By Michael Huber

As a result of the type of the finite basic teams, it hasbeen attainable in recent times to symbolize Steiner t-designs, that's t -(v, ok, 1) designs,mainly for t = 2, admitting teams of automorphisms with sufficiently strongsymmetry houses. notwithstanding, regardless of the finite basic team class, forSteiner t-designs with t > 2 every one of these characterizations have remained longstandingchallenging difficulties. in particular, the decision of all flag-transitiveSteiner t-designs with three ≤ t ≤ 6 is of specific curiosity and has been open for about40 years (cf. Delandtsheer (Geom. Dedicata forty-one, p. 147, 1992 and instruction manual of IncidenceGeometry, Elsevier technological know-how, Amsterdam, 1995, p. 273), yet most likely datingback to 1965).The current paper maintains the author's paintings (see Huber (J. Comb. conception Ser.A ninety four, 180-190, 2001; Adv. Geom. five, 195-221, 2005; J. Algebr. Comb., 2007, toappear)) of classifying all flag-transitive Steiner 3-designs and 4-designs. We provide acomplete type of all flag-transitive Steiner 5-designs and end up furthermorethat there are not any non-trivial flag-transitive Steiner 6-designs. either effects depend upon theclassification of the finite 3-homogeneous permutation teams. in addition, we surveysome of the main common effects on hugely symmetric Steiner t-designs.

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In particular, if ai is an endpoint then ψi M(w) = M(w). We denote by Mai the subtree of M(w) consisting of ai and the right subtree of ai . Thus ai is either the minimum or maximum element of Mai . Suppose that ai is the minimum element of Mai . Then replace ai with the largest element of Mai , and permute the remaining elements of Mai so that they keep their same relative order. This defines ψi M(w). Similarly suppose that ai is the maximum element of the subtree Mai with root ai . 11: (a) The min-max tree M = M(5, 10, 4, 6, 7, 2, 12, 1, 8, 11, 9, 3); (b) The transformed tree ψ7 M = M(5, 10, 4, 6, 7, 2, 1, 3, 9, 12, 11, 8) Then replace ai with the smallest element of Mai , and permute the remaining elements of Mai so that they keep their same relative order.

Wn have been defined for some 1 ≤ k < n. If the first letter of δk is greater (respectively, smaller) than wk+1 , then split δk before each letter greater (respectively, smaller) than wk+1 . Then in each compartment of δk thus formed, cyclically shift the letters one unit to the left. Let the last letter of the word thus formed be wk , and remove this last letter to obtain δk−1 . It is easily verified that this procedure simply reverses the procedure used to obtain v = ϕ(w) from w, completing the proof.

I < j) such that wi > wj . The question of whether to use I(w) or code(w) depends on the problem at hand and is clearly only a matter of convenience. Often it makes no difference which is used, such as in obtaining the next corollary. 13 Corollary. Let inv(w) denote the number of inversions of the permutation w ∈ Sn . Then q inv(w) = (1 + q)(1 + q + q 2 ) · · · (1 + q + q 2 + · · · + q n−1). 30) w∈Sn Proof. If I(w) = (a1 , . . , an ) then inv(w) = a1 + · · · + an . hence n−1 n−2 q inv(w) = a1 =0 a2 =0 w∈Sn 0 ··· q a1 +a2 +···+an an =0 = 0 n−2 n−1 q a1 q a2 =0 a1 =0 as desired.

### A census of highly symmetric combinatorial designs by Michael Huber

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