Pointed and multi-pointed partitions of type A and B
F. Chapoton1
and B. Vallette2
1Institut Camille Jordan, Universit`e Claude Bernard Lyon 1, B\hat atiment Braconnier, 21 avenue Claude Bernard, 69622 Villeurbanne Cedex France
2Laboratoire J.-A. Dieudonn`e, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex, France
2Laboratoire J.-A. Dieudonn`e, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex, France
DOI: 10.1007/s10801-006-8346-x
Abstract
The aim of this paper is to define and study pointed and multi-pointed partition posets of type A and B (in the classification of Coxeter groups). We compute their characteristic polynomials, incidence Hopf algebras and homology groups. As a corollary, we show that some operads are Koszul over \mathbb Z \mathbb{Z} .
Pages: 295–316
Keywords: keywords type A; type B; partitions
Full Text: PDF
References
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2. A. Bj\ddot orner and M. Wachs, Geometrically constructed bases for homology of partition lattices of types A, B and D. Electron. J. Combin., 11, no. 2, Research Paper 3, (2004/05), 26 pp. (electronic).
3. A. Bj\ddot orner and M. Wachs, On lexicographically shellable posets. Trans. Amer. Math. Soc., 277(1) (1983), 323-341.
4. F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad. IMRN, 8 (2001), 395-408.
5. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vol. I. Robert E. Krieger Publishing Co. Inc., Melbourne, Fla., 1981.
6. B. Fresse, Koszul duality of operads and homology of partition posets. in “Homotopy theory and its applications (Evanston, 2002)”, Contemp. Math., 346 (2004), 115-215.
7. P. Headley, On a family of hyperplane arrangements related to the affine Weyl groups. J. Algebraic Combin., 6(4) (1997), 331-338.
8. J. Pitman, Coalescent random forests. J. Combin. Theory Ser. A, 85(2) (1999), 165-193.
9. J. Riordan, Combinatorial identities. John Wiley & Sons Inc., New York, 1968.
10. W. R. Schmitt, Incidence Hopf Algebra. J. of Pure and Appl. Algebra, 96 (1994), 299-330.
11. L. J. Slater, Generalized hypergeometric functions. Cambridge University Press, Cambridge, 1966.
12. L. Solomon and H. Terao, The double Coxeter arrangement. Comment. Math. Helv., 73(2) (1998), 237- 258.
13. H. Terao, Multiderivations of Coxeter arrangements. Invent. Math., 148(3) (2002), 659-674.
14. B. Vallette, Homology of generalized partition posets. preprint arXiv:math.AT/0405312.
15. D. Zvonkine, An algebra of power series arising in the intersection theory of moduli spaces of curves and in the enumeration of ramified coverings of the sphere. preprint arXiv:math.AG/0403092.