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Rank and Durfee square

The rank of a partition is the largest number k such that the partition contains at least k parts of size at least k. For example, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4. In the Ferrers diagram or Young diagram of a partition of rank r, the r × r square of entries in the upper-left is known as the Durfee square:

The Durfee square has applications within combinatorics in the proofs of various partition identities. It also has some practical significance in the form of the h-index.

A different statistic is also sometimes called the rank of a partition (or Dyson rank), namely, the difference λ k − k {\displaystyle \lambda _{k}-k} for a partition of k parts with largest part λ k {\displaystyle \lambda _{k}}. This statistic (which is unrelated to the one described above) appears in the study of Ramanujan congruences.