The Schröder case of the generalized Delta conjecture

Abstract

We prove the Schröder case, i.e. the case 〈⋅,en−dhd〉, of the conjecture of Haglund et al. (2018) for ΔhmΔen−k−1′en in terms of decorated partially labelled Dyck paths, which we call generalized Delta conjecture. This result extends the Schröder case of the Delta conjecture proved in (D’Adderio, 2017), which in turn generalized the q,t-Schröder in Haglund (2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another combinatorial interpretation of the same polynomial in terms of a new bounce statistic. Moreover, we give two more interpretations of the same polynomial in terms of doubly decorated parallelogram polyominoes, extending some of the results in D’Adderio (2017), which in turn extended results in Aval et al., (2014). Also, we provide combinatorial bijections explaining some of the equivalences among these interpretations.