| We study additive representability of orders on multisets (of size $k$ drawn from a set of size $n$) which satisfy the condition of Independence of Equal Submultisets (IES) introduced by Sertel and Slinko (2002). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete cones. Following Fishburn (1996) and Conder and Slinko (2003), we define functions $f(n,k)$ and $g(n,k)$ which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable order satisfying the IES, respectively, from a representable order. We prove that $g(n,k)=n-1$ whenever $nge 3$ and $(n,k)ne (5,2)$. In the exceptional case, $g(5,2)=3$. We also prove that $g(n,k)le f(n,k)le n$ and establish that for small $n$ and $k$ the functions $g(n,k)$ and $f(n,k)$ coincide. |
Keywords
multiset, linear orders, Independence of Equal Submultisets, additive representability
Math Review Classification
Primary 91B16
Last Updated
30 April 2004
Length
24 pp
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