Joint Extremes of Inversions and Descents of Random Permutations

Abstract We provide asymptotic theory for the joint distribution of $$X_\textrm{inv}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mtext>inv</mml:mtext> </mml:msub> </mml:math> and $$X_\textrm{des}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mtext>des</mml:mtext> </mml:msub> </mml:math> , the numbers of inversions and descents of random permutations. Recently, [14] proved that $$X_\textrm{inv}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mtext>inv</mml:mtext> </mml:msub> </mml:math> , respectively, $$X_\textrm{des}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mtext>des</mml:mtext> </mml:msub> </mml:math> , is in the maximum domain of attraction of the Gumbel distribution. To tackle the dependency between these two permutation statistics, we use Hájek projections and a suitable quantitative Gaussian approximation. We show that $$(X_\textrm{inv}, X_\textrm{des})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mtext>inv</mml:mtext> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mtext>des</mml:mtext> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is in the maximum domain of attraction of the two-dimensional Gumbel distribution with independent margins. This result can be stated in the broader combinatorial framework of finite Coxeter groups, on which our method also yields the central limit theorem for $$(X_\textrm{inv}, X_\textrm{des})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mtext>inv</mml:mtext> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mtext>des</mml:mtext> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and various other permutation statistics as a novel contribution. In particular, signed permutation groups with random biased signs and products of classical Weyl groups are investigated.