_03d_pascal Namespace Reference

Showing a range of rows of a Pascal triangle. More...

Functions
def	binomial_array (n)
	Compute n lines of the binomial array. More...
def	binomial_coefficient (n, m, method=2)
	Compute \(n\) choose \(m\) or Combination \((n,m)\). More...
def	binomial_cumulative_distribution (p, n, x=None)
	Compute the probability of getting at most \(x\) successes in \(n\) trials, which is the sum of the first \(x\) terms of the binomial expansion \((p+q)^n\). More...
def	exponent (val, type=False)
	Return a superscript string for the given value. More...
def	binomial_expansion (e)
	Compute the binomial expansion of \((x+y)^n\). More...
def	getNextRow (curr_row)
	Computes the next row of a Pascal triangle, given the current row. More...
def	fmtRow (last_level, r, pad=True)
	Formats a row of the Pascal triangle. More...
def	pascal (first_level, last_level, center=True)
	Creates the pascal triangle between two levels. More...
def	pascal2 (first_level, last_level, center=True)
	Creates the pascal triangle between two levels, using the binomial array. More...
def	main (argv=None)
	Main function for testing. More...

Variables
dictionary	superscript_map
	dictionary for mapping characters to superscript. More...
dictionary	subscript_map
	trans
	character translation table. More...
	sub_trans

Detailed Description

Showing a range of rows of a Pascal triangle.

   0       1                                 C(0,0)
   1      1 1                            C(1,0)  C(1,1)
   2     1 2 1                        C(2,0)  C(2,1)  C(2,2)
   3    1 3 3 1                   C(3,0)  C(3,1)  C(3,2)  C(3,3)
   4   1 4 6 4 1              C(4,0)  C(4,1)  C(4,2)  C(4,3)  C(4,4)
   5  1 5 10 10 5 1       C(5,0)  C(5,1)  C(5,2)  C(5,3)  C(5,4)  C(5,5)

      0 1 2  3  4 5
  

Author

Paulo Roma

Since

13/04/2009

See also

http://en.wikipedia.org/wiki/Pascal_triangle

Function Documentation

binomial_array()

def _03d_pascal.binomial_array

(

n

)

Compute n lines of the binomial array.

Complexity is \(1 + 2\ +\ ...\ +\ n\) elements. This is arithmetic progression that sums to \({n*(n+1)}\over{2}\), which is in \(O(n^2)\).

   0 - 1
   1 - 1 1
   2 - 1 2 1
   3 - 1 3 3 1
   4 - 1 4 6 4 1
   5 - 1 5 10 10 5 1
   

Parameters

n	number of lines.

Returns

binomial array as a list of lines.

Referenced by binomial_coefficient(), binomial_cumulative_distribution(), binomial_expansion(), and pascal2().

binomial_coefficient()

def _03d_pascal.binomial_coefficient	(		n,
			m,
			method = 2
	)

Compute \(n\) choose \(m\) or Combination \((n,m)\).

Parameters

n	number of objects to choose from.
m	number of places.
method	how to get the result.

Returns

\(C{{n}\choose{m}} = {{n!} \over {(n-m)!\ m!}} = {{n (n-1) ... (n-m+1)} \over {m!}}\), number of ways of choosing m objects from n.

See also

https://en.wikipedia.org/wiki/Combination

https://docs.scipy.org/doc/scipy/reference/tutorial/special.html

https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.comb.html#scipy.special.comb

References binomial_array().

binomial_cumulative_distribution()

def _03d_pascal.binomial_cumulative_distribution	(		p,
			n,
			x = None
	)

Compute the probability of getting at most \(x\) successes in \(n\) trials, which is the sum of the first \(x\) terms of the binomial expansion \((p+q)^n\).

In other words, the probability of obtaining at most \(x\) successes in \(n\) independent trials, each of which has a probability \(p\) of success, and \(q = (1-p)\) of failure.

That is, if \(X\) denotes the number of successes, it returns:

• \(P(X \le x) = \sum_{i=0}^{x} {C {{n}\choose{i}}\ p^i\ (1-p)^{n-i}}\).

Parameters

p	probability of success.
n	number of trials.
x	number of successes.

Returns

the probability of getting at most x successes in n independent trials.

See also

https://www.khanacademy.org/math/ap-statistics/probability-ap/probability-multiplication-rule/v/getting-at-least-one-heads

http://www.wolframalpha.com/widgets/view.jsp?id=d821210668c6cc5a02db1069cc52464f

https://mat.iitm.ac.in/home/vetri/public_html/statistics/binomial.pdf

https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/visualizing-a-binomial-distribution

References binomial_array().

Referenced by main().

binomial_expansion()

def _03d_pascal.binomial_expansion

(

e

)

Compute the binomial expansion of \((x+y)^n\).

For e = 5, return:

• \(x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5\) (html)
• x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵ (unicode translation table)
• x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5 (None)

Parameters

e	exponent n.

Returns

a string representing the expansion.

See also

https://www.khanacademy.org/math/precalculus/prob-comb/prob-combinatorics-precalc/v/exactly-three-heads-in-five-flips

https://www.khanacademy.org/math/precalculus/prob-comb/combinations/v/combination-formula

https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/binomial-distribution

References binomial_array(), and exponent().

Referenced by main().

exponent()

def _03d_pascal.exponent	(		val,
			type = False
	)

Return a superscript string for the given value.

Parameters

val	a numeric string.
type	select the superscript method.

Returns

a new superscript string.

