ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 19 Sep 2011 04:55:05 -0500Can I create these sequences with Sage?http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/Let me start by confessing that I know next to nothing about math and really don't know exactly how to describe what I'm trying here; I hope the below description is adequate.
I am attempting to create a list of all possible 7-digit sequences of the numbers 1, 2, and 3 that add up to 12 (e.g., 1-2-2-3-1-1-2, 2-1-2-3-1-1-2, 2-1-2-3-1-2-1, etc.). I'm not looking merely for the number of sequences, but the actual sequences themselves. Is this something that Sage is capable of and, if so, how would I go about accomplishing this? If not any suggestions?
Thank you in advance.
GregSun, 18 Sep 2011 10:53:09 -0500http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/Answer by benjaminfjones for <p>Let me start by confessing that I know next to nothing about math and really don't know exactly how to describe what I'm trying here; I hope the below description is adequate.</p>
<p>I am attempting to create a list of all possible 7-digit sequences of the numbers 1, 2, and 3 that add up to 12 (e.g., 1-2-2-3-1-1-2, 2-1-2-3-1-1-2, 2-1-2-3-1-2-1, etc.). I'm not looking merely for the number of sequences, but the actual sequences themselves. Is this something that Sage is capable of and, if so, how would I go about accomplishing this? If not any suggestions?</p>
<p>Thank you in advance.</p>
<p>Greg</p>
http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/?answer=12657#post-id-12657There is a marvelous combinatorial class called `IntegerListsLex` that returns an iterator (a type of python object that you can iterate over) for lists of integers subject to certain constraints. Try entering `IntegerListsLex?` at the Sage command line to see the documentation.
Below is an iterator that does the job. The `length` parameter limits to sequences of length 7, the `floor` parameter limits the integers to be larger or equal to 1, the `ceiling` parameter limits the integers to be less or equal to 3.
sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[3]*7)
sage: for l in L:
....: print l
....:
[3, 3, 2, 1, 1, 1, 1]
[3, 3, 1, 2, 1, 1, 1]
[3, 3, 1, 1, 2, 1, 1]
[3, 3, 1, 1, 1, 2, 1]
etc...
The `Lex` part of the name indicates that the lists are returned in lexicographic order.
Note that if you sort the sequences, convert to tuples (so that they are hashable), and then uniquify them you get the 3 basic sequences that generate the whole collection by reordering:
sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[3]*7)
sage: LS = [ tuple(sorted(i)) for i in L ]
sage: uniq(LS)
[(1, 1, 1, 1, 2, 3, 3), (1, 1, 1, 2, 2, 2, 3), (1, 1, 2, 2, 2, 2, 2)]
-----------------------------------------------------------------------------
*Update* changed `ceiling` option to limit the sequence to 1's, 2's, and 3's
Sun, 18 Sep 2011 11:08:42 -0500http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/?answer=12657#post-id-12657Comment by Grigori for <p>There is a marvelous combinatorial class called <code>IntegerListsLex</code> that returns an iterator (a type of python object that you can iterate over) for lists of integers subject to certain constraints. Try entering <code>IntegerListsLex?</code> at the Sage command line to see the documentation. </p>
<p>Below is an iterator that does the job. The <code>length</code> parameter limits to sequences of length 7, the <code>floor</code> parameter limits the integers to be larger or equal to 1, the <code>ceiling</code> parameter limits the integers to be less or equal to 3.</p>
<pre><code>sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[3]*7)
sage: for l in L:
....: print l
....:
[3, 3, 2, 1, 1, 1, 1]
[3, 3, 1, 2, 1, 1, 1]
[3, 3, 1, 1, 2, 1, 1]
[3, 3, 1, 1, 1, 2, 1]
</code></pre>
<p>etc...</p>
<p>The <code>Lex</code> part of the name indicates that the lists are returned in lexicographic order.</p>
<p>Note that if you sort the sequences, convert to tuples (so that they are hashable), and then uniquify them you get the 3 basic sequences that generate the whole collection by reordering:</p>
<pre><code>sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[3]*7)
sage: LS = [ tuple(sorted(i)) for i in L ]
sage: uniq(LS)
[(1, 1, 1, 1, 2, 3, 3), (1, 1, 1, 2, 2, 2, 3), (1, 1, 2, 2, 2, 2, 2)]
</code></pre>
<hr/>
<p><em>Update</em> changed <code>ceiling</code> option to limit the sequence to 1's, 2's, and 3's</p>
http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/?comment=21242#post-id-21242Thanks so much for the info, Benjamin. I have Sage installed and was able to generate the lists. Looking at the result, starting with [6, 1, 1, 1, 1, 1, 1], I think I should have been clearer in my description. I want to limit the numbers used to 1, 2, and 3 (i.e., no 4, 5, 6, etc.), THEN create all possible orderings of those numbers. The results I generated include 4, 5, and 6, and are not limited to 1, 2, and 3.
I did a little background work on this, and it looks as though there are basically three series that will fulfill my criteria:
[1+1+1+1+2+3+3],
[1+1+1+2+2+2+3], and
[1+1+2+2+2+2+2] (there could be another I've missed, perhaps?).
