Geodesic
Proof of two conjectures of Brenti and Simion on Kazhdan-Lusztig polynomials
Journal of Algebraic Combinatorics, Tome 18 (2003) no. 3, pp. 171-187.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We find an explicit formula for the Kazhdan-Lusztig polynomials $P _{u}$i,a $,v _{i}$ of the symmetric group $\mathfrak S$ mathfrakS $( n)$ where, for $a, i, n \mathbb N$ mathbbN such that $1 \mathfrak S$ mathfrakS $( n)$ obtained by inserting $n$ in position $i$ in any permutation of $\mathfrak S$ mathfrakS $( n - 1)$ allowed to rise only in the first and in the last place. Our result implies, in particular, the validity of two conjectures of Brenti and Simion [4, Conjectures 4.2 and 4.3], and includes as a special case a result of Shapiro, Shapiro and Vainshtein [13, Theorem 1]. All the proofs are purely combinatorial and make no use of the geometry of the corresponding Schubert varieties.
Keywords: Kazhdan-Lusztig polynomials, symmetric group, Bruhat order 
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     author = {Caselli, Fabrizio},
     title = {Proof of two conjectures of {Brenti} and {Simion} on {Kazhdan-Lusztig} polynomials},
     journal = {Journal of Algebraic Combinatorics},
     pages = {171--187},
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     number = {3},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2003__18_3_a4/}
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Caselli, Fabrizio. Proof of two conjectures of Brenti and Simion on Kazhdan-Lusztig polynomials. Journal of Algebraic Combinatorics, Tome 18 (2003) no. 3, pp. 171-187. http://geodesic.mathdoc.fr/item/JAC_2003__18_3_a4/