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This paper deals with algorithms for producing and ordering lexical and nonlexical sequences of a given degree. The notion of " elementary operations" on positive α-sequences is introduced. Our main theorem answers the question of when two lexical sequences are adjacent. Given any lexical sequence, α ∈ L n , we can produce its adjacent successor as follows; apply one elementary operation on the tail of the longest left sequence, of even length, which gives a lexical successor α ∈ L n , then compute the fundamental sequence f = α ∧ α ∈ L m and conclude for m n that α is adjacent to α in L n . Whereas for m | n, the sequence α is adjacent to a sequence generated by f and the least element of L d , where d = n m . Thus, while right sequences control the lexicality property of an α-sequence, it turns out that left sequences control the adjacency property of lexical and nonlexical sequences.

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Following Coxeter we use barycentric coordinates in affine geometry to prove theorems on ratios of areas.In particular, we prove a version of Routh-Steiner theorem for parallelograms.