<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> -Birkhoff polytopes

In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan’s question, in this paper we define a pattern-avoiding Birkhoff polytope called a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> -Birkhoff polytope for each Coxeter element <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> of the symmetric group. We then show that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> -Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> -sorting word of the longest permutation. When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>⋯</mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> , this result recovers an affirmative answer to Davis and Sagan’s question. Another consequence of this result is that the normalized volume of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> -Birkhoff polytope is the number of the longest chains in the (type A) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> -Cambrian lattice.