combinat::lyndonWords::fromEvaluation – Lyndon words with prescribed evaluation

The combinatorial class of integral Lyndon words with prescribed evaluation

→ Examples

Superdomain

Dom::BaseDomain

Axioms

Ax::systemRep

Details:

• An integer Lyndon word is a Lyndon word whose operands are positive integers.

• The evaluation of an integer Lyndon word w is a list c such that c[i] is the number of occurrences of i in w. For example, the evaluation of [1,1,1,2,1,4,4] is [4,1,0,2]. See combinat::words::evaluation for details.

• Some functions here only deal with evaluations that are compositions. In other words the corresponding Lyndon words should contain all the integers in a range 1..k (counter examples: [2,2,3] or [1,4,3,3]).

isA – test if an object is a Lyndon word

combinat::lyndonWords::fromEvaluation::isA(any type object, <composition evaluation>)

Returns whether object is a Lyndon word.

If the optional argument is present, returns whether object is a Lyndon word with evaluation evaluation.

count – number of integer Lyndon words with prescribed evaluation

combinat::lyndonWords::fromEvaluation::count(integer list evaluation)

Returns the number of integer Lyndon words with evaluation evaluation.

list – list of the integer Lyndon words with prescribed evaluation

combinat::lyndonWords::fromEvaluation::list(composition evaluation)

Returns the list of the integer Lyndon words with evaluation evaluation.

generator – generator for the integer Lyndon words with prescribed evaluation

combinat::lyndonWords::fromEvaluation::generator(composition evaluation)

Returns a generator for the integer Lyndon words with evaluation evaluation.

Example 1:

Background:

The generation algorithm for Lyndon words with prescribed evaluation is the same as for necklaces; see combinat::necklaces for details.

Changes in MuPAD 3.1

New Function.