The sandpile model on the complete split graph: $q,t$-Schr\"oder polynomials, sawtooth polyominoes, and a cycle lemma

Abstract

This paper studies sorted recurrent configurations of the Abelian sandpile model on the complete split graph. We introduce two natural toppling processes, CTI and ITC toppling, on the recurrent configurations and use these to define two toppling delay statistics, wtopple$_{CTI}$ and wtopple$_{ITC}$. These new toppling delay statistics are time-weighted sums for the number of vertices that topple during each iteration of the toppling processes. We then introduce the bivariate $q,t$-CTI and $q,t$-ITC polynomials that are the generating functions of the bistatistics (level,wtopple$_{ITC}$) and (level,wtopple$_{CTI}$), where level is the well-established sandpile level statistic. We prove the bistatistic (level,wtopple$_{ITC}$) maps to a bistatistic (area,bounce) on Schr\"oder paths that was introduced by Egge, Haglund, Killpatrick and Kremer (2003). This establishes equality of the $q,t$-ITC polynomial and the $q,t$-Schr\"oder polynomial of those same authors. This connection allows us to relate the $q,t$-ITC polynomial to the theory of symmetric functions and also establishes symmetry of the $q,t$-ITC polynomials. We conjecture equality of the $q,t$-CTI and $q,t$-ITC polynomials. We also present and prove a characterization of sorted recurrent configurations as a new class of polyominoes that we call sawtooth polyominoes. The CTI and ITC toppling processes on sorted recurrent configurations are proven to correspond to bounce paths within the polyominoes. The main difference between the two bounce paths is the initial direction in which they travel. In addition to this, and building on the results of Aval, D'Adderio, Dukes, and Le Borgne (2016), we present a cycle lemma for a slight extension of stable configurations that allows for an enumeration of sorted recurrent configurations within the framework of the sandpile model.