Parallel Rank of Two Sandpile Models of Signed Integer Partitions

Table 1

Parallel rank in 𝑃(𝑛, 𝑑, 𝑟).

Case	Relation between 𝑛, 𝑑, and 𝑟	Parallel rank 𝑇par(0)
I	𝑑≤min{𝑟,𝑛−𝑟}	𝑛+𝑑−2
II	𝑟≥𝑑≥2(𝑛−𝑟)	𝑑+𝑟−2
III	𝑟≥𝑑≥𝑛−𝑟 and 2(𝑛−𝑟)≥𝑑≥𝑛−𝑟	2𝑛−𝑟−2
IV	𝑛−𝑟≥𝑑≥2𝑟	𝑑+𝑛−𝑟−2
V	𝑛−𝑟≥𝑑≥𝑟 and 2𝑟≥𝑑≥𝑟	𝑛+𝑟−2
VI	min{2𝑟,2(𝑛−𝑟)}≥𝑑≥max{𝑟,𝑛−𝑟}	2𝑛−𝑑−2
VII	2𝑟>𝑑≥2(𝑛−𝑟) and 𝑑≥max{𝑟,𝑛−𝑟}	2𝑟−2
VIII	2(𝑛−𝑟)>𝑑≥2𝑟 and 𝑑≥max{𝑟,𝑛−𝑟}	2(𝑛−𝑟)−2