Niveau: Supérieur, Doctorat, Bac+8
Supersolvable LL-lattices of binary trees Riccardo Biagioli and Frederic Chapoton February 28, 2005 Abstract Some posets of binary leaf-labeled trees are shown to be supersolvable lat- tices and explicit EL-labelings are given. Their characteristic polynomials are computed, recovering their known factorization in a different way. 1 Introduction The aim of this article is to study some posets on forests of binary leaf-labeled trees. These posets first appeared as an essential ingredient in the combinatorial description of the coproduct in the Hopf operad introduced by the second author in [4]. They have since been shown in [5] to have some nice properties, mainly that the characteristic polynomials of all intervals factorize completely with positive integer roots. By a theorem of Stanley [8], this factorization property is true in general for the so-called semimodular supersolvable lattices. Since these intervals are not semimodular in general, one can not use this theorem to recover the result of [5]. For a class of lattices, called LL-lattices, containing the semimodular-supersolvable ones, a theorem due to Blass and Sagan [3] generalizes Stanley's theorem. The first main theorem of our article states that these intervals are indeed lattices, which was not known before. The proof uses a new description of the intervals using admissible partitions. Our second main result is the fact that these lattices are supersolvable.
- sn el-shellable
- el-labelings has
- x1 ?
- partition lattice
- saturated chain
- el-shellable posets
- supersolvable lattices
- has
- posets
- x0 ?