Tutorial: Enumerated sets

Exercice: Poker and probability

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

{{{id=0| Suits = Set(["spades", "hearts", "diamonds", "clubs"]) /// }}}

Define similarly the set Ranks:

{{{id=1| # 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:

{{{id=2| # edit here /// }}}

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

{{{id=3| # edit here /// }}}

{{{id=4| # edit here /// }}}

{{{id=5| # edit here /// }}}

Draw a card at random:

{{{id=6| # 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:

{{{id=7| Cards = CartesianProduct(Suits, Ranks).map(tuple) /// }}}

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

{{{id=8| # edit here /// }}}

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

{{{id=9| # 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 (hint: use Partitions with max_slope):

   {{{id=10| # edit here /// }}}

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

   {{{id=11| # edit here /// }}}

   You can also use a cartesian product:

   {{{id=12| # edit here /// }}}

3. List all the Dyck words of length 6:

   {{{id=13| # edit here /// }}}

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

{{{id=14| 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 to (hint: use DisjointUnionEnumeratedSets):

   {{{id=15| # edit here /// }}}