This Sage quickstart tutorial was developed for the MAA PREP Workshop “Sage: Using Open-Source Mathematics Software with Undergraduates” (funding provided by NSF DUE 0817071). It is licensed under the Creative Commons Attribution-ShareAlike 3.0 license (CC BY-SA).

As computers are discrete and finite, topics from discrete mathematics are natural to implement and use. We’ll start with Graph Theory.

The pre-defined `graphs` object provides an abundance of examples.
Just tab to see!

```
sage: graphs.[tab]
```

Its companion `digraphs` has many built-in examples as well.

Visualizing a graph is similar to plotting functions.

```
sage: G = graphs.HeawoodGraph()
sage: plot(G)
Graphics object consisting of 36 graphics primitives
```

Defining your own graph is easy. One way is the following.

- Put a vertex next to a list (recall this concept from the programming
tutorial) with a colon, to show its adjacent vertices. For example,
to put vertex 4 next to vertices 0 and 2, use
`4:[0,2]`. - Now combine all these in curly braces (in the advanced appendix to the
programming tutorial, this is called a
*dictionary*).

```
sage: H=Graph({0:[1,2,3], 4:[0,2], 6:[1,2,3,4,5]})
sage: plot(H)
Graphics object consisting of 18 graphics primitives
```

Adjacency matrices, other graphs, and similar inputs are also recognized.

Graphs have “position” information for location of vertices. There are several different ways to compute a layout, or you can compute your own. Pre-defined graphs often come with “nice” layouts.

```
sage: H.set_pos(H.layout_circular())
sage: plot(H)
Graphics object consisting of 18 graphics primitives
```

Vertices can be lots of things, for example the codewords of an error-correcting code.

Note

Technical caveat: they need to be “immutable”, like Python’s tuples.

Here we have a matrix over the integers and a matrix of variables as vertices.

```
sage: a=matrix([[1,2],[3,4]])
sage: var('x y z w')
(x, y, z, w)
sage: b=matrix([[x,y],[z,w]])
sage: a.set_immutable()
sage: b.set_immutable()
sage: K=DiGraph({a:[b]})
sage: show(K, vertex_size=800)
```

Edges can be labeled.

```
sage: L=graphs.CycleGraph(5)
sage: for edge in L.edges():
....: u = edge[0]
....: v = edge[1]
....: L.set_edge_label(u, v, u*v)
sage: plot(L, edge_labels=True)
Graphics object consisting of 16 graphics primitives
```

There are natural connections to other areas of mathematics. Here we compute the automorphism group and eigenvalues of the skeleton of a cube.

```
sage: C = graphs.CubeGraph(3)
sage: plot(C)
Graphics object consisting of 21 graphics primitives
```

```
sage: Aut=C.automorphism_group()
sage: print "Order of automorphism group: ", Aut.order()
Order of automorphism group: 48
sage: print "Group: \n", Aut
Group:
Permutation Group with generators [('010','100')('011','101'), ('001','010')('101','110'), ('000','001')('010','011')('100','101')('110','111')]
```

```
sage: C.spectrum()
[3, 1, 1, 1, -1, -1, -1, -3]
```

There is a huge amount of LaTeX support for graphs. The following graphic shows an example of what can be done; this is the Heawood graph.

Press ‘tab’ at the next command to see all the available options.

```
sage: sage.graphs.graph_latex.GraphLatex.set_option?
```

Discrete mathematics is a broad area, and Sage has excellent support for much of it. This is largely due to the “sage-combinat” group. These developers previously developed for MuPad (as “mupad-combinat”) but switched over to Sage shortly before MuPad was sold.

