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Home > Vol 18, No 3 (2003)

Fabrizio Caselli

Università di Roma “La Sapienza” Dipartimento di Matematica “G. Castelnuovo” P. le A. Moro 5 00185 Roma, Italy

We find an explicit formula for the Kazhdan-Lusztig polynomials P _ui,a ,v _i of the symmetric group \mathfrak S \mathfrak{S} ( n) where, for a, i, n \mathbb N \mathbb{N} such that 1 \mathfrak S \mathfrak{S} ( n) obtained by inserting n in position i in any permutation of \mathfrak S \mathfrak{S} ( 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.

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