#13077: generalised Tamari lattices
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Reporter: chapoton | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: minor | Milestone: sage-5.4
Component: combinatorics | Resolution:
Keywords: poset | Work issues:
Report Upstream: N/A | Reviewers:
Authors: Frédéric Chapoton | Merged in:
Dependencies: | Stopgaps:
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Comment (by hthomas):
Salut Frédéric--
A couple of thoughts. Maybe paths_in_triangle should take a parameter
"slope" instead of a,b? Alternatively, I think the code should check that
(i,j) is actually in the rectangle between (0,0) and (a,b).
In the definition of "swap", the code as written requires m to be an
integer, but why do it like that? For arbitrary relatively prime a,b, I
think a/b is a natural choice of parameter. This matches the "rational
Catalan combinatorics" (see the two sets of Rational Catalan Combinatorics
slides at Drew Armstrong's website
http://www.math.miami.edu/~armstrong/research.html). One downside to the
further generality is that I don't know whether or not it's a lattice.
But I don't think that's really an argument against implementing it.
Maybe an appropriate analogue of Bousquet-Mélou--Fusy--Préville-Ratelle,
arXiv:1106.1498, Proposition 4 holds? But a glance, it wasn't obvious to
me that it would.
I also think it would be good to include a reference for the generalized
Tamari lattice in the documentation (and/or a more complete definition).
cheers,
Hugh
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13077#comment:7>
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