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Combinatorics

All combinatorial commands (permutations, combinations and variations, with or without repetition) have an associated icon and text command.

These commands are commonly used to calculate the number of components in a list of combinatorial selections, but they can also return the selections themselves.

Except for the special case of permutations with repetition, explained below, when the first argument of these commands is a list (shown with curly brackets) or a vector (shown in square brackets), the command returns the corresponding list of combinatorial selections from the set.

In WIRIS the elements of a list or vector are distinct, even if there are repetitions. Thus, when combinations, variations or permutations are calculated they are treated distinctly, rather than as indistinguishable, except in the case of permutations with repetition.

combinations:  Icon or , command combinations

The combinations command takes two arguments, m and n. If m and n are non-negative integers, it calculates the number of combinations of m elements taken from n in n. If m is a List or Vector and n is a non-negative integer, it returns the list of combinations of its elements taken from n in n.

Upon clicking the icon, the standard combinations symbol will appear along with two green, empty boxes. Enter the argument m in the first and the argument n in the second.

Upon clicking the icon, two boxes will appear. Enter the argument m in the top box and the argument n in the lower box.

combinations with repetition:  Icon , command combinations_with_repetition

The combinations_with_repetition command takes two arguments, m and n. If m and n are non-negative integers, it calculates the number of combinations with repetition of m elements taken from n in n. If m is a List or Vector and n a non-negative integer, it returns the list of the combinations with repetition of its elements taken from n in n.

Upon clicking the icon, the standard symbol for combinations with repetition will appear along with two green, empty boxes. Enter the argument m in the first and the argument n in the second.

variations:  Icon , command variations

The variations command takes two arguments, m and n. If m and n are non-negative integers, it calculates the number of variations of m elements taken from n in n. If m is a List or Vector and n a non-negative integer, it returns a list of the variations of its elements taken from n in n.

Upon clicking the icon, the standard variations symbol will appear along with two green, empty boxes. Enter the argument m in the first and the argument n in the second.

variations with repetition:  Icon , command variations_with_repetition

The variations_with_repetition command takes two arguments, m and n. If m and n are non-negative integers, it calculates the number of variations with repetition of m elements taken from n in n. If m is a List or Vector and n a non-negative integer, it returns the list of the variations with repetition of its elements taken from ninn.

Upon clicking the icon, the standard symbol for variations with repetition will appear along with two green, empty boxes. Enter the argument m in the first and the argument n in the second.

permutations:  Icon , command permutations

The permutations command takes one argument, n. If n is a non-negative integer, it returns the number of permutations of n elements, that is n!. If n is a List or Vector then it provides the list of all the permutations of its elements.

Clicking the icon will bring up the standard permutations symbol, containing an empty green box corresponding to the argument n.

permutations with repetition:  Icon , command permutations_with_repetition

The permutations_with_repetition command has an initial argument, n, which must be a non-negative whole number (otherwise the command has no effect) and a sequence of one or more additional arguments n1 , n2 ,..., nr .
If the additional arguments are non-negative whole numbers such as n = n1+n2+...+nr , the command will obtain the number of permutations for n elements taken from r different elements and such that the ith element repeats ni times. If these conditions are not met, the command has no effect.
In place of the sequence of additional arguments it is possible to enter a List (or a Vector) L of nelements, comprised of r different elements and such that the ith element repeats ni times. If n = n1+n2+...+nr , the command provides the list of all the different distinct permutations of L otherwise it has no effect. To calculate the set, enter the list of the elements to be combined as the second argument.

Clicking the icon, the standard symbol for permutations with repetition will appear along with two green, empty boxes. Enter the additional arguments (that is, the sequence ni , or the List or Vector) and the argument n in the second box.