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The notion of a TST-space is introduced and its connection with a parallelogram space is given. The existence of a TST-space is equivalent to the existence of a parallelogram space, which is a new characterization of a parallelogram space. The structure of a TST-space is described in terms of an abelian group.
For a simplicial complex we study the behavior of its - and -triangle under the action of barycentric subdivision. In particular we describe the - and -triangle of its barycentric subdivision . The same has been done for - and -vector of by F. Brenti, V. Welker (2008). As a consequence we show that if the entries of the -triangle of are nonnegative, then the entries of the -triangle of are also nonnegative. We conclude with a few properties of the -triangle of .
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