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►As an example, , , is a partition of .
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►The total number of partitions of is denoted by .
…For the actual partitions () for see Table 26.4.1.
►The integers whose sum is are referred to as the parts in the partition.
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►A strict shifted plane partition is an arrangement of the parts in a partition so that each row is indented one space from the previous row and there is weak decrease across rows and strict decrease down columns.
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►A descending plane partition is a strict shifted plane partition in which the number of parts in each row is strictly less than the largest part in that row and is greater than or equal to the largest part in the next row.
The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition.
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§26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
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►Table 26.4.1 gives numerical values of multinomials and partitions
for .
… is a partition of :
… is the number of set partitions of with subsets of size 1, subsets of size 2, , and subsets of size :
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►Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)).
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►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients for up to 50 and up to 25; extends Table 26.4.1 to ; tabulates Stirling numbers of the first and second kinds, and , for up to 25 and up to ; tabulates partitions
and partitions into distinct parts for up to 500.
►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts , partitions into parts , and unrestricted plane partitions up to 100.
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