I think you want to use RecursivelyEnumeratedSet in sage which consist in applying iteratively a successor function to a set of initial seeds:
sage: v = matrix(4,1,(1,0,0,0))
sage: v.set_immutable()
sage: seed = [v]
sage: T1 = matrix( ZZ, 4, [ -1,0,0,0, 2, 1,0,0, 4,0, 1,0, 4, 0,0, 1 ] )
sage: T2 = matrix( ZZ, 4, [ 1,2,0,0, 0,-1,0,0, 0,4, 1,0, 0, 4,0, 1 ] )
sage: T3 = matrix( ZZ, 4, [ 1,0,1,0, 0, 1,1,0, 0,0,-1,0, 0, 0,0, 1 ] )
sage: T4 = matrix( ZZ, 4, [ 1,6,0,6, 0, 3,0,2, 0,4, 1,4, 0,-4,0,-3 ] )
sage: T5 = matrix( ZZ, 4, [ 3,0,0,2, 6, 1,0,6, 4,0, 1,4, -4, 0,0,-3 ] )
sage: thin_group = [T1, T2, T3, T4, T5]
sage: def succ(u):
....: ans = []
....: for m in thin_group:
....: m_u = m*u
....: m_u.set_immutable()
....: ans.append(m_u)
....: return ans
....:
sage: R = RecursivelyEnumeratedSet(seed, succ, max_depth=4)
sage: R
A recursively enumerated set (breadth first search) with max_depth=4
To get a finite set, you may want to use max_depth as above or hmax as you suggest:
sage: def the_rec_enum_set(hmax):
....: def succ(u):
....: ans = []
....: for m in thin_group:
....: m_u = m*u
....: m_u.set_immutable()
....: if max(map(abs, m_u)) < hmax:
....: ans.append(m_u)
....: return ans
....: R = RecursivelyEnumeratedSet(seed, succ)
....: return R
....:
sage: the_rec_enum_set(20)
A recursively enumerated set (breadth first search)
The recursively enumerated set can be enumerated in different maner (look at the documentation). What I like the most is to create the generated graph:
sage: G = the_rec_enum_set(20).to_digraph(); G
Looped multi-digraph on 10 vertices
sage: G = the_rec_enum_set(50).to_digraph(); G
Looped multi-digraph on 37 vertices
sage: G = the_rec_enum_set(500).to_digraph(); G
Looped multi-digraph on 775 vertices
The above numbers confirm what dan_fulea obtain.
Then, you may want to use my optional package to visualize it (with dot2tex and graphviz installed):
sage -pip install slabbe
sage -i dot2tex
and then:
sage: from slabbe import TikzPicture
sage: G = the_rec_enum_set(20).to_digraph()
sage: tikz = TikzPicture.from_graph(G)
sage: tikz.pdf() # or tikz.png() or tikz.svg()
Looking at the picture, your structure looks to be symmetric (y is a successor of x if and only if x is a successor of y). If so, use
sage: R = RecursivelyEnumeratedSet(seed, succ, structure='symmetric')
which allows to enumerate all elements without keeping all of them in memory. Even better, if your structure is really a tree or a forest (that is if you are able to avoid backward arrows and any loop in the successor function), then use
sage: R = RecursivelyEnumeratedSet(seed, succ, structure='forest')
which will allow you to enumerate the vectors in parallel using fast map reduce algorithms written by Florent Hivert without keeping the elements in memory. In any case, first consult the documentation I have written:
sage: RecursivelyEnumeratedSet?
One information that is missing in the graph is the label of the edges. One way to solve this is to momentarily hide the label information inside the arrival vertex. Once, the graph is made, we put back the information into the edge. I usually do like this:
sage: seed = [(v,None)]
sage: def the_rec_enum_set(hmax):
....: def succ(node):
....: u,label = node
....: ans = []
....: for i,m in zip([1,2,3,4,5], thin_group):
....: m_u = m*u
....: m_u.set_immutable()
....: if max(map(abs, m_u)) < hmax:
....: ans.append((m_u,i))
....: return ans
....: R = RecursivelyEnumeratedSet(seed, succ)
....: return R
....:
sage: R = the_rec_enum_set(20)
sage: G = R.to_digraph()
sage: G
Looped multi-digraph on 22 vertices
sage: from slabbe.graph import digraph_move_label_to_edge
sage: GG = digraph_move_label_to_edge(G)
sage: GG
Looped digraph on 10 vertices
sage: TikzPicture.from_graph(GG).pdf()
I intend to move this feature directly in Sage once I have time. Tell me if you have any other comment, I intend to improve this module in a near future.
