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Shuffles are n-multipermutations with suit multiplicities given by a subset Rof {1,..,n-1}. Their inverses are ordered partitions of {1,..,n} whose blocksizes derive from R. These "R-permutations" depict the min length coset repsfor the quotient of S_n by the parabolic subgroup W_J, with J the complement ofR. We refer to those that blockwise avoid the pattern 312 as "312-avoidingR-permutations" and define the "parabolic R-Catalan number" to be the number ofthem. Let lambda be a partition of N with at most n parts whose set of shapecolumn lengths less than n is R. We show that the number of flagged Schurfunctions formed on the shape of lambda is this parabolic R-Catalan number, andlist over a dozen other phenomena that are enumerated by it. Let pi be anR-permutation. We view the Demazure character (key polynomial) indexed by(lambda,pi) as the sum of the content weight monomials for our "pi-Demazure"semistandard tableaux of shape lambda with entries from {1,..,n}. We show thatthe set of these tableaux is convex in Z^N if and only if pi is a 312-avoidingR-permutation. A flagged Schur function is the sum of the content weightmonomials for the semistandard tableaux of shape lambda whose entries arerow-wise bounded by a given weakly increasing n-tuple. We consider general rowbound sums for which the bounds may be any n-tuple. Reiner and Shimozono andthen Postnikov and Stanley obtained results concerning coincidences betweenflagged Schur functions and Demazure characters: when lambda is strict, theflagged Schur functions exactly coincide with the 312-avoiding Demazurecharacters. For general lambda we introduce precise indexing sets of n-tuplebounds for the row bound sums. This and our convexity results are used tosharpen their coincidence results, to extend them to general row bound sums,and to show they hold at the deeper level of coinciding underlying tableausets.

Parabolic Catalan numbers count flagged Schur functions; Convexity of tableau sets for Demazure characters