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Combinatorial enumeration of the regions of some linear arrangements

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대한수학회 (The Korean Mathematical Society)|2016.09|대한수학회보 (Bulletin of the KMS)|제53권 제5호|pp. 1281-1289

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참고문헌

키워드

hyperplane arrangementembroidered permutationlinear extensions of poset
주제분류수학

초록

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.

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.

참고문헌 (12)

  1. F. Ardila, "Computing the Tutte polynomial of a hyperplane arrangement", Pacific J. Math, 230(1), pp.1~26 (2007)
  2. C. A. Athanasiadis, "Characteristic polynomials of subspace arrangements and finite fields", Adv. Math, 122(2), pp.193~233 (1996)
  3. H. H. Crapo, "On the foundations of combinatorial theory: Combinatorial geometries" (1970), The M.I.T. Press
  4. S.-P. Eu, "On the enumeration of parking functions by leading terms", Adv. in Appl. Math, 35(4), pp.392~406 (2005)
  5. J. P. S. Kung, "Goncarov polynomials and parking functions", J. Combin. Theory Ser. A, 102(1), pp.16~37 (2003)
  6. P. Orlik, "Arrangements of Hyperplanes" (1992), Springer-Verlag
  7. A. Postnikov, "Deformations of Coxeter hyperplane arrangements", J. Combin. Theory Ser. A, 91(1-2), pp.544~597 (2000)
  8. J. Riordan, "Combinatorial identities" (1979), Robert E. Krieger Publishing Co.
  9. R. P. Stanley, "Enumerative combinatorics. Vol. I", pp.1986, Wadsworth & Brooks
  10. R. P. Stanley, "Geometric combinatorics", pp.389~496 (2007), Amer. Math. Soc.
  11. C. H. Yan, "Generalized parking functions, tree inversions, and multicolored graphs", Adv. in Appl. Math, 27(2-3), pp.641~670 (2001)
  12. T. Zaslavsky, "Facing up to arrangements: face-count formulas for partitions of space by hyperplanes", Mem. Amer. Math. Soc, 1(1)통권 : 154, pp.7~ (1975)

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