Abstract
We prove a conjecture of Stanley on thecd-index of the semisuspension of the face poset of a simplicial shelling component. We give a new signed generalization of André permutations, together with a new notion ofcd-variation for signed permutations. This generalization not only allows us to compute thecd-index of the face poset of a cube, but also occurs as a natural set of orbit representatives for a signed generalization of the Foata-Strehl commutative group action on the symmetric group. From the induction techniques used, it becomes clear that there is more than one way to define classes of permutations andcd-variation such that they allow us to compute thecd-index of the same poset.
Article PDF
Similar content being viewed by others
References
M. M. Bayer and L. J. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets.Invent. Math. 79 (1985), 143–157.
M. M. Bayer and A. Klapper, A new index for polytopes,Discrete Comput. Geom. 6 (1991), 33–47.
A. Björner, Shellable and Cohen-Macaulay partially ordered sets.Trans. Amer. Math. Soc. 260 (1980), 159–183.
R. Ehrenborg and M. Readdy, Ther-cubical lattice and a generalization of thecd-index,European J. Combin., to appear.
D. Foata and M. P. Schützenberger, Nombres d'Euler et permutations alternantes, Manuscript, University of Florida, Gainesville, FL, 1971.
D. Foata and M. P. Schützenberger, Nombres d'Euler et permutations alternantes. In: J. N. Srivastavaet al., A Survey of Combinatorial Theory, Amsterdam, North-Holland, 1973, pp. 173–187.
D. Foata and V. Strehl, Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers,Math. Z. 137 (1974), 257–264.
D. Foata and V. Strehl, Euler numbers and variations of permutations. In:Colloquio Internazionale sulle Teorie Combinatorie, 1973, Tome I (Atti Dei Convegni Lincei 17, 119–131), Accademia Nazionale dei Lincei, Rome, 1976.
N. Metropolis and G.-C. Rota, Combinatorial structure of the faces of then-cube,SIAM J. Appl. Math. 35(4) (1978), 689–694.
M. Purtill, André permutations, lexicographic shellability, and thecd-index of a convex polytope,Trans. Amer. Math. Soc. 338(1) (1993), 77–104.
R. P. Stanley, Flagf-vectors and thecd-index,Math. Z. 216 (1994), 483–499.
S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice,Adv. in Math. 104 (1994), 225–296.
S. Sundaram, The homology of partitions with an even number of blocks,J. Algebraic Combin. 4(1) (1995), 69–92.
Author information
Authors and Affiliations
Additional information
This research was supported by the UQAM Foundation.
Rights and permissions
About this article
Cite this article
Hetyei, G. On thecd-variation polynomials of André and simsun permutations. Discrete Comput Geom 16, 259–275 (1996). https://doi.org/10.1007/BF02711512
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02711512