︠8c18a7b9-ae80-471d-978c-eedeec1be08fs︠
### Data types

a = 3; # an integer
b = "3.141"; # a string
c = True; # boolean value
d = (3,1,4,1); # a tuple (not mutable)
e = [3,1,4,1]; # a list (mutable)
f = {3,1,4,1}; # a set
f={"three":3, "one":1, "four":4, "three":3}; # a dictionary
︡d9f8a207-5f6e-4a7a-b4a0-a479843cde73︡{"done":true}︡
︠51f456ed-7095-4eae-9ab7-47cb5085b50as︠
### use type(a) to check the type of a

type(a)
︡91b8897e-4e46-431d-af45-52ee70f5d54e︡{"stdout":"<type 'sage.rings.integer.Integer'>\n"}︡{"done":true}︡
︠ddeac15a-5c22-4fd9-b1af-450d3f25238f︠
### in and not in

students=["John","Jephian","Jane"];
"John" in students;
︡8649c6ef-bd5d-437e-b79f-32df1f2b3113︡
︠d97e7c6c-c1f2-41ef-82d1-6d32a9f095a1︠
### relations
### also try: >, >=, <, <=, !=

3==3
︡5b77b65b-3fd9-4709-a5ed-62397f3d7caa︡
︠73a50303-6ddb-4804-8a99-20e7c1324bf3︠
### arithmetic operators

print "23+4 =", 23+4;
print "23-4 =", 23-4;
print "23*4 =", 23*4;
print "23/4 =", 23/4;
print "23**4 =", 23**4;
print "23^4 =", 23^4;
print "23%4 =", 23%4;
print "23//4 =", 23//4;

︡1aed5918-2432-46d4-858f-a12c766e8118︡
︠2732b113-3301-41a8-81d5-7cb6194f90e3s︠
### Exercise:
###     Guess the output of the following.

type(2/3)
︡f2ce29e8-6f4f-4743-8650-f2e1eaffc44a︡{"stdout":"<type 'sage.rings.rational.Rational'>\n"}︡{"done":true}︡
︠7e942dbe-6f0e-47d4-8256-4934aa5fad5e︠
a=2;
a in [2,3,5];
︡2e70e6b4-eca4-462e-95a1-ce4627956d78︡
︠d434cfe5-7e68-4fa4-ac85-b88efbe5761c︠
a=2;
a in ["a","b","c"];
︡e9b447bb-8a6f-4f8e-83a3-4034f8cdd3a9︡
︠6ca325e1-c599-4d2e-9743-d110b842cde6︠
a=2;
type(a)==int;
︡2a3fa810-f2e1-44c0-ac61-c1d5d05e70c6︡
︠7cc2b167-74cd-466c-a2bc-ca683d545d24︠
a=int(2);
type(a)==int;
︡179283d2-1c7e-42cd-8a3b-4d56a59d0e61︡
︠24590899-6881-48ff-8286-9e611de0d08d︠
(2,3,5)==[2,3,5]
︡26dc680a-1f92-44b6-95a8-0f8b664fcb79︡
︠a8673181-bf11-47c4-9aa6-b43c3afa074c︠
2==2.0
︡0b1bc3ad-341e-43e5-8534-d4b6c4b60784︡
︠6b4dd4a1-5d60-49a3-82bf-d2c4eba4507b︠
d={"two":2, "three":3, "five":5};
print 2 in d.keys();
print 2 in d.values();
︡c58b6db7-ac1c-481e-8e73-0e028c7316e2︡
︠d6e03d8c-9fba-41b3-bd27-3d7b9e0b0902︠
50%3==0
︡1f0de80a-04ff-47b1-97b5-2752fa37710e︡
︠e5357348-508f-4980-9f9c-fe696d5104ed︠
50%3==0 or 51%3==0
︡41d1b967-b368-4c8d-9908-2b3b6beaef05︡
︠ab82e739-4032-4a7d-acf1-51ef3866964a︠
50%3==0 or 51%3==0
︡39cffc8c-0e39-488d-8241-1775b37f17bb︡
︠e4eb389a-c86b-46d5-9719-609f9f3554b1︠
[1,2,3,4]>[2,3,4]
︡be57bc93-dd10-4c48-a964-44ee1c289f8b︡
︠9277a7d9-3b33-4a59-9f26-05e86800a7ce︠
"Z">"B"
︡0004c828-c1a0-4ab5-a416-ff58a8f43d89︡
︠5be9d377-7646-4d18-97aa-b798d79958b6︠
### type conversion

a=2/2
print type(a)
print (a==1) ### This line will print True in Sage but False in C

a=Integer(a)
### Integer means an object in the integer ring in Sage;
### int is a data structure in most of programming language
print type(a)
︡e4e32d9b-fa06-46f3-8204-70459fb0bd74︡{"stdout":"<type 'sage.rings.rational.Rational'>\n"}︡{"stdout":"True\n"}︡{"stdout":"<type 'sage.rings.integer.Integer'>\n"}︡{"done":true}︡
︠e9fe6f5c-e999-4cbb-b1c4-88edd3cab504s︠
### len returns the size
### append add an element to a list

a = [3,1,4,1]
print len(a)

