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By N S Narasimha Sastry; et al
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A torsionfree coherent sheaf E over X is called semistable (respectively, stable) if the inequality Eq. 6) (respectively, Eq. 7)) holds for all coherent subsheaves V ⊂ E with 0 < rank(V ) < rank(E). A semistable sheaf is called polystable if it is a direct sum of stable sheaves. As before, a semistable vector bundle E is called strongly semistable if its iterated pullbacks by the self–map FX of X are all semistable. We recall FX is the Frobenius map of X when the characteristic of k is positive, and it is the identity map of X when the characteristic of k is zero.
We may consider internal vertices as well as leaves: there will be a ternary betweenness relation saying that one vertex is on the path joining the other two. • By embedding the trees in the plane, we may impose a circular order on the set of leaves. Many of these constructions give examples with exponential growth (roughly cn for some c > 1). Other examples. The symmetric group S on a countable set Ω has an induced action on the set of k-element subsets of Ω, for any k. These groups are oligomorphic, but except in the case k = 2, not much is known about the asymptotics of the orbit-counting sequences (see [11] for the case k = 2).
The definition speaks of orbits on Ωn , the set of all n-tuples. Let Fn∗ (G) denote the number of these orbits. It is clear that, for given n, one of these numbers is finite if and only if the others are; indeed, we have • Fn∗ = n S(n, k)Fk , where S(n, k) is the Stirling number of the seck=1 ond kind (the number of partitions of an n-set into k parts); • fn ≤ Fn ≤ n! fn . As an example for the first point, if G is highly transitive, then Fn∗ (G) = n S(n, k) = B(n), k=0 the nth Bell number (the number of partitions of an n-set).