The theory of P-partitions was developed by Stanley to understand/solve several enumerations problems and representations theory problems. Together with the work of Gessel, this led to the development of the space of quasisymmetric functions. Schur functions are naturally understood in the world of quasisymmetric functions as a sum over standard tableaux of Gessel fundamental functions.

Unrelated to this, Lascoux and Schutzenberger introduced a special class of polynomials related to Schubert varieties named flagged Schur functions. Much more recently Assaf and Searles developed the theory of slide polynomials that would decompose nicely Schubert polynomials. Together with Assaf, we noticed that these slide polynomials can be obtained as bounded P-partitions similar to Gessel Fundamental functions, and started to develop a more general theory of bounded P-partitions.

Our main theorem shows that if the bounds form a flag (only increase strictly at descents) then the bounded P partition polynomial enumerator is a positive sum of slide polynomials. This allows us to understand flag schur functions as a sum over standard tableaux of slide polynomials (completing the picture in a nice way.)

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