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Abstract If G is permutation group acting on a finite set $$\Omega $$ , then this action induces a natural action of G on the power set $$\mathscr{P}(\Omega )$$ . The number $s(G)$ of orbits in this action is an important parameter that has been used in bounding numbers of conjugacy classes in finite groups. In this context, $$\inf ({\log _2 s(G)}/{\log _2 |G|})$$ plays a role, but the precise value of this constant was unknown. We determine it where G runs over all permutation groups not containing any $${{\textrm {A}}}_l, l> 4$$ , as a composition factor.
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This content will become publicly available on January 13, 2026
Metrics on Permutations With the Same Descent Set
A permutation in a finite symmetric group on a set of ordered elements has a descent at the i-th index if the permutation value at the i-th index is greater than the permutation value that follows. The descent set of a permutation is the set of all indices where the permutation has a descent. Each finite symmetric group can be partitioned by descent sets. In this paper we study the Hamming metric and the L-infinity metric on the sets of permutations that share the same descent set for all nonempty descent sets to determine the maximum possible value that these metrics can achieve when restricted to these subsets.
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- Award ID(s):
- 2211379
- PAR ID:
- 10632662
- Publisher / Repository:
- PUMP Journal of Undergraduate Research
- Date Published:
- Journal Name:
- The PUMP Journal of Undergraduate Research
- Volume:
- 8
- ISSN:
- 2765-8724
- Page Range / eLocation ID:
- 57 to 69
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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