The triforce package is suite of tools for the numerical exploration of
triangular arrays. It allows you to manipulate well-known triangular arrays
(i.e. Pascal's triangle, the Hosoya triangle, Bell's triangle, etc.) and also
to define your own triangle via specifying generating functions.
For example, the following code allows you to define Pascal's Triangle for a
given number of rows. You can use the functionality of triforce to
investigate various numerical properties of the triangle.
>>> from triforce.triangles import PascalTriangle
>>> triangle = PascalTriangle(10)
>>>
>>> # Print the output as a triangular shape:
>>> print(triangle)
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
>>> # Extract the right-most diagonal of the triangle:
>>> print(triangle.diagonal(diagonal_index=0, direction="right"))
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
>>> # Extract the second-to-the-right-most diagonal of the triangle:
>>> print(triangle.diagonal(diagonal_index=1, direction="right"))
[1, 2, 3, 4, 5, 6, 7, 8, 9]
>>> # Extract the center entries of the triangle:
>>> print(triangle.center())
[1, 2, 6, 20, 70]
>>> # Extract row sums:
>>> print(triangle.row_sums())
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512]You can also define your own triangular array by creating a class that inherits
from the Triangle class
from triforce.triangle import Triangle
class WythoffTriangle(Triangle):
def generate_triangle(self) -> list[list[int]]:
"""Generate the Wythoff triangle up to row n using Fibonacci recurrence."""
triangle = []
for i in range(self.n):
row = []
# First element: lower Wythoff sequence
if i == 0:
row.append(0) # Starting element for Wythoff triangle
else:
row.append(lower_wythoff(i))
# Middle elements: Fill using Fibonacci recurrence
for j in range(1, i):
# Fibonacci recurrence: current element is the sum of the two elements above it
row.append(triangle[i-1][j-1] + triangle[i-1][j])
# Last element: upper Wythoff sequence
if i > 0:
row.append(upper_wythoff(i))
triangle.append(row)
return triangleOne can also visualize properties of the triangles by plotting. Here is an example of visualizing Pascal's triangle such that the even and odd terms are replaced by black and white pixels, respectively.
from triforce.plots import highlight_plot
from triforce.numerics import is_even
highlight_plot(PascalTriangle(n=510), is_even)
