J un 2 01 7 Linear Extensions and Comparable Pairs in Partial Orders
نویسندگان
چکیده
We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also show that a random interval partial order on n elements has close to a third of the pairs comparable with high probability, and the number of linear extensions is n! 2−Θ(n) with high probability.
منابع مشابه
The Average Number of Linear Extensions of a Partial Order
Kleitman and Rothschild (Trans. Amer. Math. Soc. 205 (1975), 205-220) gave an asymptotic formula for the number of partial orders with ground-set In]. We give a shorter proof of their result and extend it to count the number of pairs (P, -<), where P is a partial order on [hi and -< is a linear extension of P. This gives us an asymptotic formula for (a) the average number of linear extensions o...
متن کاملLinear Extension Diameter of Downset Lattices of 2-Dimensional Posets
The linear extension diameter of a finite poset P is the maximum distance between a pair of linear extensions of P, where the distance between two linear extensions is the number of pairs of elements of P appearing in different orders in the two linear extensions. We prove a formula for the linear extension diameter of the Boolean Lattice and characterize the diametral pairs of linear extension...
متن کاملA Ramsey theorem for partial orders with linear extensions
We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Sokić. As a bonus, our proof gives new arguments for these two results.
متن کاملON RELATIVE CENTRAL EXTENSIONS AND COVERING PAIRS
Let (G;N) be a pair of groups. In this article, first we con-struct a relative central extension for the pair (G;N) such that specialtypes of covering pair of (G;N) are homomorphic image of it. Second, weshow that every perfect pair admits at least one covering pair. Finally,among extending some properties of perfect groups to perfect pairs, wecharacterize covering pairs of a perfect pair (G;N)...
متن کاملBalanced pairs in partial orders
An-balanced pair in a partially ordered set P = (X; <) is a pair (x; y) of elements of X such that the proportion of linear extensions of P with x below y lies between and 1 ?. The 1=3 ? 2=3 Conjecture states that, in every nite partial order P , not a chain, there is a 1 3-balanced pair. This was rst conjectured in a 1968 paper of Kislitsyn, and remains unsolved. We survey progress towards a r...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2017