Weighted Ehrhart Series and a Type-$$\textsf{B}$$ Analogue of a Formula of MacMahon

Abstract We present a formula for a generalisation of the Eulerian polynomial, namely the generating polynomial of the joint distribution of major index and descent statistic over the set of signed multiset permutations. It has a description in terms of the $${h^*\text {-polynomial}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>h</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mtext>-polynomial</mml:mtext> </mml:mrow> </mml:math> of a certain polytope. Moreover, we associate a family of polytopes to (generalised) Eulerian polynomials of types $$\textsf{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> and $$\textsf{B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> . Using this connection, properties of the generalised Eulerian numbers of types $$\textsf{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> and $$\textsf{B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> , such as palindromicity and unimodality, are reflected in certain properties of the associated polytope. We also present results on generalising the connection between descent polynomials and polytopes to coloured (multiset) permutations.