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In mathematics, solid partitions are natural generalizations of partitions and plane partitions defined by Percy Alexander MacMahon. A solid partition of n {\displaystyle n} is a three-dimensional array of non-negative integers n i , j , k {\displaystyle n_{i,j,k}} such that
and
Let p 3 {\displaystyle p_{3}} denote the number of solid partitions of n {\displaystyle n}. As the definition of solid partitions involves three-dimensional arrays of numbers, they are also called three-dimensional partitions in notation where plane partitions are two-dimensional partitions and partitions are one-dimensional partitions. Solid partitions and their higher-dimensional generalizations are discussed in the book by Andrews.