Generate the universe of rankings given the input partition
A ranking, an ordering, a matrix of rankings, a matrix of orderings or a number
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
The universe of rankings must be returned as orderings (default) or rankings?
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.
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|
Antonio D'Ambrosio firstname.lastname@example.org
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.
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
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