Opened 4 years ago

Closed 4 years ago

## #24560 closed enhancement (fixed)

Reported by: Owned by: chapoton major sage-8.2 combinatorics tscrim, stumpc5, darij Frédéric Chapoton Darij Grinberg N/A 2b61458 2b614588ac014d88bf4592a5a2411a500df54c04

### Description

that are currently somewhat hidden

### comment:1 Changed 4 years ago by chapoton

• Branch set to u/chapoton/24560
• Commit set to 7406b309e697faf79583345463cec6ebe75ad4e4
• Status changed from new to needs_review

New commits:

 ​7406b30 allow easy access to m-Tamari lattices

### comment:2 Changed 4 years ago by darij

    For a general slope m, the elements are Dyck
paths in the (m n+1 \times n)-rectangle.

The elements are :func:Dyck paths<sage.combinat.dyck_word.DyckWord>
in the (a \times b)-rectangle. The order relation depends on m.


Which is true: mn+1, or ab?

Version 0, edited 4 years ago by darij (next)

### comment:3 Changed 4 years ago by chapoton

Well, a=m*n+1 and b=n in the case of interest here

### comment:4 follow-up: ↓ 5 Changed 4 years ago by darij

Oops, I derped. Looks good. But:

• In the class doc of GeneralizedTamariLattice, I'd replace a \geq b \times m by a \geq bm. The \times is misleading, as it makes me think of a rectangle.
• Is the lattice property of a GeneralizedTamariLattice still a conjecture?

### comment:5 in reply to: ↑ 4 Changed 4 years ago by chapoton

Oops, I derped. Looks good. But:

• In the class doc of GeneralizedTamariLattice, I'd replace a \geq b \times m by a \geq bm. The \times is misleading, as it makes me think of a rectangle.

ok

• Is the lattice property of a GeneralizedTamariLattice still a conjecture?

Well, I think so.

### comment:6 Changed 4 years ago by darij

Actually, it seems to follow from https://arxiv.org/pdf/1406.3787.pdf , Theorem 1, doesn't it? (Where v is the Christoffel path, i.e., the (a,b)-Dyck path that keeps closest to the diagonal.)

### comment:7 Changed 4 years ago by chapoton

Of course, the m-Tamari lattices themselves are known to be lattices.

In general no, and this is more subtle. In this sage procedure, the generalisation goes beyond that of Preville-Ratelle and Viennot, because shape and slope become independant parameters.

### comment:8 Changed 4 years ago by git

• Commit changed from 7406b309e697faf79583345463cec6ebe75ad4e4 to 2b614588ac014d88bf4592a5a2411a500df54c04

Branch pushed to git repo; I updated commit sha1. New commits:

 ​2b61458 one detail about times

### comment:9 Changed 4 years ago by darij

Ah, they aren't corner-to-corner paths...

### comment:10 Changed 4 years ago by darij

• Reviewers set to Darij Grinberg
• Status changed from needs_review to positive_review

Thanks

### comment:12 Changed 4 years ago by vbraun

• Branch changed from u/chapoton/24560 to 2b614588ac014d88bf4592a5a2411a500df54c04
• Resolution set to fixed
• Status changed from positive_review to closed
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