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Tutorial: Enumerated sets

Exercice: Poker and probability

A poker card is characterized by a suit (“spades”, “hearts”, “diamonds”, “clubs”) and a rank (\(2, 3, \dots, 9\), “Jack”, “Queen”, “King” and “Ace”). Here is the way to define suits in sage:

sage: Suits = Set(["spades", "hearts", "diamonds", "clubs"])

Define similarly the set Ranks:

sage: # edit here

The deck of card is the cartesian product of the set of suits by the set of ranks. Define a set Cards accordingly:

sage: # edit here

Use the method .cardinality() to compute the number of suits, ranks and cards:

sage: # edit here

sage: # edit here

sage: # edit here

Draw a card at random:

sage: # edit here

Cards are (currently) returned as lists. To be able to build a set of cards, we need them to be hashable. Let’s redefine the set of cards by transforming cards to tuples:

sage: Cards = CartesianProduct(Suits, Ranks).map(tuple)

Use Subsets to draw a hand of five cards at random:

sage: # edit here

Use .cardinality() to compute the number of hands, check the result with binomial:

sage: # edit here

To go further, see exercises 38, 39, 40 in Calcul Mathématique avec Sage (version 1.0) page 255.

Using existing Enumerated Sets

1. List all the strict partitions of \(5\) (hint: use Partitions with max_slope):

   sage: # edit here
   

2. List all the vectors of 0 and 1 of length 5 (hint: use IntegerVectors with max_part):

   sage: # edit here
   

   You can also use a cartesian product:

   sage: # edit here
   

3. List all the Dyck words of length 6:

   sage: # edit here
   

Here is the way to print the standard tableaux of size $4$:

sage: for t in StandardTableaux(3): t.pp(); print
1  2  3

1  3
2

1  2
3

1
2
3

1. Define the set of all the partitions of \(1\) to \(5\) (hint: use DisjointUnionEnumeratedSets):

   sage: # edit here