TAILIEUCHUNG - Báo cáo toán học: " Permutations which are the union of an increasing and a decreasing subsequence"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Permutations which are the union of an increasing and a decreasing subsequence. | Permutations which are the union of an increasing and a decreasing subsequence . Atkinson School of Mathematical and Computational Sciences North Haugh St Andrews Fife KY16 9SS UK mda@ Abstract It is shown that there are I2 pn Lo 2n m 1I2 permutations n 0 which are the union of an increasing sequence and a decreasing sequence. 1991 Mathematics Subject Classification 05A15 Submitted December 1 1997 Accepted January 10 1998 1 Introduction Merge permutations permutations which are the union of two increasing subsequences have been studied for many years 3 . It is known that they are characterised by the property of having no decreasing subsequence of length 3 and that there are 2 J n 1 such permutations of length n. Recently there has been some interest in permutations which are the union of an increasing subsequence with a decreasing subsequence. We call such permutations skew-merged. Stankova 4 proved that a permutation is skew-merged if and only if it has no subsequence abcd ordered in the same way as 2143 or 3412. In 1 2 the more general problem of partitioning a permutation into given numbers of increasing and decreasing subsequences is considered. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 5 1998 R6 2 y x Figure 1 The graph of a skew-merged permutation This paper solves the enumeration problem for skew-merged permutations. The proof yields another proof of Stankova s result. Finally a corollary allows the skew-merged enumeration result to be compared with the enumeration of merge permutations. 2 Points and colours We shall consider sets of points a b in the T -plane. Our point sets will always have the property that there is no duplicated first coordinate or second coordinate. In view of that condition there are two natural total orders on points. If P a b and Q c d are points then we define if a c and P Q if b d. Two sets of points S T are said to be order isomorphic if there is a one-to one correspondence between them which respects both

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