On the Schur-positivity of various sets of set partitions
A symmetric function is called Schur-positive if it admits an expansion in the Schur basis with nonnegative coefficients. In this paper, we study the Schur positivity of symmetric functions naturally associated with set partitions, with respect to two different notions of descent. In the first case, the Schur expansion involves hook-shaped Young diagrams, and the corresponding coefficients are given by Touchard-Riordan polynomials, which enumerate matchings by their number of crossings. In the second case, the Schur functions correspond to two-rows Young diagrams, and the coefficients are partial sums of associated Bell numbers. A key ingredient of our approach in the second case is the notion of a removable singleton, defined algebraically and shown to admit an equivalent combinatorial interpretation via jeu-de-taquin rectification of skew tableaux. As an application, we establish Schur positivity for various classes of symmetric functions indexed by non-crossing partitions and partitions with a given number of parts. We provide an explicit combinatorial description of the tableaux that contribute to the Schur expansion, and we connects the obtained coefficients to some known integer sequences.