tailieunhanh - Báo cáo toán học: "Updown numbers and the initial monomials of the slope variety"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Updown numbers and the initial monomials of the slope variety. | Updown numbers and the initial monomials of the slope variety Jeremy L. Martin Department of Mathematics University of Kansas Lawrence KS 66047 USA jmartin@ Jennifer D. Wagner Department of Mathematics and Statistics Washburn University Topeka KS 66621 USA j Submitted May 28 2009 Accepted Jun 28 2009 Published Jul 9 2009 Mathematics Subject Classifications 05A15 14N20 Abstract Let In be the ideal of all algebraic relations on the slopes of the n lines formed by placing n points in a plane and connecting each pair of points with a line. Under each of two natural term orders the ideal of In is generated by monomials corresponding to permutations satisfying a certain pattern-avoidance condition. We show bijectively that these permutations are enumerated by the updown or Euler numbers thereby obtaining a formula for the number of generators of the initial ideal of In in each degree. The symbol N will denote the set of positive integers. For integers m n we put n 1 2 . n and m n m m 1 . n . The set of all permutations of an integer set P will be denoted p and the nth symmetric group is Sn S n . We will write each permutation w G Sp as a word with n P digits w w1. .wn where w1 . wn P. If necessary for clarity we will separate the digits with commas. Concatenation will also be denoted with commas for instance if w 12 and w 34 then w w 5 12345. The reversal w of w1w2 . wn-1wn is the word wnwn-1. w2w1. A subword of a permutation w G Sp is a word w i j wiwi 1 wj where i j G n . The subword is proper if w i j w. We write w w if the digits of w are in the same relative order as those of w for instance 58462 35241. Definition 1. Let P c N with n P 2. A permutation w G Sp is a G-word if it satisfies the two conditions G1 w1 max P and wn max P w1 and Partially supported by an NSA Young Investigator s Grant THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R82 1 G2 If n 4 then w2 wn-1. It is an R-word if it satisfies the two conditions R1 wi .