Involutions in Coxeter groups

Abstract We combinatorially characterize the number $$\textrm{cc}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>cc</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count the number of conjugacy classes of reflections. Moreover, we provide formulae for finite and affine types, besides computing $$\textrm{cc}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>cc</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> for all triangle groups and RACGs.