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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Standard paths in another composition poset. | Standard paths in another composition poset Jan Snellman Department of Mathematics Stockholm University SE-10691 Stockholm Sweden Jan.Snellman@math.su.se Submitted Oct 8 2003 Accepted Oct 17 2004 Published Oct 26 2004 Abstract Bergeron Bousquet-Melou and Dulucq 1 enumerated paths in the Hasse diagram of the following poset the underlying set is that of all compositions and a composition 1 covers another composition A if 1 can be obtained from A by adding 1 to one of the parts of A or by inserting a part of size 1 into A. We employ the methods they developed in order to study the same problem for the following poset which is of interest because of its relation to non-commutative term orders the underlying set is the same but 1 covers A if 1 can be obtained from A by adding 1 to one of the parts of A or by inserting a part of size 1 at the left or at the right of A. We calculate generating functions for standard paths of fixed width and for standard paths of height 2. 1 Definition of standard paths By a composition P we mean a sequence of positive integers pi p2 pk which are the parts of P. We define the length h P of P as the number of parts and the weight PI Pi pk as the sum of its parts. If P has weight n then P is a composition of n and we write P 1 n. We say that a composition Q covers a composition P if Q is obtained from P either by adding 1 to a part of P or by inserting a part of size 1 to the left or by inserting a part of size 1 to the right. Thus P p1 p2 pk is covered by 1 1 pi .ipk 2. p 1 3. and for 1 i k pi pi 1 pk . Extending this relation by transitivity makes the set of all compositions into a partially ordered set which we denote by N. This is in accordance with the notations in the author s THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R76 1 article A poset classifying non-commutative term orders 3 where N was used for the following isomorphic poset of words the underlying set is X the free associative monoid on X xi x2 x3 . and mi xi xir is .
