rational exponent for finite words

[staroste]

Upgrade exponent of a FiniteWord_class to calculate rational exponents.

Now:

sage: Word('1231').exponent()
1

Expected:

sage: Word('1231').exponent()
4/3

The goal

sage: Word('1231').exponent()
4/3
sage: Word('1231').exponent(integer=True)
1

Also, upgrade (deprecate, review, and/or change name) the method primitive() to be in accordance with the terminology used in literature (the word is usually called the period instead of the primitive)

sage: Word('1231').exponent()
4/3
sage: Word('1231').period()
word: 123

Upgrade influenced methods (primitive_length(), ...).

CC: @videlec @seblabbe

Component: combinatorics

Issue created by migration from https://trac.sagemath.org/ticket/12226

[seblabbe]

comment:2

Should not period be an integer?

... I already had the idea of making period an alias of minimal_period:

sage: Word('1231').minimal_period()
3

What do you think?

[staroste]

comment:3

Replying to @seblabbe:

Should not period be an integer?

You are right, I found both versions in the literature (integer in Lothaire, word for instance in http://www.springerlink.com/content/t59071u33573168v/ )

When hearing period without context, I think an integer is expected. Although a word is definitely more useful in papers and you have the integer by taking just the length of the word.

Is something like that stupid?

sage: Word('1231').period(word=True)
word: 123
sage: Word('1231').period()
3

What do you think?

I think an alias is not necessary. There is no other period, and anyone looking for that notion will use "period" anyway.

[seblabbe]

comment:4

I think an alias is not necessary. There is no other period, and anyone looking for that notion will use "period" anyway.

If 3 is a period, than 6, 9, 12 are period as well. But I agree that 3 is the period.

I would rather suggest the following :

sage: Word('1231').period_word()
word: 123
sage: Word('1231').period()
3

[seblabbe]

The goal

sage: Word('1231').exponent()
4/3
sage: Word('1231').exponent(integer=True)
1

I would prefer the argument to speak about the use or not of the fraction :

sage: Word('1231').exponent() # default
1
sage: Word('1231').exponent(allow_fraction=True)
4/3

or

sage: Word('1231').exponent(allow_fraction=False)
1
sage: Word('1231').exponent() # default
4/3