Crossings and nestings of matchings and partitions

Abstract

We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket n right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, as well as over all matchings on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 2 n right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[2n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a corollary, the number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-noncrossing partitions is equal to the number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-crossing (or with no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-nesting).