7 References

An exact formula for the move-to-front rule for self-organizing lists. (19) where (m) = jfT 2 B n : m 2 m (T)gj. It is easily seen that (m) depends only on the unordered set m ] and equals (m) = m Y i=0 g i ((m]) : (20) Therefore (19) equals n X m=jRj (?1) m?jRj X UR jUj=m (U) = X UR (?1) jUj?jRj (U): (21) But (U) is, by (20), precisely the number of trees that x at least the points in U. By MM obius inversion, (21) reduces to n (R). Remarks: 1. For n 2, the second largest eigenvalue is the sum which leaves out the consecutive pair fi;i+1g with the smallest total weight. Its multiplicity (assuming no ties) is 2 = 1. 2. As in the case of MTF, when the weights are uniform the eigenvalues of MTR are the numbers The multiplicity of the eigenvalue m=n is the number of trees which x exactly m points. 22 short, n is the number of admissible closeness relations (in Fine's terminology) on n]. Fine gave a method of calculating n but did not produce an explicit formula like our (15). 3. It is easy to show that n satisses the following recursive relationship with respect to n : (18) We feel that the simplest method for calculating n is to calculate n iteratively and then use (17) iteratively to get n. 4. Combining (17) and (18) gives a simple recurrence relation satissed by (n): 2(n + 1) n = (7n ? 5) n?1 + 2(2n ? 1) n?2 ; n 2: We now give a tree-based description of the spectral structure of MTR.