This document is one of More SageMath Tutorials. You may edit it on github. \(\def\NN{\mathbb{N}}\) \(\def\ZZ{\mathbb{Z}}\) \(\def\QQ{\mathbb{Q}}\) \(\def\RR{\mathbb{R}}\) \(\def\CC{\mathbb{C}}\)

# 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¶

List all the strict partitions of \(5\) (hint: use

`Partitions`

with`max_slope`

):sage: # edit here

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

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
```

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

`DisjointUnionEnumeratedSets`

):sage: # edit here