Sandpiles and order structure of integer partitions
The sandpile model is closely related to integer partitions. For a partition \(a=(a_1,\dots, a_n)\) of the positive integer \(n= \sum^n_{i=1} a_i\), \(a_1\geq a_2\geq\cdots\geq a_n\geq 0\), the movement of a grain of sand in the corresponding Ferrers diagram is defined by application of one of the following two rules. Rule 1: \[ a_1,\dots, a_i,a_{i+1},\dots, a_n\to a_1,\dots, a_i-1, a_{i+1}+ 1,\dots, a_n\quad\text{if }a_i- a_{i+1}\geq 2, \] and Rule 2: \[ a_1,\dots, p+1,\underbrace{p,\dots,p,}_{k\text{\;times}} p-1,\dots, a_n\to a_1,\dots,\underbrace{p,\dots,p,}_{k+2\text{ times}}\dots,a_n. \] \textit{T. Brylawski} [Discrete Math. 6, 201-219 (1973; Zbl 0283.06003)] has introduced and studied the order \(L_B\) consisting of all partitions obtained from the partition \((n,0,\dots, 0)\) by applying the above rules. A partition \(b\) is smaller than a partition \(a\) if \(b\) can be obtained from \(a\). In this paper the authors consider a modification of Rule 2, preserving Rule 1. They define and study orders associated to these models. In particular, they show that such orders are suborders of \(L_B\) and characterize their unique fixed points and their longest chains.
scientific article; zbMATH DE number 1742047
scientific article