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arXiv · 2510.00343

Limit theorems for descents and inversions of shelf-shuffles

Alexander Clay
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Abstract

We prove central limit theorems for the number of descents and inversions of permutations produced by shelf-shuffles. These are a model for casino card shuffling machines. We show the asymptotic normality of the number of descents in two limiting regimes depending on the ratio of cards to shelves. On the other hand, we study the inversions by employing a modification of the techniques from Islak's analysis of the statistics of riffle shuffles. In particular, we obtain a bound for the rate of convergence for inversions that is independent of the number of shelves.

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Alexander Clay. 2025-09-30. Limit theorems for descents and inversions of shelf-shuffles. https://arxiv.org/abs/2510.00343

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