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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)

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You probably know this, but I figure it's worth reminding people that π is an involution if and only if P and Q (the tableaux under RSK) are equal. –  Jonah Ostroff Nov 10 '10 at 14:13
    
Agreed. I edited accordingly. –  Vasu vineet Nov 11 '10 at 2:32
    
@Vasu-vineet, while you do have a link to a page explaining the Robinson-Schensten-(Knuth) RSK correspondence, you never specifically state the full name of the algorithm. Others reading this page would be a bit mystified and may not bother clicking through to read the details of concept which establishes a bijective correspondence between elements of the symmetric group $S_n$ and pairs of standard Young tableaux of the same shape. Also, your wikipedia link has malformed characters in it and doesn't work. Try en.wikipedia.org/wiki/Robinson-Schensted_algorithm –  sleepless in beantown Nov 11 '10 at 2:56
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1 Answer 1

For a version of a), see C. Greene, An extension of Schensted's theorem, Advances in Mathematics, 1974. Note: the result is beautiful, but the statement is a bit delicate.

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This looks interesting. I'll take a look at what this offers. Thanks. –  Vasu vineet Nov 11 '10 at 2:33
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