The toric ℎ-vectors of partially ordered sets

Abstract

An explicit formula for the toric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vector of an Eulerian poset in terms of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold c bold d"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">c</mml:mi> <mml:mi mathvariant="bold">d</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {cd}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vector. For this, another proof based on Fine’s algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g Subscript n slash 2 Baseline left-parenthesis upper P right-parenthesis equals g Subscript n slash 2 Baseline left-parenthesis upper P Superscript asterisk Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>P</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">g_{n/2}(P)=g_{n/2}(P^*)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is the only equation relating the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vectors of posets and their duals. A result on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vectors of oriented matroids is given. A simple formula for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold c bold d"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">c</mml:mi> <mml:mi mathvariant="bold">d</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {cd}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-index in terms of the flag <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vector is derived.