Having those basic series (I apologize if my terminology is confusing), I want to generate ALL POSSIBLE ORDERINGS of each. For example, for the first series - the one with two 3's - there is the original sequence (1+1+1+1+2+3+3), but those could be reordered as (1+1+1+1+3+2+3), (1+1+1+1+3+3+2), (2+1+1+1+1+3+3), etc,. So, what I want to generate are all possible orderings of those three sequences.
Is this possible within Sage?
Again, thanks for the assistance. Sun, 18 Sep 2011 12:57:45 -0500http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/?comment=21242#post-id-21242Comment by Grigori for <p>There is a marvelous combinatorial class called <code>IntegerListsLex</code> that returns an iterator (a type of python object that you can iterate over) for lists of integers subject to certain constraints. Try entering <code>IntegerListsLex?</code> at the Sage command line to see the documentation. </p>
<p>Below is an iterator that does the job. The <code>length</code> parameter limits to sequences of length 7, the <code>floor</code> parameter limits the integers to be larger or equal to 1, the <code>ceiling</code> parameter limits the integers to be less or equal to 3.</p>
<pre><code>sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[3]*7)
sage: for l in L:
....: print l
....:
[3, 3, 2, 1, 1, 1, 1]
[3, 3, 1, 2, 1, 1, 1]
[3, 3, 1, 1, 2, 1, 1]
[3, 3, 1, 1, 1, 2, 1]
</code></pre>
<p>etc...</p>
<p>The <code>Lex</code> part of the name indicates that the lists are returned in lexicographic order.</p>
<p>Note that if you sort the sequences, convert to tuples (so that they are hashable), and then uniquify them you get the 3 basic sequences that generate the whole collection by reordering:</p>
<pre><code>sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[3]*7)
sage: LS = [ tuple(sorted(i)) for i in L ]
sage: uniq(LS)
[(1, 1, 1, 1, 2, 3, 3), (1, 1, 1, 2, 2, 2, 3), (1, 1, 2, 2, 2, 2, 2)]
</code></pre>
<hr/>
<p><em>Update</em> changed <code>ceiling</code> option to limit the sequence to 1's, 2's, and 3's</p>
http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/?comment=21240#post-id-21240That seems to have done the trick; thanks so much, Benjamin!Sun, 18 Sep 2011 18:15:08 -0500http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/?comment=21240#post-id-21240Comment by benjaminfjones for <p>There is a marvelous combinatorial class called <code>IntegerListsLex</code> that returns an iterator (a type of python object that you can iterate over) for lists of integers subject to certain constraints. Try entering <code>IntegerListsLex?</code> at the Sage command line to see the documentation. </p>
<p>Below is an iterator that does the job. The <code>length</code> parameter limits to sequences of length 7, the <code>floor</code> parameter limits the integers to be larger or equal to 1, the <code>ceiling</code> parameter limits the integers to be less or equal to 3.</p>
<pre><code>sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[3]*7)
sage: for l in L:
....: print l
....:
[3, 3, 2, 1, 1, 1, 1]
[3, 3, 1, 2, 1, 1, 1]
[3, 3, 1, 1, 2, 1, 1]
[3, 3, 1, 1, 1, 2, 1]
</code></pre>
<p>etc...</p>
<p>The <code>Lex</code> part of the name indicates that the lists are returned in lexicographic order.</p>
<p>Note that if you sort the sequences, convert to tuples (so that they are hashable), and then uniquify them you get the 3 basic sequences that generate the whole collection by reordering:</p>
<pre><code>sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[3]*7)
sage: LS = [ tuple(sorted(i)) for i in L ]
sage: uniq(LS)
[(1, 1, 1, 1, 2, 3, 3), (1, 1, 1, 2, 2, 2, 3), (1, 1, 2, 2, 2, 2, 2)]
</code></pre>
<hr/>
<p><em>Update</em> changed <code>ceiling</code> option to limit the sequence to 1's, 2's, and 3's</p>
http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/?comment=21241#post-id-21241Sure, it's an easy change. I somehow missed that part of your question. I've updated the answer now.Sun, 18 Sep 2011 16:53:45 -0500http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/?comment=21241#post-id-21241Answer by John Palmieri for <p>Let me start by confessing that I know next to nothing about math and really don't know exactly how to describe what I'm trying here; I hope the below description is adequate.</p>
<p>I am attempting to create a list of all possible 7-digit sequences of the numbers 1, 2, and 3 that add up to 12 (e.g., 1-2-2-3-1-1-2, 2-1-2-3-1-1-2, 2-1-2-3-1-2-1, etc.). I'm not looking merely for the number of sequences, but the actual sequences themselves. Is this something that Sage is capable of and, if so, how would I go about accomplishing this? If not any suggestions?</p>
<p>Thank you in advance.</p>
<p>Greg</p>
http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/?answer=12658#post-id-12658A *composition* of a nonnegative integer `n` is an ordered list of positive integers with total sum `n`, so you can call `Compositions`, specifying the maximum term:
sage: P = Compositions(4, max_part=2)
sage: P.list()
[[2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]
sage: list(P)
[[2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]
sage: list(Compositions(12, length=7, max_part=3))
[[3, 3, 2, 1, 1, 1, 1], [3, 3, 1, 2, 1, 1, 1], [3, 3, 1, 1, 2, 1, 1], ...
Mon, 19 Sep 2011 04:55:05 -0500http://ask.sagemath.org/question/8330/can-i-create-these-sequences-with-sage/?answer=12658#post-id-12658