Parallelogram polyominoes, partially labelled Dyck paths, and the Delta conjecture (FULL VERSION)

2017/12/23 by D'Adderio, Michele, Iraci, Alessandro
#05E05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.08787

Abstract

We introduce area, bounce and dinv statistics on decorated parallelogram polyominoes, and prove that some of their q,t-enumerators match ⟨ Δ_hm e_n+1,s_k+1,1^n-k⟩, extending in this way the work in (Aval et al. 2014). Also, we provide a bijective connection between decorated parallelogram polyominoes and decorated labelled Dyck paths, which allows us to prove the combinatorial interpretation of the coefficient ⟨ Δ_e_m+n-k-1'e_m+n,h_m h_n⟩ predicted by the Delta conjecture in (Haglund et al. 2015). Finally, we define a statistic pmaj on partially labelled Dyck paths, which provides another conjectural combinatorial interpretation of Δ_h_ℓΔ_e_n-k-1'e_n, cf. (Haglund et al. 2015). This is the full version of (D'Adderio, Iraci 2017) arXiv:1711.03923.

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