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Identities Involoving Some Numbers Related to Dyck Paths

Combinatorics · math.CO · posed by SUN Yi-dong · open

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Statement

and for r ≥3 ∑i=0n(−1)n−i(ni)Um+k+2+i,k+i(r)={2nif m=2n,2n(52n+2r−1)if m=2n+1,(1.11)\sum_{i=0}^{n}(-1)^{n-i}\left(\begin{array}{l}n\\i \end{array}\right)U_{m+k+2+i,k+i}^{(r)}=\left\{\begin{array}{ll}2^{n}&\text{if }m=2n,\\2^{n}\left(\frac{5}{2}n+2r-1\right)&\text{if }m=2n+1,\end{array}\right.\quad(1.11)

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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: Reconstructed statement. Let

    C(x)=1−1−4x2x=1+x+2x2+5x3+⋯C(x)=\frac{1-\sqrt{1-4x}}{2x}=1+x+2x^2+5x^3+\cdots

    be the Catalan generating function, and define Sun’s Dyck-path numbers UN,K(r)U_{N,K}^{(r)} by

    UN,K(r)=[xN] x2C(x)2r−1(x(C(x)−x))K=[xN−K−2] C(x)2r−1(C(x)−x)K.U_{N,K}^{(r)}=[x^N]\,x^2C(x)^{2r-1}\bigl(x(C(x)-x)\bigr)^K =[x^{N-K-2}]\,C(x)^{2r-1}(C(x)-x)^K .

    This is the Riordan-array form of the U(r)U^{(r)}-family used in the cited Dyck-path identities.

    For all integers n,k≥0n,k\ge0, r≥3r\ge3,

    ∑i=0n(−1)n−i(ni)Um+k+2+i,k+i(r)={2n,m=2n,2n(52n+2r−1),m=2n+1.\sum_{i=0}^{n}(-1)^{n-i}\binom ni U_{m+k+2+i,k+i}^{(r)} = \begin{cases} 2^n,&m=2n,\\[2mm] 2^n\left(\frac52n+2r-1\right),&m=2n+1. \end{cases}

    Result: Put H(x)=C(x)−xH(x)=C(x)-x. Then

    H(x)=1+2x2+5x3+O(x4),C(x)2r−1=1+(2r−1)x+O(x2).H(x)=1+2x^2+5x^3+O(x^4),\qquad C(x)^{2r-1}=1+(2r-1)x+O(x^2).

    Using the definition of UN,K(r)U_{N,K}^{(r)},

    S=∑i=0n(−1)n−i(ni)[xm] C(x)2r−1H(x)k+i=[xm] C(x)2r−1H(x)k(H(x)−1)n.\begin{aligned} S &=\sum_{i=0}^n(-1)^{n-i}\binom ni [x^m]\,C(x)^{2r-1}H(x)^{k+i}\\ &=[x^m]\,C(x)^{2r-1}H(x)^k(H(x)-1)^n . \end{aligned}

    Now

    H(x)−1=2x2+5x3+O(x4),H(x)-1=2x^2+5x^3+O(x^4),

    so

    (H(x)−1)n=2nx2n+5n 2n−1x2n+1+O(x2n+2).(H(x)-1)^n =2^n x^{2n}+5n\,2^{n-1}x^{2n+1}+O(x^{2n+2}).

    Also H(x)k=1+O(x2)H(x)^k=1+O(x^2), so it does not affect the coefficients of x2nx^{2n} or x2n+1x^{2n+1}.

    Thus

    [x2n] C(x)2r−1H(x)k(H(x)−1)n=2n,[x^{2n}]\,C(x)^{2r-1}H(x)^k(H(x)-1)^n=2^n,

    and

    [x2n+1] C(x)2r−1H(x)k(H(x)−1)n=2n(2r−1)+5n2n−1=2n(52n+2r−1).[x^{2n+1}]\,C(x)^{2r-1}H(x)^k(H(x)-1)^n =2^n(2r-1)+5n2^{n-1} =2^n\left(\frac52 n+2r-1\right).

    This proves both claimed cases.

    Verification audit: the proof uses exactly the reconstructed U(r)U^{(r)}-definition, all n,k≥0n,k\ge0, r≥3r\ge3, and no condition on kk beyond nonnegativity. The conclusion matches (1.11).

    Citation: No prior proof found. Definition/source context: Sun Yi-dong, “Identities Involoving Some Numbers Related to Dyck Paths,” Journal of Mathematical Research and Exposition, 2005; Yidong Sun, “The statistic ‘number of udu’s’ in Dyck paths,” Discrete Mathematics 287 (2004), 177–186.

  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 correctly reduces the alternating binomial sum to a coefficient of

    C(x)2r−1(C(x)−x)k(C(x)−x−1)n,C(x)^{2r-1}(C(x)-x)^k\bigl(C(x)-x-1\bigr)^n,

    using the stated Riordan-array definition of U(r)U^{(r)}. The low-order expansions then determine exactly the coefficients of x2nx^{2n} and x2n+1x^{2n+1}, yielding the two claimed values. The argument is uniform in k≥0k\ge0 and r≥3r\ge3, including n=0n=0.

    I found no prior published proof or stronger result in the available literature search.

    Novelty assessment

    TYPE1

    Classification rationale: The accepted proof is a very short coefficient-extraction argument using only the first two nonzero terms of C(x)−x−1C(x)-x-1. Even if the identity was not previously proved in print, it is an immediate, routine consequence of the generating-function definition of Sun’s UN,K(r)U^{(r)}_{N,K}. It is not substantial enough for a standalone combinatorics paper.

    Literature check: I searched the exact misspelled title, corrected title, formula fragments such as 5n/2+2r−15n/2+2r-1, Um+k+2+i,k+iU_{m+k+2+i,k+i}, and generating-function fragments involving C(x)−xC(x)-x and x2C(x)2r−1x^2C(x)^{2r-1}. I also checked Semantic Scholar/OpenAlex/Crossref records and citation trails for Sun’s 2005 paper and the related 2004 Discrete Mathematics paper. Semantic Scholar lists only two citations to the 2005 paper: Sun’s “number of udu’s” paper and Chao-Jen Wang’s 2011 thesis on Goulden–Jackson cluster methods; neither appears to prove this conjectural identity. No open-access note, thesis, survey, OEIS-style source, or later paper located contained this exact resolution or a stronger one.

    Citation: Sun Yi-dong, “Identities Involoving Some Numbers Related to Dyck Paths,” Journal of Mathematical Research and Exposition 25(3) (2005), 441–446. Related context: Yidong Sun, “The statistic ‘number of udu’s’ in Dyck paths,” Discrete Mathematics 287 (2004), 177–186.

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