Shards and noncrossing tree partitions Alexander Clifton and Peter - PowerPoint PPT Presentation

Shards and noncrossing tree partitions Alexander Clifton and Peter Dillery August 4, 2016 Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 1 / 48

Outline 1 Broad overview 2 What is a noncrossing tree partition? 3 Lattice theory 4 The structure of noncrossing tree partitions 1 Grading 2 Self-duality 3 Enumerative results 5 Defining a CU-labeling of Bic p T q 6 Shard intersection order of Bic p T q 1 Describing ψ p B q 2 Describing ψ p C q X ψ p D q 3 Putting it all together 7 Further enumerative results Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 2 / 48

Broad Overview Fix a tree T embedded in a disk with exactly its leaves on the boundary and whose interior vertices (the vertices not on the boundary) have degree at least 3. T = Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 3 / 48

Broad Overview We obtain the following diagram of posets defined from T : ψ Bic( T ) Ψ(Bic( T )) φ ? ψ − → Ψ( − → FG ( T ) FG ( T )) ∼ NCP( T ) Goal: Understand the combinatorics of NCP p T q Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 4 / 48

Noncrossing tree partitions: Overview The poset NCP p T q is called the noncrossing tree partitions of T . In this part of the talk, we will discuss our research of the following properties of NCP p T q : 1 NCP p T q is a lattice 2 NCP p T q is graded (conjecture) 3 NCP p T q is not self-dual 4 How to count the maximal chains in NCP p T q Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 5 / 48

What is NCP p T q ? For a tree T , a segment s “ p v 0 , . . . , v t q “ r v 0 , v t s with t ě 1 is a sequence of interior vertices of T that takes a “sharp” turn at each v i . In particular, the interior vertices of T are not segments. Example In the tree below, p 1 , 5 q and p 2 , 4 , 6 q are segments. The sequence p 1 , 3 q is not a segment. 6 1 5 2 4 3 Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 6 / 48

A noncrossing partition B “ p B 1 , . . . , B k q is a set partition of the interior vertices of T where the vertices in B i can be connected by red admissible curves (i.e. curves whose endpoints define segments of T and leave their endpoints to the right), where any pair of such curves can only agree at their endpoints, and red admissible curves connecting vertices of B i do not cross those of B j for i ‰ j . We let NCP p T q denote the poset of noncrossing tree partitions ordered by refinement. Example B “ tt 1 , 4 , 6 u , t 2 , 3 u , t 5 uu is an element of NCP p T q . 6 1 5 2 4 3 Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 7 / 48

Theorem (Garver-McConville) The poset NCP p T q is a lattice. Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 8 / 48

Lattice Theory Before we talk about the structural properties of NCP p T q , we need to discuss the relevant lattice theory. Definition A lattice is called congruence-uniform if it can be constructed from a single point using interval doublings. Here is an example of a lattice constructed from interval doublings: Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 9 / 48

Lattice Theory Theorem A lattice is congruence-uniform if and only if it admits an edge labeling known as a CU-labeling . In fact, the colors on the edges of the picture above form a CU-labeling, where the color set is ordered s ď t if the color s appears before t in the sequence of doublings. Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 10 / 48

Lattice Theory L a lattice Ψ( L ) λ a CU-labeling of L Shard intersection order x ∈ L Ψ p L q consists of sets k ľ ψ p x q “ t labels appearing between y i and x u i “ 1 where t y i u k i “ 1 is the set of elements immediately below x in L . The partial ordering on Ψ p L q is inclusion. We call the interval r Ź k i “ 1 y i , x s the facial interval corresponding to x . Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 11 / 48

Back to NCP p T q Theorem (Garver-McConville) For a tree T , NCP p T q is isomorphic to Ψ pÝ Ý Ñ FG p T qq . This brings us to one of the main objects in our project: Conjecture The lattice NCP p T q is graded by the number of blocks in a partition. Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 12 / 48

