Combinatorics
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package math
Compute the beta function B(a, b) for the following two cases: (1) when a or b are integers and (2) when a or b are integers + 1/2.
Compute the beta function B(a, b) for the following two cases: (1) when a or b are integers and (2) when a or b are integers + 1/2.
the first parameter, a real number satisfying (1) or (2)
the second parameter, a real number satisfying (1) or (2)
http://mathworld.wolfram.com/BetaFunction.html
Compute trinomial coefficients: n choose (k, l), combinations of n things, (k, l) at a time, using Pascal's Tetrahedron.
Compute trinomial coefficients: n choose (k, l), combinations of n things, (k, l) at a time, using Pascal's Tetrahedron. Ex: Given n balls, counts ways in which k are chosen for group 1 and l are chosen for group 2.
the total number of items
the of items to choose
the of items to choose (requires 0 <= k + l <= n)
http://people.sju.edu/~pklingsb/bintrin.pdf
Compute binomial coefficients: n choose k, combinations of n things, k at a time, using Pascal's Triangle.
Compute binomial coefficients: n choose k, combinations of n things, k at a time, using Pascal's Triangle.
the total number of items
the of items to choose (requires k <= n)
http://www.mathsisfun.com/pascals-triangle.html
Compute n choose k (combinations of n things, k at a time).
Compute n choose k (combinations of n things, k at a time). A more efficient implementation is given below.
the total number of items
the of items to choose (requires k <= n)
Compute 'k' factorial (k!) using three techniques (requires k <= 170).
Compute 'k' factorial (k!) using three techniques (requires k <= 170).
the nonnegative integer-valued argument to the factorial function
Show the flaw by printing the error message.
Show the flaw by printing the error message.
the method where the error occurred
the error message
Compute the gamma function gamma(a) using the Lanczos Approximation.
Compute the gamma function gamma(a) using the Lanczos Approximation.
the parameter, a real number
http://en.wikipedia.org/wiki/Lanczos_approximation
Compute Gauss's Hypergeometric function 2F1(z; a, b, c) via an approximation.
Compute Gauss's Hypergeometric function 2F1(z; a, b, c) via an approximation.
the variable, a real/complex number s.t. |z| < 1
the first paramater, a real/complex number
the second parameter, a real/complex number
the third parameter, a real/complex number, may not be a negative integer
Compute the incomplete beta function B(z; a, b), a generalization of the beta function (z = 1).
Compute the incomplete beta function B(z; a, b), a generalization of the beta function (z = 1).
the variable, a real/complex number s.t. 0 <= |z| <= 1
the first parameter, a real/complex number > 0
the second parameter, a real/complex number > 0
http://mathworld.wolfram.com/IncompleteBetaFunction.html
Table of all factorials that can be represented as a long (64-bit) integer
Compute the natural log factorial (ln (k!) so k! = exp (logfac (k).
Compute the natural log factorial (ln (k!) so k! = exp (logfac (k). The formula is a log transformation of Ramanujan's Factorial Approximation.
the value to take the log factorial of
For small 'k', compute 'k' factorial by iterative multiplication.
For small 'k', compute 'k' factorial by iterative multiplication.
k! = k * (k-1) * ... * 2 * 1
the nonnegative integer-valued argument to the factorial function
Compute k! using Mortici's Factorial Approximation (more accurate than Stirling's 2nd Order Factorial Approximation).
Compute k! using Mortici's Factorial Approximation (more accurate than Stirling's 2nd Order Factorial Approximation).
the nonnegative integer-valued argument to the factorial function
http://but.unitbv.ro/BU2010/Series%20III/BULETIN%20III%20PDF/Mathematics/Mortici.pdf
Initial part of Pascal's Tetrahedron, precomputed to speed calculations (Trinomial Coefficients)
Initial part of Pascal's Tetrahedron, precomputed to speed calculations (Trinomial Coefficients)
https://sites.google.com/site/pascalloids/pascal-s-pyramid-3-var
Initial part of Pascal's Triangle, precomputed to speed calculations (Binomial Coefficients)
Compute permutations of k items selected from n total items.
Compute permutations of k items selected from n total items.
the total number of items
the of items selected
Compute the regularized (incomplete) beta function I(z; a, b).
Compute the regularized (incomplete) beta function I(z; a, b).
the variable, a real/complex number s.t. 0 <= |z| <= 1
the first parameter, a real/complex number > 0
the second parameter, a real/complex number > 0
http://mathworld.wolfram.com/RegularizedBetaFunction.html
Compute k! using Ramanujan's Factorial Approximation (more accurate than Mortici's Factorial Approximation).
Compute k! using Ramanujan's Factorial Approximation (more accurate than Mortici's Factorial Approximation).
the nonnegative integer-valued argument to the factorial function
http://files.ele-math.com/articles/jmi-05-53.pdf
Compute the kth degree rising factorial of x.
Compute the kth degree rising factorial of x. When x = 1, this is the regular factorial function k!.
the number of factors in the product
the base number to start the product
Compute k! using Stirling's 2nd Order Factorial Approximation.
Compute k! using Stirling's 2nd Order Factorial Approximation.
the nonnegative integer-valued argument to the factorial function
http://en.wikipedia.org/wiki/Stirling%27s_approximation
The
Combinatoricsobject provides several common combinatorics functions, such as factorial permutations, combinations, gamma and beta functions.