# combinatorics - Man Page

Combinatorial functions in the Tcl Math Library

## Synopsis

`package require `

**Tcl 8.2**

`package require `

**math ?1.2.3?**

**::math::ln_Gamma** *z*

**::math::factorial** *x*

**::math::choose** *n k*

**::math::Beta** *z w*

## Description

The **math** package contains implementations of several functions useful in combinatorial problems.

## Commands

**::math::ln_Gamma***z*Returns the natural logarithm of the Gamma function for the argument

*z*.The Gamma function is defined as the improper integral from zero to positive infinity of

t**(x-1)*exp(-t) dt

The approximation used in the Tcl Math Library is from Lanczos, *ISIAM J. Numerical Analysis, series B,* volume 1, p. 86. For "**x** > 1", the absolute error of the result is claimed to be smaller than 5.5*10**-10 -- that is, the resulting value of Gamma when

exp( ln_Gamma( x) )

is computed is expected to be precise to better than nine significant figures.

**::math::factorial***x*Returns the factorial of the argument

*x*.For integer

*x*, 0 <=*x*<= 12, an exact integer result is returned.For integer

*x*, 13 <=*x*<= 21, an exact floating-point result is returned on machines with IEEE floating point.For integer

*x*, 22 <=*x*<= 170, the result is exact to 1 ULP.For real

*x*,*x*>= 0, the result is approximated by computing*Gamma(x+1)*using the**::math::ln_Gamma**function, and the result is expected to be precise to better than nine significant figures.It is an error to present

*x*<= -1 or*x*> 170, or a value of*x*that is not numeric.**::math::choose***n k*Returns the binomial coefficient

*C(n, k)*C(n,k) = n! / k! (n-k)!

If both parameters are integers and the result fits in 32 bits, the result is rounded to an integer.

Integer results are exact up to at least

*n*= 34. Floating point results are precise to better than nine significant figures.**::math::Beta***z w*Returns the Beta function of the parameters

*z*and*w*.Beta(z,w) = Beta(w,z) = Gamma(z) * Gamma(w) / Gamma(z+w)

Results are returned as a floating point number precise to better than nine significant digits provided that

*w*and*z*are both at least 1.

## Bugs, Ideas, Feedback

This document, and the package it describes, will undoubtedly contain bugs and other problems. Please report such in the category *math* of the *Tcllib Trackers* [http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas for enhancements you may have for either package and/or documentation.

When proposing code changes, please provide *unified diffs*, i.e the output of **diff -u**.

Note further that *attachments* are strongly preferred over inlined patches. Attachments can be made by going to the **Edit** form of the ticket immediately after its creation, and then using the left-most button in the secondary navigation bar.

## Category

Mathematics