The 𝐬-Eulerian polynomials have only real roots

Abstract

We study the roots of generalized Eulerian polynomials via a novel approach. We interpret Eulerian polynomials as the generating polynomials of a statistic over inversion sequences. Inversion sequences (also known as Lehmer codes or subexcedant functions) were recently generalized by Savage and Schuster, to arbitrary sequences <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold s"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">s</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of positive integers, which they called <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold s"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">s</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-inversion sequences. Our object of study is the generating polynomial of the <italic>ascent</italic> statistic over the set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold s"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">s</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-inversion sequences of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Since this ascent statistic over inversion sequences is equidistributed with the descent statistic over permutations, we call this generalized polynomial the <italic><inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold s"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">s</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Eulerian polynomial</italic>. The main result of this paper is that, for any sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold s"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">s</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of positive integers, the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold s"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">s</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Eulerian polynomial has only real roots. This result is first shown to generalize several existing results about the real-rootedness of various Eulerian polynomials. We then show that it can be used to settle a conjecture of Brenti, that Eulerian polynomials for all finite Coxeter groups have only real roots, and partially settle a conjecture of Dilks, Petersen, Stembridge on type B affine Eulerian polynomials. It is then extended to several <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-analogs. We show that the MacMahon–Carlitz <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Eulerian polynomial has only real roots whenever <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a positive real number, confirming a conjecture of Chow and Gessel. The same holds true for the hyperoctahedral group and the wreath product groups, confirming further conjectures of Chow and Gessel, and Chow and Mansour, respectively. Our results have interesting geometric consequences as well.