tailieunhanh - Báo cáo toán học: "COMPOSITION SUM IDENTITIES RELATED TO THE DISTRIBUTION OF COORDINATE VALUES IN A DISCRETE SIMPLEX"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: COMPOSITION SUM IDENTITIES RELATED TO THE DISTRIBUTION OF COORDINATE VALUES IN A DISCRETE SIMPLEX. | COMPOSITION SUM IDENTITIES RELATED TO THE DISTRIBUTION OF COORDINATE VALUES IN A DISCRETE SIMPLEX. R. MILSON DEPT. MATHEMATICS STATISTICS DALHOUSIE UNIVERSITY HALIFAX . B3H 3J5 CANADA MILSON@ Submitted March 27 2000 Accepted April 13 2000. AMS Subject Classifications 05A19 05A20. Abstract. Utilizing spectral residues of parameterized recursively defined sequences we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first and second order ordinary linear differential equations. Regarding the first class the corresponding identities amount to a proof of the exponential formula of labelled counting. The identities in the second class can be used to establish certain geometric properties of the simplex of bounded ordered integer tuples. We present three theorems that support the conclusion that the inner dimensions of such an order simplex are in a certain sense more ample than the outer dimensions. As well we give an algebraic proof of a bijection between two families of subsets in the order simplex and inquire as to the possibility of establishing this bijection by combinatorial rather than by algebraic methods. 1. Introduction The present paper is a discussion of composition sum identities that may be obtained by utilizing spectral residues of parameterized recursively defined sequences. Here we are using the term composition sum to refer to a sum whose index runs over all ordered lists of positive integers P1 P2 . . . pi that such that for a fixed n P1 . Pl n. Spectral residues will be discussed in detail below. Compositions sums are a useful device and composition sum identities are frequently encountered in combinatorics. For example the Stirling numbers of both kinds have a This research supported by a Dalhousie University grant. 1 2 THE ELECTRONIC JOURNAL OF COMBINATORICS 7 2000 R20 natural representation by means of such sums 4 51 60 8l n