univranks | R Documentation |

Generate the universe of rankings given the input partition

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

`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? |

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 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.

```
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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