Permutations with restricted cycle lengths

We find exact and asymptotic formulas for the number of pairs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> -cycles <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>s</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>z</mml:mi> </mml:math> such that all cycles of the product <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>·</mml:mo> <mml:mi>z</mml:mi> </mml:mrow> </mml:math> have lengths from a given set of integers. We then apply one of these results to prove a surprisingly high lower bound for the number of permutations whose block transposition distance from the identity is at least <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> .