Durfee square

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In number theory, a Durfee square is an attribute of an integer partition. Durfee square_sentence_0

A partition of n has a Durfee square of side s if s is the largest number such that the partition contains at least s parts with values ≥ s. An equivalent, but more visual, definition is that the Durfee square is the largest square that is contained within a partition's Ferrers diagram. Durfee square_sentence_1

The side-length of the Durfee square is known as the rank of the partition. Durfee square_sentence_2

The Durfee symbol consists of the two partitions represented by the points to the right or below the Durfee square. Durfee square_sentence_3

Examples Durfee square_section_0

The partition 4 + 3 + 3 + 2 + 1 + 1: Durfee square_sentence_4

Durfee square_description_list_0

has a Durfee square of side 3 (in red) because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4. Durfee square_sentence_5

Its Durfee symbol consists of the 2 partitions 1 and 3+1. Durfee square_sentence_6

History Durfee square_section_1

Durfee squares are named after William Pitt Durfee, a student of English mathematician James Joseph Sylvester. Durfee square_sentence_7

In a letter to Arthur Cayley in 1883, Sylvester wrote: Durfee square_sentence_8

Properties Durfee square_section_2

See also Durfee square_section_3

Durfee square_unordered_list_1

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Durfee square.