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 .
