Data on Kazhdan-Lusztig polynomials up to S_11 suggests superexponential growth of extremal coefficients, near-universal unimodality, and a conjectured closed family (1+v+...+v^l)^{k-1}.
Pattern Avoidance and the Bruhat Order
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abstract
The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. A method for determining non-isomorphic principal order ideals is described and applied for small lengths. The permutations with boolean principal order ideals are characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, the permutations whose principal order ideals have a form related to boolean posets are also completely described. It is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed. Finally, the Bruhat order in types B and D is studied, and the elements with boolean principal order ideals are characterized and enumerated by length.
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Big data approach to Kazhdan-Lusztig polynomials
Data on Kazhdan-Lusztig polynomials up to S_11 suggests superexponential growth of extremal coefficients, near-universal unimodality, and a conjectured closed family (1+v+...+v^l)^{k-1}.