Preferential Arrangements

Combinatorics of counting finite fuzzy subsets

“…Denote by J n = n k=1 k!S(n, k). Then J n is the number of ordered partitions of an n-element set and has been the subject of many investigations mentioned in the Introduction, for instance [1,5,2,6]. The following lemma gives a connection between F (n) and J n .…”

“…Since permutations are involved in the determination of the number of fuzzy subsets the generating function is an exponential function. A generating function for J n is given by f (x) = 1/(2 − e x ), see for instance [2,6]. We derive a generating function for F (n) = 4J n − 1 using f (x).…”

“…Also seemingly he was aware of their combinatorial properties. Later the same idea appeared as preferential arrangements based on choice of indifference in Gross [2]; that is, each object is to be placed in exactly one indifference set with the sets themselves totally ranked in order of strict transitive preference. Thus, in some sense preferential arrangements can be thought of as a pre-cursor to Table 1 The number of preferential fuzzy subsets of X n n 0 1 2 3 4 5 6 7 8 9 F (n) 1 3 11 51 299 2163 18 731 189 171 2 183 339 28 349 043 Zadeh's fuzzy sets [13] in a different context (as far as combinatorics is concerned).…”

A Recursive Partitioning Method for the Prediction of Preference Rankings Based Upon Kemeny Distances

“…As τ X is the rank correlation coefficient associated to the Kemeny distance in a one-to-one manner, Emond and Mason defined the consensus ranking as that ranking (or those rankings) that maximizes the weighted average τ X rank correlation coefficients between the candidate to be the consensus ranking and all other rankings, where the weights are given positive numbers expressing the relative importance of judge k. In other words, the consensus ranking is the ranking for which the (weighted) average of the τ X rank correlation coefficient between itself and all the rankings stated by a set of judges is maximum with respect to the same measure computed between any other ranking belonging to the universe of rankings and all the rankings stated by a set of judges. Finding the consensus ranking is known to be a NP-hard problem (Gross, 1962). In fact, the number of orderings of n objects is closely approximated by:…”

Voting power when using preference ballots

“…The kth element of this sequence, a(k), is given by i = l k! times the coefficient of x k in (2 -e x)-1; [5] k ~ r!S(k,r), [6] where S(k, r) are the Stifling numbers of the second kind; or a number of other elegant formulas (see [8,10]). …”