tailieunhanh - Báo cáo toán học: "Combinatorics of Partial Derivatives"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Combinatorics of Partial Derivatives. | Combinatorics of Partial Derivatives Michael Hardy School of Mathematics University of Minnesota Minneapolis MN 55455 USA hardy@ Submitted Oct 1 2005 Accepted Dec 24 2005 Published Jan 7 2006 Mathematics Subject Classifications 05A15 05A18 11B73 05-02 Abstract The natural forms of the Leibniz rule for the kth derivative of a product and of Faà di Bruno s formula for the kth derivative of a composition involve the differential operator dk dx-1 dxk rather than dk dxk with no assumptions about whether the variables x1 . Xk are all distinct or all identical or partitioned into several distinguishable classes of indistinguishable variables. Coefficients appearing in forms of these identities in which some variables are indistinguishable are just multiplicities of indistinguishable terms in particular if all variables are distinct then all coefficients are 1 . The computation of the multiplicities in this generalization of Faàa di Bruno s formula is a combinatorial enumeration problem that although completely elementary seems to have been neglected. We apply the results to cumulants of probability distributions. 1 Introduction Both the well-known Leibniz rule dk k a d u dxk ịt W dx 0 v 7 dk - v dxk 1 and the celebrated formula of Francesco Faa di Bruno XIf y E Jf 1 m y n dj dxk 2-- I mi k mk m1 mk . L dxmi j mj 0 2 where the sum is over all k-tuples m1 . mk of non-negative integers satisfying the constraint m1 2m2 3m3 kmk k are formulas for kth derivatives of functions of functions of x. That is what the left sides of these identities share in common. The THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R1 1 right sides of both identities are sums whose terms have products of higher derivatives with respect to x as factors. All mathematicians know the combinatorial interpretation of the coefficients in the Leibniz rule the number of size- subsets of a size-fe set and all combinatorialists know the combinatorial interpretation of the coefficients in Faa di .