12 Rudolf Ahlswede ,

One of the basic results in extremal set theory was discovered in [1] and rediscovered in [2]: For a given number of k-element subsets of an n-set the shadow, that is, the set of ( k 1)-element subsets contained in at least one of the specified k-element subsets, is minimal, if the k-element subsets are chosen as an initial segment in the squashed order (see [10]; called colex order in [liD, that is, a kelement subset A precedes a k-element subset B, if the largest element in A A B is in B. A closely related result was discovered in [3] and rediscovered in [5]: For a given number u C [0,2 n] of arbitrary subsets of an n-set the "Hamming distance 1"-boundary is minimal for the initial segment of size u, also called in short "u-th initial segment", in the H-order (of [3]), that is, if one chooses all subsets of cardinality less than n k (k suitable) and all remaining subsets of cardinality n k, whose complements are in the initial segment of the squashed order. In this paper we consider sequences and subsequences rather than sets and subsets. n The basic objects are X n = 1-I x for X = {0,1} and n E N, and operations of deletion Vi, V and of insertion Ai, A. Here Vi (resp. Ai) means tha t letter i (i = 0,1) is deleted (resp. inserted) and V (resp. A) means the deletion (resp. insertion) of any letter. So for A C X n we get the down shadow