A basis for the non-crossing partition lattice top homology
Eliana Zoque
Universidad de los Andes Departamento de Matemáticas Carrera 1#18A-10 Bogotá Colombia South America Carrera 1#18A-10 Bogotá Colombia South America
DOI: 10.1007/s10801-006-7395-5
Abstract
We find a basis for the top homology of the non-crossing partition lattice T n. Though T n is not a geometric lattice, we are able to adapt techniques of Björner (A. Björner, On the homology of geometric lattices. Algebra Universalis 14 (1982), no. 1, 107-128) to find a basis with C n - 1 elements that are in bijection with binary trees. Then we analyze the action of the dihedral group on this basis.
Pages: 231–242
Keywords: keywords non-crossing partition; binary trees; homology group; Catalan numbers; representation matrix; dihedral group; stack-sortable permutations
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References
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2. A. Bj\ddot orner, “Orderings of Coxeter groups,” Combinatorics and algebra (Boulder, Colo., 1983), 175-195, Contemp. Math., 34, Amer. Math. Soc., Providence, RI, 1984.
3. A. Bj\ddot orner, “Shellable and Cohen-Macaulay partially ordered sets,” Trans. Amer. Math. Soc. 260, 159-183.
4. P. Edelman, “Chain enumeration and noncrossing partitions,” Discrete Math. 31(2), (1980), 171-180.
5. P. Edelman, “Multichains, noncrossing partitions and trees,” Discrete Math. 40(2/3), (1982), 171-179.
6. P. Edelman, R. Simion, “Chains in the lattice of noncrossing partitions,” Discrete Math. 126(1/3), (1994), 107-119.
7. D. Knuth, The Art of Computer Programming, Addison-Wesley Pub. Co., Upper Saddle River, NJ, 1973.
8. G. Kreweras, “Sur les partitions non croisÈes d'un cycle,” (French) Discrete Math. 1(4), (1972), 333-350.
9. C. Montenegro, The fixed point non-crossing partition lattices, Preprint, 1993.
10. V. Reiner, “Non-crossing partitions for classical reflection groups,” Discrete Math. 177(1/3), (1997), 195-222.
11. D. Rotem and Y. L. Varol, “Generation of binary trees from ballot sequences,” J. Assoc. Comput. Mach. 25(3), (1978), 396-404.
12. M. Wachs, “On the (co)homology of the partition lattice and the free Lie algebra,” Discrete Math. 193 (1998), 287-319.