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In their 1987 paper "Balanced Tableaux", Edelman and Greene construct a bijection between standard young tableaux with staircase shape $(n-1,n-2, \dots , 1)$ and reduced decompositions of the reverse permutation $(n,n-1, \dots, 1)$. The bijection is constructed using jeu de taquin and evacuation paths in forward direction and a modified version of RSK in the reverse.

David Little constructed a more general bijection between reduced decompositions of an arbitrary permutation and standard young tableaux of specific shapes in his 2003 paper "Combinatorial aspects of the Lascoux-Schutzenberger tree". His construction is, to my eyes, completely different, using line diagrams and circle diagrams (I don't know if these are standard terms). In the paper, he conjectures that his bijection, when specialized to the reverse permutation, recovers the Edelman-Greene bijection. It is know that the sets on which the bijections act coincide.

My questions are the following:

  1. What is the current research status of Little's conjecture? Is it known whether the bijections are the same? Little states that he will investigate it further, but I am unaware of any new results.
  2. Are there any other bijections between standard young tableaux with staircase shape and reduced decompositions of the reverse permutation? I am aware of Felsner's generalization of the Edelman-Greene bijection, but I am restricting my question to the case of the reverse permutation.
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2 Answers 2

up vote 4 down vote accepted

Question one now has an answer. Benjamin Young and I have proved that the two bijections are the same in this case. A reference is available on the arXiv. More generally, we have shown the Little map is the same map as the recording tableau of Edelman-Greene insertion, the RSK variant referred to above.

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I suppose it's bad form for me to upvote this answer? –  Benjamin Young Nov 4 '12 at 16:26
    
Probably worse form for me to make it :) –  Zach Hamaker Nov 4 '12 at 17:30

There is perhaps more computational evidence that the two bijections coincide (in the form of larger samples). However, to the best of my knowledge the conjecture is still open. I'd be interested to hear if you can make any progress in this direction.

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