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This is the first of three papers that develop structures which are countedby a "parabolic" generalization of Catalan numbers. Fix a subset R of{1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes aredetermined by R. These are the "inverses" of (parabolic) multipermutationswhose multiplicities are determined by R. The standard forms of the orderedpartitions are refered to as "R-permutations". The notion of 312-avoidance isextended from permutations to R-permutations. Let lambda be a partition of Nsuch that the set of column lengths in its shape is R or R union {n}. Fix anR-permutation pi. The type A Demazure character (key polynomial) in x_1, ..,x_n that is indexed by lambda and pi can be described as the sum of the weightmonomials for some of the semistandard Young tableau of shape lambda that areused to describe the Schur function indexed by lambda. Descriptions of these"Demazure" tableaux developed by the authors in earlier papers are used toprove that the set of these tableaux is convex in Z^N if and only if pi isR-312-avoiding if and only if the tableau set is the entire principal idealgenerated by the key of pi. These papers were inspired by results of Reiner andShimozono and by Postnikov and Stanley concerning coincidences between Demazurecharacters and flagged Schur functions. This convexity result is used in thenext paper to deepen those results from the level of polynomials to the levelof tableau sets. The R-parabolic Catalan number is defined to be the number ofR-312-avoiding permutations. These special R-permutations are reformulated as"R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gaplessR-tuples"; the latter n-tuples arise in multiple contexts in these papers.

Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers