Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Counting Simplexes in R 3. | Counting Simplexes in R3 Claude Laflamme Department of Mathematics and Statistics University of Calgary Calgary Alberta Canada T2N 1N4 email laf@ István Szalkaiy Department of Mathematics and Computer Science University of Veszprám H-8201 Veszprám Hungary email szalkai@ szalkai@ Submitted September 26 1997 Accepted January 22 1998. AMS Subject Classification 05D99 05B35. Abstract A finite set of vectors S c ift is called a simplex iff S is linearly dependent but all its proper subsets are independent. This concept arises in particular from stoichiometry. We are interested in this paper in the number of simplexes contained in some H c ift which we denote by simp H . This investigation is particularly interesting for H spanning K and containing no collinear vectors. Our main result shows that for any H c K3 of fixed size not equal to 3 4 or 7 and such that H spans K3 and contains no collinear vectors simp H is minimal if and only if H is contained in two planes intersecting in H and one of which is of size exactly 3. The minimal configurations for H 3 4 7 are also completely described. The general problem for K remains open. The research of the first author was partially supported by NSERC of Canada. yThe research of the second author was partially supported by the Fund Peregrinatio I of MOL Co. Hungary Grant no. 3 1994. 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 5 1998 R40 1 Introduction 2 Simplexes are used for example in stoichiometry when hnding minimal reactions and mechanisms or for hnding dimensionless groups in dimensional analysis see 3 . To explain the notion of minimal reaction let the chemical species A1 A2 . An consist of elements E1 E2 . Em as Aj J2 1 ai j Ei ai j e N for j 1 2 . n. Writing Aj for the vector a1 j a2 j . amj T we know that there might exists a chemical reaction between the species Aj j e Sg for any S c 1 2 . ng if and only if the homogeneous linear equation X X Aj 0 1 j2S has a non
