univranks: Generate the universe of rankings In ConsRank: Compute the Median Ranking(s) According to the Kemeny's Axiomatic Approach

Description

Generate the universe of rankings given the input partition

Usage

 `1` ```univranks(X, k = NULL, ordering = TRUE) ```

Arguments

 `X` A ranking, an ordering, a matrix of rankings, a matrix of orderings or a number `k` Optional: the number of the non-empty subsets. It has to be used only if X is anumber. The default value is NULL, In this case the universe of rankings with n=X items are computed `ordering` The universe of rankings must be returned as orderings (default) or rankings?

Details

The function should be used with small numbers because it can generate a large number of permutations. The use of X greater than 9, of X matrices with more than 9 columns as input is not reccomended.

Value

a "list" containing the following components:

 Runiv The universe of rankings Cuniv A list containing: R The universe of rankings in terms of rankings; Parts for each ranking in input the produced rankings Univinbuckets the universe of rankings within each bucket

Author(s)

Antonio D'Ambrosio antdambr@unina.it

`stirling2` Stirling number of second kind.

`rank2order` Convert rankings into orderings.

`order2rank` Convert orderings into ranks.

`partitions` Generate partitions of n items constrained into k non empty subsets.

Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```S2<-stirling2(4,4)\$SM[4,] #indicates in how many ways 4 objects #can be placed, respectively, into 1, 2, #3 or 4 non-empty subsets. CardConstr<-factorial(c(1,2,3,4))*S2 #the cardinality of rankings #constrained into 1, 2, 3 and 4 #buckets Card<-sum(CardConstr) #Cardinality of the universe of rankings with 4 #objects U<-univranks(4)\$Runiv #the universe of rankings with four objects # we know that the universe counts 75 #different rankings Uk<-univranks(4,2)\$Runiv #the universe of rankings of four objects #constrained into k=2 buckets, we know they are 14 Up<-univranks(c(1,4,3,1))\$Runiv #the universe of rankings with 4 objects #for which the first and the fourth item #are tied ```

ConsRank documentation built on Sept. 28, 2021, 5:07 p.m.