Geodesic
Bijection between bigrassmannian permutations maximal below a permutation and its essential set
The electronic journal of combinatorics, Tome 17 (2010)
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Bigrassmannian permutations are known as permutations which have precisely one left descent and one right descent. They play an important role in the study of Bruhat order. Fulton introduced the essential set of a permutation and studied its combinatorics. As a consequence of his work, it turns out that the essential set of bigrassmannian permutations consists of precisely one element. In this article, we generalize this observation for essential sets of arbitrary permutations. Our main theorem says that there exists a bijection between bigrassmanian permutations maximal below a permutation and its essential set. For the proof, we make use of two equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.
DOI : 10.37236/476
Classification : 05A05, 20B30
Mots-clés : bigrassmanian permutations, Bruhat order, essential set 
@article{10_37236_476,
     author = {Masato Kobayashi},
     title = {Bijection between bigrassmannian permutations maximal below a permutation and its essential set},
     journal = {The electronic journal of combinatorics},
     year = {2010},
     volume = {17},
     doi = {10.37236/476},
     zbl = {1188.05002},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/476/}
}
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Masato Kobayashi. Bijection between bigrassmannian permutations maximal below a permutation and its essential set. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/476

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