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Abstract:In the siblings (or brotherhood) variant of the coupon collector's problem, a main collector draws coupons until her own album is complete and passes every duplicate down a chain of siblings; the $j$th collector is then left with $U_j^N$ empty places, $j\ge 2$. It has been conjectured [stated as an open problem in the work that introduced the model] that, for every fixed number of coupon types $N$ and every $j\ge 2$, the expected deficit $\E[U_j^N]$ is maximized by the equiprobable coupon distribution. We prove this in a sharp, finite-$N$ form: $\E[U_j^N]$ is strictly larger at the uniform vector than at any other probability vector, and indeed strictly increases along every ray running from an arbitrary distribution toward the uniform one. The proof is exact and elementary in its ingredients. An inclusion--exclusion step turns the governing Poissonized integral into a one-dimensional integral with a separable integrand; a single integration by parts then rewrites the radial derivative of $\E[U_j^N]$ as a positively weighted covariance of an increasing function, whose sign is settled by Chebyshev's correlation inequality. We show that $\E[U_j^N]$ is \emph{not} Schur-concave, so that no majorization or pairwise-smoothing argument can yield the result, and we explain why the recent variance-extremality method of Long~[Long, arXiv:2604.25108, 2026] does not transfer. As by-products we obtain a finite closed form for $\E[U_j^N]$ over subsets of the coupon set and the exact Hessian of $\E[U_j^N]$ at the uniform vector. The argument extends without change to all real $j>1$.

Submission history

From: Aristides Doumas [view email]
[v1] Fri, 19 Jun 2026 16:47:42 UTC (9 KB)