Let $\lambda_n=\{n1,n2,\dots,1\}$ be a partition of staircase shape. Let $f(n,k)=\#\{\mu=k\mu\leq\lambda_n\}$ and $g(n,(i,j))=\#\{\mu(i,j)\notin \mu,~\mu\leq \lambda_n\}$, where $(i,j)$ denotes the $(i,j)$ entry of the Ferrers diagram of $\lambda_n$. Can anyone give some references about $f(n,k)$ and $g(n,(i,j))$ or similar problems of contained partitions?
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1$\begingroup$ For the first question, you are asking for an enumeration of Dyck paths by area. The generating function for this is known, but getting particular values might be a little messy. For the second problem, you can get a sum by adding the counts which pass through particular points. math.stackexchange.com/questions/64664/… I don't know whether the sum simplifies. $\endgroup$– Douglas ZareMar 24, 2014 at 16:42

$\begingroup$ Thanks very much for your comments and references, which are very helpful to me. $\endgroup$– xmchenhitMar 25, 2014 at 1:16

$\begingroup$ See my post [The number of partitions between two fixed partitions][1] [1]: mathoverflow.net/questions/161148/… $\endgroup$– Harry HuangApr 4, 2014 at 4:48
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