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Recent developments in the theory of Stirling numbers

Combinatorics · math.CO · posed by István Mező · open

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Statement

Find the asymptotics of the maximizing index of Er(n,k)E_{r}(n,k) .

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  • Recent developments in the theory of Stirling numbers
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  1. exploration by a model · #1

    GPT-5.5 xhigh

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    Problem: The phrase is ambiguous because several inequivalent objects are called rr-Eulerian numbers. In the Stirling-number context I reconstruct Mező’s Er(n,k)E_r(n,k) as the coefficients of

    Pn,r(x)=∑kEr(n,k)xk=(1−x)n+r+1∑m≥0(m+1)n(m+2)rxm,P_{n,r}(x)=\sum_k E_r(n,k)x^k =(1-x)^{n+r+1}\sum_{m\ge0}(m+1)^n(m+2)^r x^m,

    with r≥0r\ge0, n≥1n\ge1. Let κn,r\kappa_{n,r} be any index maximizing Er(n,k)E_r(n,k).

    Result: For every fixed rr,

    ∣κn,r−(n+r−12−rn+r)∣<1.\left|\kappa_{n,r}-\left(\frac{n+r-1}{2}-\frac{r}{n+r}\right)\right|<1.

    Hence

    κn,r=n2+Or(1),more preciselyκn,r=n+r−12+O(1).\kappa_{n,r}=\frac n2+O_r(1), \qquad\text{more precisely}\qquad \kappa_{n,r}=\frac{n+r-1}{2}+O(1).

    Proof sketch. Put d=n+rd=n+r. The numerator Pn,rP_{n,r} is real-rooted with nonnegative coefficients: from

    Pn+1,r=(1+dx)Pn,r+x(1−x)Pn,r′,P_{n+1,r}=(1+dx)P_{n,r}+x(1-x)P'_{n,r},

    and the analogous rr-step recurrence

    Pn,r+1=(2+(d−1)x)Pn,r+x(1−x)Pn,r′,P_{n,r+1}=(2+(d-1)x)P_{n,r}+x(1-x)P'_{n,r},

    real-rootedness follows inductively from P1,0=1P_{1,0}=1, since the operator (a+cx)Q+x(1−x)Q′(a+cx)Q+x(1-x)Q', a>0, c≥deg⁡Qa>0,\ c\ge\deg Q, preserves negative real zeros by the standard interlacing sign argument.

    Thus the normalized coefficients form a Poisson-binomial distribution. Its mean is

    μn,r=Pn,r′(1)Pn,r(1).\mu_{n,r}=\frac{P'_{n,r}(1)}{P_{n,r}(1)}.

    Since

    (m+1)n(m+2)r=md+(n+2r)md−1+O(md−2),(m+1)^n(m+2)^r=m^d+(n+2r)m^{d-1}+O(m^{d-2}),

    one obtains

    Pn,r(1)=d!,Pn,r′(1)=d!(d+12−n+2rd),P_{n,r}(1)=d!,\qquad P'_{n,r}(1)=d!\left(\frac{d+1}{2}-\frac{n+2r}{d}\right),

    so

    μn,r=n+r−12−rn+r.\mu_{n,r}=\frac{n+r-1}{2}-\frac{r}{n+r}.

    Darroch’s theorem for sums of independent Bernoulli variables gives every mode within distance <1<1 of the mean, proving the claim.

    Verification audit: the result applies to all n≥1n\ge1, r≥0r\ge0, allows multiple maximizing indices, and gives the claimed asymptotic without adding hypotheses beyond the reconstructed definition.

    Citation: Darroch, J. N., “On the distribution of the number of successes in independent trials,” Ann. Math. Statist. 35 (1964), 1317–1321.

  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 KNOWN

    PASS

    The claimed argument attacks the right object, assuming Mező’s standard generating-function definition of Er(n,k)E_r(n,k). The proof is mathematically sound: the recurrences imply real-rootedness with nonnegative coefficients, hence the normalized row is Poisson-binomial; the mean computation from the top two coefficients of (m+1)n(m+2)r(m+1)^n(m+2)^r gives

    μ=n+r−12−rn+r,\mu=\frac{n+r-1}{2}-\frac{r}{n+r},

    and Darroch’s theorem puts every mode within <1<1 of this mean. This gives a stronger-than-asymptotic localization of the maximizing index. I did not find an existing published statement of this mode asymptotic in the literature searches.

    Novelty assessment

    KNOWN

    Classification rationale: The stated asymptotic location of the maximizing index is already subsumed by later work on Eulerian recurrences. Hwang–Chern–Duh treat the recurrence family containing these rr-Eulerian polynomials (up to shifts/initial conditions) and prove asymptotic normality with mean n/2+Or(1)n/2+O_r(1) and variance n/12+O(n)n/12+O(n). Together with known real-rootedness/log-concavity, this gives the maximizing index asymptotic n/2+Or(1)n/2+O_r(1) (indeed n/2+o(n)n/2+o(n) already answers Mező’s stated problem). The accepted proof’s sharper “within 1 of the mean” form is a routine Darroch-theorem corollary once real-rootedness and the mean are known.

    Literature check: I checked the Mező/RIMS problem context, OEIS entries for rr-Eulerian/Li Shanlan numbers, and later arXiv/open literature on Eulerian recurrences. The key later source is Hwang–Chern–Duh, “An asymptotic distribution theory for Eulerian recurrences with applications,” especially Theorem 1 and §§4.2, 4.5.2, where rr-Eulerian numbers/related initial-condition variants are explicitly included and the CLT N(n/2,n/12)\mathscr N(n/2,n/12) is established. Liu–Wang gives the standard real-rootedness machinery for these Eulerian-type recurrences; Darroch then localizes modes near the mean.

    Citation: H.-K. Hwang, H.-H. Chern, G.-H. Duh, “An asymptotic distribution theory for Eulerian recurrences with applications,” arXiv:1807.01412, §§4.2, 4.5.2. See also L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007), 542–560; J. N. Darroch, Ann. Math. Statist. 35 (1964), 1317–1321.

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