Decorated Dyck paths, polyominoes, and the Delta conjecture

<p>We discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case the main results of both Haglund (“A proof of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q comma t"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">q,t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schröder conjecture”, 2004) and Aval et al. (“Statistics on parallelogram polyominoes and a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q comma t"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">q,t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-analogue of the Narayana numbers”, 2014). This settles in particular the cases <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="mathematical left-angle dot comma e Subscript n minus d Baseline h Subscript d Baseline mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨<! -- ⟨ --></mml:mo> <mml:mo>⋅<! -- ⋅ --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−<! -- − --></mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩<! -- ⟩ --></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle \cdot ,e_{n-d}h_d\rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="mathematical left-angle dot comma h Subscript n minus d Baseline h Subscript d Baseline mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨<! -- ⟨ --></mml:mo> <mml:mo>⋅<! -- ⋅ --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−<! -- − --></mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩<! -- ⟩ --></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle \cdot ,h_{n-d}h_d\rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Delta conjecture of Haglund, Remmel and Wilson (“The delta conjecture”, 2018). Along the way, we introduce some new statistics, formulate some new conjectures, prove some new identities of symmetric functions, and answer a few open problems in the literature (e.g., from Aval, Bergeron and Garsia [2015], Haglund, Remmel and Wilson [2018], and Zabrocki [2019]). The main technical tool is a new identity in the theory of Macdonald polynomials that extends a theorem of Haglund in “A proof of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="q comma t"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">q,t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schröder conjecture” (2004). </p>