Rank of a partition

<p>In <a href="/facts/Number_theory/zhoa7zrc">number theory</a> and <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, the rank of an <a href="/facts/Integer_partition/LtyXWqwH">integer partition</a> is a certain number associated with the partition. In fact at least two different definitions of rank appear in the literature.  The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition. The concept was introduced by <a href="/facts/Freeman_Dyson/kSRxTh2r">Freeman Dyson</a> in a paper published in the journal <a href="/facts/Eureka_(University_of_Cambridge_magazine)/X6max3H3">Eureka</a>.  It was presented in the context of a study of certain <a href="/facts/Congruence_relation/TV3njpqF">congruence</a> properties of the <a href="/facts/Partition_function_(number_theory)/mBUDy5vU">partition function</a> discovered by the Indian mathematical genius <a href="/facts/Srinivasa_Ramanujan/HBPr4bzd">Srinivasa Ramanujan</a>.  A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be the size of the <a href="/facts/Durfee_square/BdebsgX3">Durfee square</a> of the partition.
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Rank of a partition open-in-new

Rank of a partition