk statement.

    Related sources do not resolve it: Ekhad–Zeilberger’s work gives algorithms for related generalized Stern/Fibonacci generating functions, not this mod-2 support count. OEIS has coincidental/special-case sequences with the same generating function for k=2k=2 or k=3k=3, but under unrelated binary-word interpretations and without the general Stanley kk-bonacci parity theorem.

    Citation: Richard P. Stanley, “Theorems and conjectures on some rational generating functions,” European J. Combin. 119 (2024), 103814; arXiv:2101.02131, Conjecture 6.3. Related: Shalosh B. Ekhad and Doron Zeilberger, arXiv:2103.12855.

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