Referenced by binomial_expansion().

fmtRow()

def _03d_pascal.fmtRow	(		last_level,
			r,
			pad = True
	)

Formats a row of the Pascal triangle.

Parameters

last_level	last_level.
r	row to be printed.
pad	whether to pad the row with blanks to the left.

Returns

string representing the row.

Referenced by pascal(), and pascal2().

getNextRow()

def _03d_pascal.getNextRow

(

curr_row

)

Computes the next row of a Pascal triangle, given the current row.

Parameters

curr_row

current row.

Returns

next_row: list(map(sum, zip([0] + curr_row, curr_row + [0])))

Referenced by pascal().

main()

def _03d_pascal.main

(

argv = None

)

Main function for testing.

Parameters

argv	first and last levels of the Pascal triangle.

References binomial_cumulative_distribution(), binomial_expansion(), _01d_dec2bin.input, pascal(), and pascal2().

pascal()

def _03d_pascal.pascal	(		first_level,
			last_level,
			center = True
	)

Creates the pascal triangle between two levels.

Parameters

first_level	first level to be printed.
last_level	height of the Pascal triangle.
center	whether to center the lines.

See also

https://www.cut-the-knot.org/arithmetic/combinatorics/PascalTriangleProperties.shtml

https://www.johndcook.com/blog/2016/07/05/distribution-of-numbers-in-pascals-triangle/

References fmtRow(), and getNextRow().

Referenced by main().

pascal2()

def _03d_pascal.pascal2	(		first_level,
			last_level,
			center = True
	)

Creates the pascal triangle between two levels, using the binomial array.

The lines are centered in respect to the last line of the triangle. The output is written to a docx file.

Parameters

first_level	first level to be printed.
last_level	height of the Pascal triangle.
center	whether to center the lines.

References binomial_array(), and fmtRow().

Referenced by main().

Variable Documentation

sub_trans

dictionary _03d_pascal.sub_trans

Initial value:

=  str.maketrans(
        ''.join(subscript_map.keys()),
        ''.join(subscript_map.values()))

subscript_map

dictionary _03d_pascal.subscript_map

Initial value:

=  {
    "0": "₀", "1": "₁", "2": "₂", "3": "₃", "4": "₄", "5": "₅", "6": "₆",
    "7": "₇", "8": "₈", "9": "₉", "a": "ₐ", "b": "♭", "c": "꜀", "d": "ᑯ",
    "e": "ₑ", "f": "բ", "g": "₉", "h": "ₕ", "i": "ᵢ", "j": "ⱼ", "k": "ₖ",
    "l": "ₗ", "m": "ₘ", "n": "ₙ", "o": "ₒ", "p": "ₚ", "q": "૧", "r": "ᵣ",
    "s": "ₛ", "t": "ₜ", "u": "ᵤ", "v": "ᵥ", "w": "w", "x": "ₓ", "y": "ᵧ",
    "z": "₂", "A": "ₐ", "B": "₈", "C": "C", "D": "D", "E": "ₑ", "F": "բ",
    "G": "G", "H": "ₕ", "I": "ᵢ", "J": "ⱼ", "K": "ₖ", "L": "ₗ", "M": "ₘ",
    "N": "ₙ", "O": "ₒ", "P": "ₚ", "Q": "Q", "R": "ᵣ", "S": "ₛ", "T": "ₜ",
    "U": "ᵤ", "V": "ᵥ", "W": "w", "X": "ₓ", "Y": "ᵧ", "Z": "Z", "+": "₊",
    "-": "₋", "=": "₌", "(": "₍", ")": "₎"}

superscript_map

dictionary _03d_pascal.superscript_map

Initial value:

=  {
    "0": "⁰", "1": "¹", "2": "²", "3": "³", "4": "⁴", "5": "⁵", "6": "⁶",
    "7": "⁷", "8": "⁸", "9": "⁹", "a": "ᵃ", "b": "ᵇ", "c": "ᶜ", "d": "ᵈ",
    "e": "ᵉ", "f": "ᶠ", "g": "ᵍ", "h": "ʰ", "i": "ᶦ", "j": "ʲ", "k": "ᵏ",
    "l": "ˡ", "m": "ᵐ", "n": "ⁿ", "o": "ᵒ", "p": "ᵖ", "q": "۹", "r": "ʳ",
    "s": "ˢ", "t": "ᵗ", "u": "ᵘ", "v": "ᵛ", "w": "ʷ", "x": "ˣ", "y": "ʸ",
    "z": "ᶻ", "A": "ᴬ", "B": "ᴮ", "C": "ᶜ", "D": "ᴰ", "E": "ᴱ", "F": "ᶠ",
    "G": "ᴳ", "H": "ᴴ", "I": "ᴵ", "J": "ᴶ", "K": "ᴷ", "L": "ᴸ", "M": "ᴹ",
    "N": "ᴺ", "O": "ᴼ", "P": "ᴾ", "Q": "Q", "R": "ᴿ", "S": "ˢ", "T": "ᵀ",
    "U": "ᵁ", "V": "ⱽ", "W": "ᵂ", "X": "ˣ", "Y": "ʸ", "Z": "ᶻ", "+": "⁺",
    "-": "⁻", "=": "⁼", "(": "⁽", ")": "⁾"}

dictionary for mapping characters to superscript.

trans

dictionary _03d_pascal.trans

Initial value:

=  str.maketrans(
        ''.join(superscript_map.keys()),
        ''.join(superscript_map.values()))

character translation table.