Provided by: tcllib_1.21+dfsg-1_all 
NAME
math::combinatorics - Combinatorial functions in the Tcl Math Library
SYNOPSIS
package require Tcl 8.2
package require math ?1.2.3?
package require Tcl 8.6
package require TclOO
package require math::combinatorics ?2.0?
::math::ln_Gamma z
::math::factorial x
::math::choose n k
::math::Beta z w
::math::combinatorics::permutations n
::math::combinatorics::variations n k
::math::combinatorics::combinations n k
::math::combinatorics::derangements n
::math::combinatorics::catalan n
::math::combinatorics::firstStirling n m
::math::combinatorics::secondStirling n m
::math::combinatorics::partitionP n
::math::combinatorics::list-permutations n
::math::combinatorics::list-variations n k
::math::combinatorics::list-combinations n k
::math::combinatorics::list-derangements n
::math::combinatorics::list-powerset n
::math::combinatorics::permutationObj new/create NAME n
$perm next
$perm reset
$perm setElements elements
$perm setElements
::math::combinatorics::combinationObj new/create NAME n k
$combin next
$combin reset
$combin setElements elements
$combin setElements
_________________________________________________________________________________________________
DESCRIPTION
The math package contains implementations of several functions useful in combinatorial
problems. The math::combinatorics extends the collections based on features in Tcl 8.6.
Note: the meaning of the partitionP function, Catalan and Stirling numbers is explained on
the MathWorld website [http://mathworld.wolfram.com]
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.
::math::combinatorics::permutations n
Return the number of permutations of n items. The returned number is always an
integer, it is not limited by the range of 32-or 64-bits integers using the
arbitrary precision integers available in Tcl 8.5 and later.
int n The number of items to be permuted.
::math::combinatorics::variations n k
Return the number of variations k items selected from the total of n items. The
order of the items is taken into account.
int n The number of items to be selected from.
int k The number of items to be selected in each variation.
::math::combinatorics::combinations n k
Return the number of combinations of k items selected from the total of n items.
The order of the items is not important.
int n The number of items to be selected from.
int k The number of items to be selected in each combination.
::math::combinatorics::derangements n
Return the number of derangements of n items. A derangement is a permutation where
each item is displaced from the original position.
int n The number of items to be rearranged.
::math::combinatorics::catalan n
Return the n'th Catalan number. The number n is expected to be 1 or larger. These
numbers occur in various combinatorial problems.
int n The index of the Catalan number
::math::combinatorics::firstStirling n m
Calculate a Stirling number of the first kind (signed version, m cycles in a
permutation of n items)
int n Number of items
int m Number of cycles
::math::combinatorics::secondStirling n m
Calculate a Stirling number of the second kind (m non-empty subsets from n items)
int n Number of items
int m Number of subsets
::math::combinatorics::partitionP n
Calculate the number of ways an integer n can be written as the sum of positive
integers.
int n Number in question
::math::combinatorics::list-permutations n
Return the list of permutations of the numbers 0, ..., n-1.
int n The number of items to be permuted.
::math::combinatorics::list-variations n k
Return the list of variations of k numbers selected from the numbers 0, ..., n-1.
The order of the items is taken into account.
int n The number of items to be selected from.
int k The number of items to be selected in each variation.
::math::combinatorics::list-combinations n k
Return the list of combinations of k numbers selected from the numbers 0, ..., n-1.
The order of the items is ignored.
int n The number of items to be selected from.
int k The number of items to be selected in each combination.
::math::combinatorics::list-derangements n
Return the list of derangements of the numbers 0, ..., n-1.
int n The number of items to be rearranged.
::math::combinatorics::list-powerset n
Return the list of all subsets of the numbers 0, ..., n-1.
int n The number of items to be rearranged.
::math::combinatorics::permutationObj new/create NAME n
Create a TclOO object for returning permutations one by one. If the last
permutation has been reached an empty list is returned.
int n The number of items to be rearranged.
$perm next
Return the next permutation of n objects.
$perm reset
Reset the object, so that the command next returns the complete list again.
$perm setElements elements
Register a list of items to be permuted, using the nextElements command.
list elements
The list of n items that will be permuted.
$perm setElements
Return the next permulation of the registered items.
::math::combinatorics::combinationObj new/create NAME n k
Create a TclOO object for returning combinations one by one. If the last
combination has been reached an empty list is returned.
int n The number of items to be rearranged.
$combin next
Return the next combination of n objects.
$combin reset
Reset the object, so that the command next returns the complete list again.
$combin setElements elements
Register a list of items to be permuted, using the nextElements command.
list elements
The list of n items that will be permuted.
$combin setElements
Return the next combination of the registered items.
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