If λ = (λ
1,…, λm
) where λ
1,…,λm
are nonnegative integers with λ
1 ≥…≥ λ
m, then λ is a partition of |λ| = λ
1 + …+λm
, and we write λ
⊢ |λ|. The non-zero λi
's are the parts of λ, so λ
1 is the largest part, and ℓ(λ) is the number of parts of λ. Two partitions with the same parts, so they differ only in number of zeros, are the same. The set of all partitions, including the partition of 0 (with 0 parts) is denoted by The conjugate of λ, denoted by , is the partition (μ
1,…, μk
), in which μi
is the number of λ's that are ≥i , for i = 1,…, k, where k=
λ
1.