Conjecture The lattice NCP p T q is graded by the number of blocks in a partition. How we want to prove this conjecture: Show that every covering relation in NCP p T q is given by merging two blocks of a partition (which is what happens with NC p n q ). To do this, it suffices to show that if we can merge m blocks of B , m ě 3, then we can merge m ´ 1 blocks. To show the above, we work with Ý Ý Ñ FG p T q . We know that B corresponds to a facial interval in Ý Ý Ñ FG p T q . We want to show that it is contained in a facial interval “one dimension lower” than the entire lattice. B ÞÑ ψ p x q „ r a, x s Ĺ r a 1 , x 1 s Ĺ Ý Ý Ñ FG p T q Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 13 / 48

Corollaries of conjecture and further structure Garver and McConville defined a bijection NCP p T q called the Kreweras Complement . The Kreweras complement sends a partition with m blocks to a partition with # V o p T q ` 1 ´ m blocks. A corollary of this map and the previous conjecture is the following: Corollary The lattice NCP p T q is rank-symmetric. The above property is shared by NC p n q . A natural question to ask is: How many of the nice properties of NC p n q carry over to NCP p T q ? We provide a partial answer here: Theorem In general, NCP p T q is not self-dual. Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 14 / 48

We conclude our discussion of NCP p T q with a method of calculating the number of maximal chains, denoted mc p T q . We will exploit the following fact in order to obtain recursions: let t a i u n i “ 1 be the set of coatoms of NCP p T q ; then n ÿ mc pr ˆ mc p T q “ 0 , a i sq . i “ 1 From here, we can note that r ˆ 0 , a i s is isomorphic to the product of two noncrossing tree partitions of smaller trees, as shown by the following picture: Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 15 / 48

6 1 5 a i = 2 4 3 We have that r ˆ 0 , a i s – NCP p T 1 q ˆ NCP p T 2 q , where T 2 = T 1 = Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 16 / 48

Using this method, we can count the maximal chains of the k th star-graph , denoted S k , which is the family of trees of the following form: k = # of edges S 5 attached to central vertex We get that mc p S k q “ k ! F k ` 1 , where F k ` 1 is the p k ` 1 q th fibonacci 2 number. Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 17 / 48

Segments S 1 , S 2 P Seg p T q whose composition is also in Seg p T q are composable . Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 18 / 48

A subset B Ă Seg p T q is closed if for any composable S 1 , S 2 P B , we have S 1 9 S 2 P B . Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 19 / 48

A subset B Ă Seg p T q is closed if for any composable S 1 , S 2 P B , we have S 1 9 S 2 P B . A subset B Ă Seg p T q is biclosed if both B and B C are closed. Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 19 / 48

A subset B Ă Seg p T q is closed if for any composable S 1 , S 2 P B , we have S 1 9 S 2 P B . A subset B Ă Seg p T q is biclosed if both B and B C are closed. Bic p T q is a poset who elements are biclosed sets B Ă Seg p T q , partially ordered by inclusion. Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 19 / 48

We will explicitly demonstrate the CU-labeling for Bic p T q . Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 20 / 48

For a segment r a, c s with vertex b in between, we say that r a, b s and r b, c s constitute a break of r a, c s Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 21 / 48

For a segment r a, c s with vertex b in between, we say that r a, b s and r b, c s constitute a break of r a, c s Each of r a, b s and r b, c s is a split of r a, c s corresponding to that break. Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 21 / 48

Recall that a CU-labeling is a map λ : t covering relations of Bic p T qu Ñ P for some poset P of labels. Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 22 / 48

Recall that a CU-labeling is a map λ : t covering relations of Bic p T qu Ñ P for some poset P of labels. We choose P with elements of the form S ∆ where S P Seg p T q and ∆ is a set of splits of S . The partial ordering is given by S ∆ ě Q µ if S contains Q . Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 22 / 48