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)

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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)

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.)

Robinson–Schensted correspondence - Wikipedia

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.)