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Poisson approximation for large permutation groups

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arxiv 2408.06611 v2 pith:FHIR543H submitted 2024-08-13 math.PR math.CO

Persi Diaconis , Nathan Tung This is my paper · ORCID
classification math.PRmath.CO
keywords blockscyclespermutespoissonapproximationaspectscompounddependence
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abstract

Let $G_{k,n}$ be a group of permutations of $kn$ objects which permutes things independently in disjoint blocks of size $k$ and then permutes the blocks. We investigate the probabilistic and/or enumerative aspects of random elements of $G_{k,n}$. This includes novel limit theorems for fixed points, cycles of various lengths, number of cycles and inversions. The limits are compound Poisson distributions with interesting dependence structure.

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  1. Counting the number of group orbits by marrying the Burnside process with importance sampling

    math.PR 2025-01 conditional novelty 6.0 of 10

    A new algorithm estimates orbit counts by multiplying estimates of orbit-count ratios from Burnside process samples, and estimates k(U_n(F_q)) for q=2,3 up to n=32.

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