Permutation from a pair of standard tableaux

[seblabbe]

1. In sage 3.4, the Robinson Schensted algorithm is coded for a permutation :

sage: p = Permutation([3, 6, 5, 2, 7, 4, 1])
sage: p.robinson_schensted()
[[[1, 4, 7], [2, 5], [3], [6]], [[1, 2, 5], [3, 6], [4], [7]]]

Since this algorithm is invertible, it would be nice to allow to construct a permutation from a pair of standard tableaux of the same shape.

2. The Robinson-Schensted is broken on the empty permutation. It should simply return a pair of empty tableaux :

sage: p=Permutation([])
sage: p.robinson_schensted()
Traceback (most recent call last):
...
ValueError: invalid tableau

CC: @sagetrac-sage-combinat

Component: combinatorics

Keywords: robinson schensted

Issue created by migration from https://trac.sagemath.org/ticket/5551

[hivert]

comment:3

Dear Sebastien,

It's good to have this ! Thanks. There are three little problems:

1. The documentation says:

-  a pair of two standard tableaux of the same shape.  
   The right tableau must be over the integers 1 to n,  
   where n is its size.

As far as I know a tableau with entries from 1 to n is what is called a standard tableau. When there are repeated entries the usual terminology is semi standard tableaux or young tableaux. See eg: http://en.wikipedia.org/wiki/Young_diagram

2. why not call it robinson_schented_inv ?

[seblabbe]

comment:4

Dear Florent,

Thanks for your quick answer,

Replying to @hivert:

  Dear Sebastien,

It's good to have this ! Thanks.

Cool!

There are three little problems:

Did you forget the third one or is this a joke meaning you are of the second type of mathematician?

1. The documentation says:

-  a pair of two standard tableaux of the same shape.  
   The right tableau must be over the integers 1 to n,  
   where n is its size.

As far as I know a tableau with entries from 1 to n is what is called a standard tableau.

And increasing in row and column. Right. I agree. I should remove the second sentence above.

So this leads me to a related question. From a permutation, Robinson-Schensted Algo gives a pair of standard tableaux :

sage: p = Permutation([3, 6, 5, 2, 7, 4, 1])
sage: p.robinson_schensted()
[[[1, 4, 7], [2, 5], [3], [6]], [[1, 2, 5], [3, 6], [4], [7]]]

But from the following "non bijective" permutation, we obtain a pair of tableaux (p,q) where p is semi-standard and q is standard. Well, we can say more about p : there are no repeated entry, only possible weigth 1 and 0.

sage: p = Permutation([3, 6, 5, 2, 117, 4, 1])
sage: p.robinson_schensted()
[[[1, 4, 117], [2, 5], [3], [6]], [[1, 2, 5], [3, 6], [4], [7]]]
sage: t1,t2 = _
sage: t1.weight()
[1, 1, 1, 1, 1, 1, 0, 0, 0, ..., 0, 0, 1]
sage: t2.weight()
[1, 1, 1, 1, 1, 1, 1]

Should from_tableaux handle the above pair of tableaux? Actually it does :

sage: p = Permutation([3, 6, 5, 2, 117, 4, 1]) ; p
[3, 6, 5, 2, 117, 4, 1]
sage: import sage.combinat.permutation as permutation
sage: p.robinson_schensted()
[[[1, 4, 117], [2, 5], [3], [6]], [[1, 2, 5], [3, 6], [4], [7]]]
sage: permutation.from_tableaux(*_)
[3, 6, 5, 2, 117, 4, 1]

Then, what should the input of from_tableaux say? Should it say simply a pair of standard tableaux as robinson_shensted doc string says it returns a pair of standard tableaux? Should it say that it handles a pair (p, q) of tableaux where p is semi-standard (weigth 0 or 1) and q is standard?

When there are repeated entries the usual terminology is semi standard tableaux or young
tableaux. See eg: http://en.wikipedia.org/wiki/Young_diagram

2. why not call it robinson_schented_inv ?

There is a section in permutation.py containing functions constructing a permutation from different objects :

#############################
# Constructing Permutations #
#############################
def from_permutation_group_element(pge):
def from_rank(n, rank):
def from_inversion_vector(iv):
def from_cycles(n, cycles):
def from_lehmer_code(lehmer):
def from_reduced_word(rw):

The name from_tableaux was then natural. Also, I coded the function for a colleague used to mathematica and he told me in mathematica they use TableauxToPermutation and PermutationToTableaux. I don't mind change it to robinson_schensted_inv or robinson_schensted_inverse(). Do I?

[hivert]

comment:5

Did you forget the third one or is this a joke meaning you are of the second type of mathematician?

Both !!! Every people doing combinatorics should know that he is of the second type ! Do you know how to count the number of twin prime numbers ? If not you are of the second type !

Then, what should the input of from_tableaux say? Should it say simply a pair of standard tableaux as robinson_shensted doc string says it returns a pair of standard tableaux? Should it say that it handles a pair (p, q) of tableaux where p is semi-standard (weigth 0 or 1) and q is standard?

Obviously you never tried the following one:

sage: p = Permutation([1,3,2,2,4,3])
sage: p.robinson_schensted()
[[[1, 2, 2, 3], [3, 4]], [[1, 2, 4, 5], [3, 6]]]
sage: permutation.from_tableaux(*_)
[1, 3, 2, 2, 4, 3]

Yes !!! Do it !!! Try It !!!

Does it answer your question ?

There is a section in permutation.py containing functions constructing a permutation from different objects :
[...]
The name from_tableaux was then natural. Also, I coded the function for a colleague used to mathematica and he told me in mathematica they use TableauxToPermutation and PermutationToTableaux. I don't mind change it to robinson_schensted_inv or robinson_schensted_inverse(). Do I?

In MuPAD we had schensted and schenstedInv. I'd rather have either robinson_schensted_inv}}/{{{inverse() of from_tableaux_pair.
I think you should ask for a vote on sage-combinat-devel.

Finally, my third remark is that you should raise a ValueError with a proper explanation of what is wrong if the two tableaux are not of the same shape:

sage: permutation.from_tableaux(Tableau([[1,2,3]]), Tableau([[1,2]]))
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)

/home/averell/.sage/temp/tomahawk/7383/_home_averell__sage_init_sage_0.py in <module>()

/usr/local/sage/sage/local/lib/python2.5/site-packages/sage/combinat/permutation.pyc in from_tableaux(p, q)
   3112     p = map(list, p)
   3113     for n in range(size, 0, -1):
-> 3114         i,j = d[n]
   3115         x = p[i][j]
   3116         del p[i][j]

KeyError: 3

Cheers,

Florent

[seblabbe]

Attachment: permutation_from_tableaux-5551-submitted-sl.patch.gz

Against sage 3.4. This patch is part of sage-combinat tree.

[seblabbe]

comment:6

I improved the patch after Florent's comments. I just uploaded it.

slabbe

[hivert]

Attachment: permutation_from_tableaux-5551-review-fh.patch.gz