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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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What is the symbol between $\lambda$ and $n$? I'm not familiar with it. – Nicolas Villanueva May 14 '11 at 2:27
So $\lambda$ partitions $n$, and $\mu$ would partition $(n+1)$ s.t. the Young Tableau of $\mu$ is the Young Tableau of $\lambda$ with an extra square? – Nicolas Villanueva May 14 '11 at 2:30
Essentially you would have to find all the integer partitions of $n$ and sum up (# of unique integers in the partition) + 1 [since you can legally add a square to the end of the Tableau]. I'm sorry for writing this out little by little, just trying to get a hold of the question in my head. – Nicolas Villanueva May 14 '11 at 2:38
Nicolas: you're on the right track. – Michael Lugo May 14 '11 at 2:42

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 $k$.

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