Sage can work with basic combinatorial structures like combinations and permutations.

```
sage: pets = ['dog', 'cat', 'snake', 'spider']
sage: C=Combinations(pets)
sage: C.list()
[[], ['dog'], ['cat'], ['snake'], ['spider'], ['dog', 'cat'], ['dog', 'snake'], ['dog', 'spider'], ['cat', 'snake'], ['cat', 'spider'], ['snake', 'spider'], ['dog', 'cat', 'snake'], ['dog', 'cat', 'spider'], ['dog', 'snake', 'spider'], ['cat', 'snake', 'spider'], ['dog', 'cat', 'snake', 'spider']]
```

```
sage: for pair in Combinations(pets, 2):
....: print "The " + pair[0] + " chases the " + pair[1] + "."
The dog chases the cat.
The dog chases the snake.
The dog chases the spider.
The cat chases the snake.
The cat chases the spider.
The snake chases the spider.
```

```
sage: for pair in Permutations(pets, 2):
....: print pair
['dog', 'cat']
['dog', 'snake']
['dog', 'spider']
['cat', 'dog']
['cat', 'snake']
['cat', 'spider']
['snake', 'dog']
['snake', 'cat']
['snake', 'spider']
['spider', 'dog']
['spider', 'cat']
['spider', 'snake']
```

Of course, we often want these for numbers, and these are present as well. Some are familiar:

```
sage: Permutations(5).cardinality()
120
```

Others somewhat less so:

```
sage: D = Derangements([1,1,2,2,3,4,5])
sage: D.list()[:5]
[[2, 2, 1, 1, 4, 5, 3], [2, 2, 1, 1, 5, 3, 4], [2, 2, 1, 3, 1, 5, 4], [2, 2, 1, 3, 4, 5, 1], [2, 2, 1, 3, 5, 1, 4]]
```

And some somewhat more advanced – in this case, symmetric polynomials.

```
sage: s = SymmetricFunctions(QQ).schur()
sage: a = s([2,1])
sage: a.expand(3)
x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2
```

Various functions related to this are available as well.

```
sage: binomial(25,3)
2300
```

```
sage: multinomial(24,3,5)
589024800
```

```
sage: falling_factorial(10,4)
5040
```

Do you recognize this famous identity?

```
sage: var('k,n')
(k, n)
sage: sum(binomial(n,k),k,0,n)
2^n
```

This is also briefly mentioned in the *Number theory quickstart*. Sage has a number of good pedagogical resources
for cryptography.

```
sage: # Two objects to make/use encryption scheme
sage: #
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: bin = BinaryStrings()
sage: #
sage: # Convert English to binary
sage: #
sage: P = bin.encoding("Encrypt this using S-DES!")
sage: print "Binary plaintext: ", P, "\n"
sage: #
sage: # Choose a random key
sage: #
sage: K = sdes.list_to_string(sdes.random_key())
sage: print "Random key: ", K, "\n"
sage: #
sage: # Encrypt with Simplified DES
sage: #
sage: C = sdes(P, K, algorithm="encrypt")
sage: print "Encrypted: ", C, "\n"
sage: #
sage: # Decrypt for the round-trip
sage: #
sage: plaintxt = sdes(C, K, algorithm="decrypt")
sage: print "Decrypted: ", plaintxt, "\n"
sage: #
sage: # Verify easily
sage: #
sage: print "Verify encryption/decryption: ", P == plaintxt
Binary plaintext: 01000101011011100110001101110010011110010111000001110100001000000111010001101000011010010111001100100000011101010111001101101001011011100110011100100000010100110010110101000100010001010101001100100001
Random key: 0100000011
Encrypted: 00100001100001010011000111000110010000011011101011111011100011011111101111110111110010101000010010001101101010101000010011001010100001010111000010001101000011001001111111110100001000010000110001011000
Decrypted: 01000101011011100110001101110010011110010111000001110100001000000111010001101000011010010111001100100000011101010111001101101001011011100110011100100000010100110010110101000100010001010101001100100001
Verify encryption/decryption: True
```

Here is a brief example of a linear binary code (group code).

Start with a generator matrix over \(\ZZ/2\ZZ\).

```
sage: G = matrix(GF(2), [[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
```

```
sage: C.is_self_dual()
False
```

```
sage: D = C.dual_code()
sage: D
Linear code of length 7, dimension 3 over Finite Field of size 2
```

```
sage: D.basis()
[(1, 0, 1, 0, 1, 0, 1), (0, 1, 1, 0, 0, 1, 1), (0, 0, 0, 1, 1, 1, 1)]
```

```
sage: D.permutation_automorphism_group()
Permutation Group with generators [(4,5)(6,7), (4,6)(5,7), (2,3)(6,7), (2,4)(3,5), (1,2)(5,6)]
```