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How many pairs (λ,μ) of partitions of integers are there such that λ⊢n,and the Young diagram of μ is obtained from the Young diagram of λ by adding a single square?

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    Nicolas: you're o$n$ the right track.2011-05-14

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This is A000070 in the OEIS, following the explanations given there by Jon Perry or Thomas Wieder. (Both are pretty clearly equivalent to your problem but are not stated exactly the same way.) In particular, if we call the answer to your question $f(n)$, then $f(n) = \sum_{k=1}^n p(k)$ where $p(k)$ is the number of partitions of $k$.