It is well known via the RSK-correspondence that the length of the longest decreasing subsequence in a permutation $\pi \in S_n$ is the length of the longest column of the insertion tableau of $\pi$. ( The insertion tableau and the recording tableau produced by this algorithm have the same shape)
My question is what else can be gleaned from the RSK correspondence in terms of ,say,
a) the length of the next longest decreasing subsequence in $\pi$ ?
b) the number of longest decreasing subsequences in $\pi$, given the fact that there is exactly one column of maximum length ?
c) can we say more about the above two questions if we knew that $\pi$ was an involution? (If $\pi$ happens to be an involution, then insertion tableau and recording tableau produced are equal)