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International audienceWe investigate the enumeration of non-crossing tree realizations of integer sequences, and we consider a special case in four parameters, that can be seen as a four-dimensional tetrahedron that generalizes Pascal’s triangle and the Catalan numbers. This work is motivated by the study of ambiguities in categorial grammars

We consider certain applications of proof theory to the study of algebraic categories. The case usually studied in the literature is that of free categories with an additional structure. In this paper, we consider several problems in nonfree categories, such as the problem of full coherence, the problem of dependency of diagrams, the problem of description of arbitrary natural transformations, which show that the applications of proof theory to categories may go much farther. Bibliography: 18 titles.