Designs, permutations, and transitive groups
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A notion of t-designs in the symmetric group on n letters was introduced by Godsil in 1988. In particular t-transitive sets of permutations form a t-design. We derive special lower bounds for t=1 and t=2 by a power moment method. For general n,t we give a For n≥ 4 and t=2, this bound is strong enough to show a lower bound on the size of such t-designs of n(n-1)… (n-t+1), which is best possible when sharply t-transitive sets of permutations exist. This shows, in particular, that tight 2-designs do not exist.
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