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Central Delannoy Numbers and Balanced Cohen- Macaulay Complexes
Gábor Hetyei
Department of Mathematics and Statistics, The University of North Carolina at Charlotte, Charlotte, NC 28223, USA
ghetyei@uncc.edu
Annals of Combinatorics 10 (4) p.443-462 December, 2006
AMS Subject Classification: 13F55, 05A15, 16E65, 33C45
Abstract:
We introduce a new join operation on colored simplicial complexes that preserves the Cohen-Macaulay property. An example of this operation puts the connection between the central Delannoy numbers and Legendre polynomials in a wider context.
Keywords: balanced simplicial complex, Delannoy numbers, Cohen-Macaulay property

References:

1. C. Banderier and S. Schwer, Why Delannoy numbers?, J. Statist. Plann. Inference 135 (1) (2005) 40-54.

2. A. Björner, P. Frankl, and R. Stanley, The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem, Combinatorica 7 (1) (1987) 23-34.

3. A. Björner and M. Wachs, Bruhat order of Coxeter groups and shellability, Adv. Math. 43 (1982) 87-100.

4. A. Björner and M. Wachs, On lexicographically shellable posets, Trans. Amer. Math. Soc. 277 (1983) 323-341.

5. T.S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its Applications 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978.

6. L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974.

7. H. Delannoy, Employ d'échiquier pour la résolution de certains problemes de probabilités, Assoc. Franc. Bordeaux 24 (1895) 70-90.

8. I.J. Good, Legendre polynomials and trinomial random walks, Proc. Cambridge Philos. Soc. 54 (1958) 39-42.

9. G. Hetyei, Orthogonal polynomials represented by CW-spheres, Electron. J. Combin. 11 (2) (2004) #R4, 28 pp.

10. D.F. Lawden, On the solution of a linear difference equation, Math. Gaz. 36 (1952) 193-196.

11. L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Math. 26 (1963) 223-229.

12. M. Skandera, Dumont's statistic on words, Electron. J. Combin. 8 (2001) #R11.

13. N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/.

14. R.P. Stanley, Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc. 249 (1) (1979) 139-157.

15. R.P. Stanley, Combinatorics and Commutative Algebra, Second Edition, Progress in Mathematics 41, Birkhäuser, Boston, 1996.

16. R.P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge University Press, Cambridge, 1999.

17. R.A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (1) (2003) 19 pp.

18. H.S.Wilf, generatingfunctionology, Second Edition, Academic Press, Boston, MA, 1994.