Refined restricted involutions

Define I n k ( α ) to be the set of involutions of { 1 , 2 , … , n } with exactly k fixed points which avoid the pattern α ∈ S i , for some i ≥ 2 , and define I n k ( 0̸ ; α ) to be the set of involutions of { 1 , 2 , … , n } with exactly k fixed points which contain the pattern α ∈ S i , for some i ≥ 2 , exactly once. Let i n k ( α ) be the number of elements in I n k ( α ) and let i n k ( 0̸ ; α ) be the number of elements in I n k ( 0̸ ; α ) . We investigate I n k ( α ) and I n k ( 0̸ ; α ) for all α ∈ S 3 . In particular, we show that i n k ( 132 ) = i n k ( 213 ) = i n k ( 321 ) , i n k ( 231 ) = i n k ( 312 ) , i n k ( 0̸ ; 132 ) = i n k ( 0̸ ; 213 ) , and i n k ( 0̸ ; 231 ) = i n k ( 0̸ ; 312 ) for all 0 ≤ k ≤ n .