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See: Binomial theorem
{$\binom{n}{i}$} counts the number of Young diagrams that fit within a rectangle with sides i and n-i.
Attach:BinomialTheoremInTermsOfYoungDiagrams.png Δ
The fact that the rows of the Young diagram are organized in increasing order means that a single square determines whether the Young diagram is built from those which do not use that column or those that do.