General_Pascals_Triangle.ipynb

Kernel: Python 3 (system-wide)

General Pascal's Triangle

def make_triangle(op:"function", rep:"function", gen_row:list, num_rows:int, aux:int, sp:int=1):
    last_row = gen_row
    ch = len(rep(gen_row[0], aux))
    t = 0
    if (ch % 2) != (sp % 2) and sp != -1:
        sp += 1
    if sp < 0:
        t = sp * -1
    ol = (abs(ch - sp) // 2) + (ch if sp >= ch else sp)
    for i in range(num_rows + 1):
        print((" " * ol * (num_rows - i) if sp >= 0 else "") + (" " * sp if sp >= 0 else "\t"*t).join(rep(element,aux) for element in last_row))
        last_row = [gen_row[0]] + [op(last_row[i], last_row[i+1], aux) for i in range(len(last_row) - 1)] + [gen_row[-1]]

$\mathbb{Z}_n$

# these define the semigroup structure and how the elements are displayed

def op_Z_n(a,b,aux):
    return (a + b) % aux

def rep_Z_n(a,aux=1):
    return str(a)

# Z_4
make_triangle(op_Z_n, rep_Z_n, gen_row=[1], num_rows=10, aux=4, sp=3)

                    1
                  1   1
                1   2   1
              1   3   3   1
            1   0   2   0   1
          1   1   2   2   1   1
        1   2   3   0   3   2   1
      1   3   1   3   3   1   3   1
    1   0   0   0   2   0   0   0   1
  1   1   0   0   2   2   0   0   1   1
1   2   1   0   2   0   2   0   1   2   1

# Z_7
make_triangle(op_Z_n, rep_Z_n, gen_row=[1], num_rows=15, aux=7, sp=3)

                              1
                            1   1
                          1   2   1
                        1   3   3   1
                      1   4   6   4   1
                    1   5   3   3   5   1
                  1   6   1   6   1   6   1
                1   0   0   0   0   0   0   1
              1   1   0   0   0   0   0   1   1
            1   2   1   0   0   0   0   1   2   1
          1   3   3   1   0   0   0   1   3   3   1
        1   4   6   4   1   0   0   1   4   6   4   1
      1   5   3   3   5   1   0   1   5   3   3   5   1
    1   6   1   6   1   6   1   1   6   1   6   1   6   1
  1   0   0   0   0   0   0   2   0   0   0   0   0   0   1
1   1   0   0   0   0   0   2   2   0   0   0   0   0   1   1

$D_n$

def op_D_n(a,b,aux):  # r^ks^e --> (k,e)
    return ((a[0] + b[0] * (1 - 2 * a[1])) % aux, (a[1] + b[1]) % 2)

def rep_D_n(a,aux=1):
    if a == (0,0):
        return "  e   "
    return (f"r^{a[0]}" if a[0] > 1 else {0:"   ", 1:"  r"}[a[0]]) + {0:"   ", 1:"s  "}[a[1]]

# D_3

make_triangle(op_D_n, rep_D_n, [(1,1),(2,0)], num_rows=12, aux = 3, sp=2)

                                                  rs    r^2   
                                              rs    r^2s    r^2   
                                          rs    r^2        s    r^2   
                                      rs    r^2s    r^2s      rs    r^2   
                                  rs    r^2       e       r     r^2s    r^2   
                              rs    r^2s    r^2       r        s       s    r^2   
                          rs    r^2        s      e       rs      e       rs    r^2   
                      rs    r^2s    r^2s       s      rs      rs      rs    r^2s    r^2   
                  rs    r^2       e     r^2     r^2       e       e     r^2        s    r^2   
              rs    r^2s    r^2     r^2       r     r^2       e     r^2     r^2s      rs    r^2   
          rs    r^2        s      r       e       e     r^2     r^2       rs      r     r^2s    r^2   
      rs    r^2s    r^2s    r^2s      r       e     r^2       r        s       s       s       s    r^2   
  rs    r^2       e       e       rs      r     r^2       e       rs      e       e       e       rs    r^2   

$S_n$

a,b=eval(input())
p=a+b
o=[]
for i in range(1,1+max(map(max,p))):
    if not any(i in t for t in o):
        u=(i,)
        m=i
        while True:
            for t in p[::-1]:
                if m in t:
                    m=t[(t.index(m)+1)%len(t)]
            if m == i:
                o+=[u]
                break
            u+=(m,)
print(o)

a,b=eval(input())
p=a+b
o=[]
for i in range(1,1+max(map(max,p))):
    if not any(i in t for t in o):
        u=(i,)
        m=i
        while True:
            for t in p[::-1]:
                if m in t:
                    m=t[(t.index(m)+1)%len(t)]
            if m == i:
                o+=[u]
                break
            u+=(m,)
print(o)

def op_S_n(a,b,aux):
    a.reverse()
    b.reverse()
    out = []
    for i in range(1,aux+1):
        if any(i in tup for tup in out):
            continue
        build = [i]
        maps_to = i
        while True:
            for tup in b + a:
                if maps_to in tup:
                    maps_to = tup[(tup.index(maps_to) + 1) % len(tup)]
            if maps_to == i:
                out += [tuple(build)]
                break
            build += [maps_to]
    return out

from math import floor, ceil
def rep_S_n(a,aux):
    s = "".join("(" + " ".join(str(el) for el in tup) + ")" for tup in a if len(tup) != 1)
    s = s if s != "" else "(1)"
    pad = (aux // 2 + 1) * 5 - len(s)
    return " " * ceil(pad / 2) + s + " " * floor(pad/2)

# S_5

make_triangle(op_S_n, rep_S_n, [[(1,2,3,4)]], num_rows=7, aux = 5, sp=0)

                                                           (1 2 3 4)   
                                                   (1 2 3 4)       (1 2 3 4)   
                                           (1 2 3 4)       (1 3)(2 4)      (1 2 3 4)   
                                   (1 2 3 4)       (1 4 3 2)       (1 4 3 2)       (1 2 3 4)   
                           (1 2 3 4)          (1)          (1 3)(2 4)         (1)          (1 2 3 4)   
                   (1 2 3 4)       (1 2 3 4)       (1 3)(2 4)      (1 3)(2 4)      (1 2 3 4)       (1 2 3 4)   
           (1 2 3 4)       (1 3)(2 4)      (1 4 3 2)          (1)          (1 4 3 2)       (1 3)(2 4)      (1 2 3 4)   
   (1 2 3 4)       (1 4 3 2)       (1 2 3 4)       (1 4 3 2)       (1 4 3 2)       (1 2 3 4)       (1 4 3 2)       (1 2 3 4)