# Generalization's of Greene's Theorem for the Robinson-Schensted correspondence

One important property of the Robinson-Schensted correspondence (RS) is that the longest increasing subsequence of the permutation $\sigma$ is $\lambda_1$, the first entry of the shape $\lambda(\sigma)$ of the tableaux associated to $\sigma$. Greene's theorem (sorry for the paywall) generalizes this result to $k$-increasing sequences and the sum $\lambda_1 + \dots + \lambda_k$.

Are there generalizations of Greene's theorem to other insertion algorithms? Such algorithms would include RSK, Hecke insertion, Edelman-Greene insertion and any other variants you are aware of. In particular, it seems a generalization of this is known for the full RSK correspondence, but I am having difficulty finding a reference. Moreover, I was told of an article by Haiman on generalizations of RS that may include such results, but have never found it.

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The boycott be thanked, there is no paywall on Advances in Mathematics anymore. It's a good question, and I wouldn't be surprised if the straightforward RSK analogue (interpret the matrix of nonnegative integers as a "poset", with $k$ elements in every cell filled with integer $k$) would work as soon as one would sort out the failure of antisymmetry in these "posets". – darij grinberg Mar 15 '13 at 17:44
I think the article of Haiman that you are thinking of is "Dual equivalence with applications, including a conjecture of Proctor", sciencedirect.com/science/article/pii/0012365X9290368P – Michael Joyce Mar 15 '13 at 17:48
We considered trying to construct Greene invariants for Hecke insertion but everything we tried turned out to be trivial. This is likely because the Hecke inserted tableaux, while uniquely determined, may be K-Knuth equivalent to other tableaux of different shape. – Matt Samuel Sep 5 '15 at 21:33
@MattSamuel Interesting. Have you looked at Patrias and Pylyavskyy (arxiv.org/abs/1410.7683)? – Zach Hamaker Sep 5 '15 at 21:38
Yes. I'm the Samuel in citation 7! – Matt Samuel Sep 5 '15 at 22:50

For RSK the answer is "well known". You can find the statements neatly arranged in an article by Christian Krattenthaler http://arxiv.org/abs/math/0510676.

I think the right framework for this question is Sergey Fomin's theory of dual graded graphs. However, I don't think there are many other insertion algorithms where the Greene-Kleitman invariant is known. One is the insertion algorithm for shifted tableaux, and another, easy one is the pair (BinTree, BinWord).

In fact, whenever you have such a Greene-Kleitman invariant and whenever this invariant behaves well with respect to "promotion", you are in a good position to get a result parallel to http://arxiv.org/abs/math/0604140. For the pair (BinTree, BinWord) this is indeed the case (and interesting), but I never managed to write it up due to time constraints...

For Edelman-Greene the story is slightly different I think. If I recall correctly you can say at least a little bit about the shape of the word by staring long enough at the article by Christian Stump and Luis Serrano http://arxiv.org/abs/1009.4690 or myself http://arxiv.org/abs/1009.3919.

EDIT:

The Kleitman Greene invariants for some insertion algorithms (for the standard case, i.e., where the words are permutations) are described in Sergey Fomin's paper "Schensted algorithms for dual graded graphs":

1) Theorem 4.4.4: Young-Fibonacci insertion (due to Tom Roby and Sergey Fomin, perhaps the invariant for Janvier Nzeutchap's algorithm is different).

2) Just below Proposition 4.5.2: Shifted insertion (attributed to Worley and Bruce Sagan, see Richard Stanley's answer for the description in the semistandard case due to Luis Serrano)

3) Proposition 4.6.2: (BinTree, BinWord)-insertion (independently due to Xavier Viennot)

I'd be interested in learning about Kleitman-Greene invariants for other insertion algorithms. In particular, is it known for domino insertion (as described by Marc van Leeuwen, see also this paper by Thomas Lam)

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If nothing else, you will be able to say something about the shape of the words in that article. – Marc van Leeuwen Mar 16 '13 at 14:03
Sorry, I don't understand? – Martin Rubey Mar 16 '13 at 15:39