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The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced in [6], [7]. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions.We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers. * This paper is part of the author's Ph.D. thesis written under the direction of Prof. F. Brenti at the Univ. "la Sapienza" of Rome, Italy.

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MSC 2010: 05E10 (Primary); 20F55 (Secondary).Abstract We study the parabolic Kazhdan-Lusztig polynomials for the quasi-minuscule quotients of Weyl groups. We give explicit closed combinatorial formulas for the parabolic Kazhdan-Lusztig polynomials of type q. Our study implies that these are always either zero or a monic power of q, and that they are not combinatorial invariants. We conjecture a combinatorial interpretation for the parabolic Kazhdan-Lusztig polynomials of type −1.