a.append(5)
a.append(9)
print a
print len(a)
︡243b5e86-7fdb-4375-8a49-9127180c26b7︡{"stdout":"4\n"}︡{"stdout":"[3, 1, 4, 1, 5, 9]\n"}︡{"stdout":"6\n"}︡{"done":true}︡
︠93e12b35-d850-426c-bc1f-a047d8ff3abb︠
### check repeated elments
### set converse a list to a set

a = [3,1,4,1]
print set(a)
print len(set(a)) < len(a)
︡fc35d79d-c50b-495b-b7e3-e586c411dbcb︡{"stdout":"set([1, 3, 4])\n"}︡{"stdout":"True\n"}︡{"done":true}︡
︠2fb0f9c4-0ae2-429d-bb7c-f4724cf3fbc8s︠
### Combinations(list,k) returns all k-combinations in list

counter = 0
for com in Combinations(range(5),2):
    counter += 1;
    print com;

print "---"
print "total count is", counter
print "5 choose 2 is", binomial(5,2)
︡6332726a-ac02-4bb2-9d15-9267222a380f︡{"stdout":"[0, 1]\n[0, 2]\n[0, 3]\n[0, 4]\n[1, 2]\n[1, 3]\n[1, 4]\n[2, 3]\n[2, 4]\n[3, 4]\n"}︡{"stdout":"---\n"}︡{"stdout":"total count is 10\n"}︡{"stdout":"5 choose 2 is 10\n"}︡{"done":true}︡
︠ec8d067c-4ade-4e50-b956-23eb953e503e︠
︡5413c436-e52a-492e-b8fd-96dedb5b51d9︡
︠9f20d4a5-6626-47fd-9308-d8da5f58503di︠
%md

## Project 1
Suppose `positions` is a list of the form `[(x1,y1),(x2,y2),...]`.

Write a function to check `positions` is non-attacking or not.
︡1491331a-ff0e-4231-ae95-9c18acf6be1f︡{"done":true,"md":"\n## Project 1\nSuppose `positions` is a list of the form `[(x1,y1),(x2,y2),...]`.\n\nWrite a function to check `positions` is non-attacking or not."}
︠e8a40d27-ed26-4b3c-81aa-4bc1b950f580︠
def is_nonattacking(positions):
    ### positions is a list of the form [(x1,y1),(x2,y2),...]
    ### create two lists xcors and ycors to records the x-coordinates and y-coordinates.
    ### len(set(l)) < len(l)   allows you to check if l contains any repeated elements

    ### Your answers:
    xcors = []
    ycors = []
    for pos in positions:
        xcors.append(pos[0]);
        ycors.append(pos[1]);
    if len(set(xcors)) < len(xcors):
        return False;
    if len(set(ycors)) < len(ycors):
        return False;
    ### if nothing happens
    return True;

### test;
positions1=[(0,0),(0,1)]
positions2=[(0,0),(1,1)]
print "your answer is {}, {}=correct answer".format(is_nonattacking(positions1),False)
print "your answer is {}, {}=correct answer".format(is_nonattacking(positions2),True)
︡3c9220c1-7ec6-4b90-bd23-f2cb34d4bbe8︡{"stdout":"your answer is False, False=correct answer\n"}︡{"stdout":"your answer is True, True=correct answer\n"}︡{"done":true}︡
︠9521f063-083a-4bb6-b448-aef313add05ci︠
%md
## Project 2
Suppose `B` is a list all positions on a board.

For example `B = [(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)]` for the $3\times 3$ derangement.

Write a function to compute number of ways to put $k$ rooks on $B$.
︡bc1a6203-7757-4240-9efd-494b052c3ed8︡{"done":true,"md":"## Project 2\nSuppose `B` is a list all positions on a board.\n\nFor example `B = [(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)]` for the $3\\times 3$ derangement.\n\nWrite a function to compute number of ways to put $k$ rooks on $B$."}
︠19071467-d0cd-4343-9c78-e4b15712975cs︠
def rook_placements(k,B):
    ### Hint: use Combinations(B,k)

    ### You answers:
    counter = 0;
    for positions in Combinations(B,k):
        if is_nonattacking(positions):
            counter += 1;
    return counter;

### test;
n = 10
B = [(i,i) for i in range(n)] + [(i,(i+1)%n) for i in range(n)]
for k in range(n+1):
    print "your answer for r_10,{} is {}, {}=correct answer".format(k,rook_placements(k,B),2*n/(2*n-k)*binomial(2*n-k,k))
︡b02a5f35-b21e-4bde-9453-bd5a7584f213︡{"stdout":"your answer for r_10,0 is 1, 1=correct answer\nyour answer for r_10,1 is 20, 20=correct answer\nyour answer for r_10,2 is 170, 170=correct answer\nyour answer for r_10,3 is 800, 800=correct answer\nyour answer for r_10,4 is 2275, 2275=correct answer\nyour answer for r_10,5 is 4004, 4004=correct answer"}︡{"stdout":"\nyour answer for r_10,6 is 4290, 4290=correct answer"}︡{"stdout":"\nyour answer for r_10,7 is 2640, 2640=correct answer"}︡{"stdout":"\nyour answer for r_10,8 is 825, 825=correct answer"}︡{"stdout":"\nyour answer for r_10,9 is 100, 100=correct answer"}︡{"stdout":"\nyour answer for r_10,10 is 2, 2=correct answer"}︡{"stdout":"\n"}︡{"done":true}︡
︠6a6551c2-dba9-4378-809b-1fc31e17c0aa︠