Polygon Dissections Complexes are Shellable
Abstract
All dissections of a convex (mn + 2)-gons into (m + 2)-gons arefacets of a simplicial complex. This complex is introduced by S. Fomin and A.V. Zelevinsky in [7]. We reprove the result of E. Tzanaki about shellability of such complex by nding a concrete shelling order. Also, we use this shelling order to nd a combinatorial interpretation of h-vector and to describe the
generating facets of these complexes.
References
[1] C. A. Athanasiadis. On a refinement of the generalized Catalan numbers for Weyl group. Trans. Amer. Math. Soc., 357 (1)(2005), 179-196
[2] C. A. Athanasiadis and E. Tzanaki. Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes. Israel J. Math., 167(1)(2008), 177-191.
[3] A. Bjorner. Topological methods. In: R. L. Graham, M. Grotschel and L. Lovasz (Eds.). Handbook of combinatorics (pp. 1819{1872), Elsevier, Amsterdam, 1995.
[4] A. Bjorner. Shellable and Cohen-Macaulay partially ordered sets. Trans. Amer. Math. Soc., 260(1)(1980), 159-183.
[5] A. Bjorner and M. L.Wachs. Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc., 348(4)(1996), 1299-1327.
[6] J. Cigler. Some remarks on Catalan families. European J. Combin., 8(3)(1987), 261-267.
[7] S. Fomin and A. Zelevinsky. Y -systems and generalized associahedra. Ann. Math., 158(3)(2003), 977-1018.
[8] J. Jonsson. Simplicial complexes of graphs. Lecture Notes in Mathematics, 1928. Springer-Verlag, Berlin, 2008.
[9] D. E. Knuth. The art of computer programming Vol. 1: Fundamental algorithms. Addison-Wesley Publishing Co., 1969
[10] J. R. Munkres. Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park CA, 1984.
[11] J. H. Przytycki and A. S. Sikora. Polygon dissections and Euler, Fuss, Kirkman, and Cayley numbers. J. Combin. Theory Ser. A, 92(1)(2000), 68-76.
[12] A. Postnikov. Permutohedra, associahedra, and beyond. Int. Math. Res. Notices, 2009(6)(2009), 1026-1106.
[13] B. Rhoades. Alexander duality and rational associahedra SIAM J. Discrete Math., 29(1)(2015), 431-460.
[14] R.P. Stanley. Catalan numbers. Cambridge University Press, New York, 2015.
[15] E. Tzanaki. Polygon dissections and some generalizations of cluster complexes. J. Combin. Theory Ser. A, 113(6)(2006), 1189-1198.
[16] G. M. Ziegler. Lectures on polytopes. Graduate Texts in Mathematics, 152. Springer-Verlag, New York, 1995
[2] C. A. Athanasiadis and E. Tzanaki. Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes. Israel J. Math., 167(1)(2008), 177-191.
[3] A. Bjorner. Topological methods. In: R. L. Graham, M. Grotschel and L. Lovasz (Eds.). Handbook of combinatorics (pp. 1819{1872), Elsevier, Amsterdam, 1995.
[4] A. Bjorner. Shellable and Cohen-Macaulay partially ordered sets. Trans. Amer. Math. Soc., 260(1)(1980), 159-183.
[5] A. Bjorner and M. L.Wachs. Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc., 348(4)(1996), 1299-1327.
[6] J. Cigler. Some remarks on Catalan families. European J. Combin., 8(3)(1987), 261-267.
[7] S. Fomin and A. Zelevinsky. Y -systems and generalized associahedra. Ann. Math., 158(3)(2003), 977-1018.
[8] J. Jonsson. Simplicial complexes of graphs. Lecture Notes in Mathematics, 1928. Springer-Verlag, Berlin, 2008.
[9] D. E. Knuth. The art of computer programming Vol. 1: Fundamental algorithms. Addison-Wesley Publishing Co., 1969
[10] J. R. Munkres. Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park CA, 1984.
[11] J. H. Przytycki and A. S. Sikora. Polygon dissections and Euler, Fuss, Kirkman, and Cayley numbers. J. Combin. Theory Ser. A, 92(1)(2000), 68-76.
[12] A. Postnikov. Permutohedra, associahedra, and beyond. Int. Math. Res. Notices, 2009(6)(2009), 1026-1106.
[13] B. Rhoades. Alexander duality and rational associahedra SIAM J. Discrete Math., 29(1)(2015), 431-460.
[14] R.P. Stanley. Catalan numbers. Cambridge University Press, New York, 2015.
[15] E. Tzanaki. Polygon dissections and some generalizations of cluster complexes. J. Combin. Theory Ser. A, 113(6)(2006), 1189-1198.
[16] G. M. Ziegler. Lectures on polytopes. Graduate Texts in Mathematics, 152. Springer-Verlag, New York, 1995
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2017-04-12
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