Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials) - Robert A. Proctor; Matthew J. Willis (Discrete Mathematics & Theoretical Computer Science, 2017)
Fix an integer partition lambda that has no more than n parts. Let beta be a weakly increasing n-tuple with entries from {1,..,n}. The flagged Schur function indexed by lambda and beta is a polynomial generating function in x_1, .., x_n for certain semistandard tableaux of shape lambda. Let pi be an n-permutation. The type A Demazure character (key polynomial, Demazure polynomial) indexed by lambda and pi is another such polynomial generating function. Reiner and Shimozono and then Postnikov and Stanley studied coincidences between these two families of polynomials. Here their results are sharpened by the specification of unique representatives for the equivalence classes of indexes for both families of polynomials, extended by the consideration of more general beta, and deepened by proving that the polynomial coincidences also hold at the level of the underlying tableau sets. Let R be the set of lengths of columns in the shape of lambda that are less than n. Ordered set partitions of {1,..,n} with block sizes determined by R, called R-permutations, are used to describe the minimal length representatives for the parabolic quotient of the nth symmetric group specified by the set {1,..,n-1}\R. The notion of 312-avoidance is generalized from n-permutations to these set partitions. The R-parabolic Catalan number is defined to be the number of these. Every flagged Schur function arises as a Demazure polynomial. Those Demazure polynomials are precisely indexed by the R-312-avoiding R-permutations. Hence the number of flagged Schur functions that are distinct as polynomials is shown to be the R-parabolic Catalan number. The projecting and lifting processes that relate the notions of 312-avoidance and of R-312-avoidance are described with maps developed for other purposes.