Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài:q-Exponential Families. | -Exponential Families Kent E. Morrison Department of Mathematics California Polytechnic State University San Luis Obispo CA 93407 kmorriso@ Submitted Mar 17 2004 Accepted May 28 2004 Published Jun 11 2004 MR Subject Classifications 05A15 05A30 Abstract We develop an analog of the exponential families of Wilf in which the label sets are finite dimensional vector spaces over a finite field rather than finite sets of positive integers. The essential features of exponential families are preserved including the exponential formula relating the deck enumerator and the hand enumerator. 1 Introduction and Definitions In this paper we analogize Wilf s approach to labelled counting in 9 based on exponential families. Notation and definitions follow his as closely as possible. This work is further elaboration of the subset-subspace analogy that has long been a rich source in enumerative combinatorics. See Kung 4 for an historical survey. Let Fg the finite field of order q and FgN the vector space of countable dimension over Fg whose elements are infinite sequences 1 a2 . with a finite number of non-zero entries. Let e1 e2 . be the standard basis and define En to be the span of e1 . en. Let P be an abstract set of pictures. Definition 1 A q-card Q V p is a pair consisting of a subspace V c Fg N and a picture p E P. V is the label space space and the dimension of Q is dim V. A card is standard if its label space is En. Definition 2 A q-hand is a finite set of q-cards whose label spaces form a direct sum decomposition of En for some n. The dimension of the hand is n. Definition 3 A q-card Q V p is a relabeling of Q V p if dim V dim V. Thus the pictures must be the same. The standard relabeling of Q V p is Q EdimV p . THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R36 1 Definition 4 A q-deck D is a finite set of standard q-cards whose dimensions are the same and whose pictures are different. The dimension of the deck dim D is the common dimension of the cards. .
