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Hansson, Mikael
Journal of algebraic combinatorics, 12/2016, Volume: 44, Issue: 4Journal Article
Let I n be the set of involutions in the symmetric group S n , and for A ⊆ { 0 , 1 , … , n } , let F n A = { σ ∈ I n ∣ σ has exactly a fixed points for some a ∈ A } . We give a complete characterisation of the sets A for which F n A , with the order induced by the Bruhat order on S n , is a graded poset. In particular, we prove that F n { 1 } (i.e. the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When F n A is graded, we give its rank function. We also give a short, new proof of the EL-shellability of F n { 0 } (i.e. the set of fixed-point-free involutions), recently proved by Can et al.
