[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.b150404

COMBINATORIAL ENUMERATION OF THE REGIONS OF SOME LINEAR ARRANGEMENTS

Seo, Seunghyun (Department of Mathematics Education Kangwon National University)

Publication Information

Bulletin of the Korean Mathematical Society / v.53, no.5, 2016 , pp. 1281-1289 More about this Journal

Abstract

Richard Stanley suggested the problem of finding combinatorial proofs of formulas for counting regions of certain hyperplane arrangements defined by hyperplanes of the form $x_i=0$ , $x_i=x_j$ , and $x_i=2x_j$ that were found using the finite field method. We give such proofs, using embroidered permutations and linear extensions of posets.

Keywords

hyperplane arrangement; embroidered permutation; linear extensions of poset;

Citations & Related Records

Reference

1	F. Ardila, Computing the Tutte polynomial of a hyperplane arrangement, Pacific J. Math. 230 (2007), no. 1, 1-26. DOI
2	C. A. Athanasiadis, Characteristic polynomials of subspace arrangements and finite fields, Adv. Math. 122 (1996), no. 2, 193-233. DOI
3	H. H. Crapo and G.-C. Rota, On the foundations of combinatorial theory: Combinatorial geometries, The M.I.T. Press, Cambridge, Mass.-London, preliminary edition, 1970.
4	S.-P. Eu, T.-S. Fu, and C.-J. Lai, On the enumeration of parking functions by leading terms, Adv. in Appl. Math. 35 (2005), no. 4, 392-406. DOI
5	J. P. S. Kung and C. Yan, Goncarov polynomials and parking functions, J. Combin. Theory Ser. A 102 (2003), no. 1, 16-37. DOI
6	P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin, 1992.
7	A. Postnikov and R. P. Stanley, Deformations of Coxeter hyperplane arrangements, J. Combin. Theory Ser. A 91 (2000), no. 1-2, 544-597. DOI
8	J. Riordan, Combinatorial identities, Robert E. Krieger Publishing Co., Huntington, N.Y., 1979.
9	R. P. Stanley, Enumerative combinatorics. Vol. I, TheWadsworth & Brooks/Cole Math- ematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986.
10	R. P. Stanley, An introduction to hyperplane arrangements, In Geometric combinatorics, volume 13 of IAS/Park City Math. Ser., pages 389-496. Amer. Math. Soc., Providence, RI, 2007.
11	C. H. Yan, Generalized parking functions, tree inversions, and multicolored graphs, Adv. in Appl. Math. 27 (2001), no. 2-3, 641-670. DOI
12	T. Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), issue 1, no. 154, vii+102 pp.