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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 BicpTq 6 Shard intersection order of BicpTq 1 Describing ψpBq 2 Describing ψpCq X ψpDq 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 =

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Broad Overview

We obtain the following diagram of posets defined from T:

Bic(T) Ψ(Bic(T)) − → FG(T) Ψ(− → FG(T)) ψ ψ φ ? NCP(T) ∼

Goal: Understand the combinatorics of NCPpTq

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Noncrossing tree partitions: Overview

The poset NCPpTq is called the noncrossing tree partitions of T. In this part of the talk, we will discuss our research of the following properties of NCPpTq:

1 NCPpTq is a lattice 2 NCPpTq is graded (conjecture) 3 NCPpTq is not self-dual 4 How to count the maximal chains in NCPpTq Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 5 / 48

What is NCPpTq?

For a tree T, a segment s “ pv0, . . . , vtq “ rv0, vts with t ě 1 is a sequence of interior vertices of T that takes a “sharp” turn at each vi. In particular, the interior vertices of T are not segments.

Example

In the tree below, p1, 5q and p2, 4, 6q are segments. The sequence p1, 3q is not a segment.

2 3 4 5 6 1

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A noncrossing partition B “ pB1, . . . , Bkq is a set partition of the interior vertices of T where the vertices in Bi 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

• f such curves can only agree at their endpoints, and

red admissible curves connecting vertices of Bi do not cross those of Bj for i ‰ j. We let NCPpTq denote the poset of noncrossing tree partitions ordered by refinement.

Example

B “ tt1, 4, 6u, t2, 3u, t5uu is an element of NCPpTq.

2 3 4 5 6 1

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Theorem (Garver-McConville)

The poset NCPpTq is a lattice.

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Lattice Theory

Before we talk about the structural properties of NCPpTq, 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:

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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.

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Lattice Theory

L a lattice λ a CU-labeling of L x ∈ L

Ψ(L)

Shard intersection order

ΨpLq consists of sets ψpxq “ tlabels appearing between

k

ľ

i“1

yi and xu where tyiuk

i“1 is the set of elements immediately below x in L. The

partial ordering on ΨpLq is inclusion. We call the interval rŹk

i“1 yi, xs

the facial interval corresponding to x.

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Back to NCPpTq

Theorem (Garver-McConville)

For a tree T, NCPpTq is isomorphic to ΨpÝ Ý Ñ FGpTqq. This brings us to one of the main objects in our project:

Conjecture

The lattice NCPpTq is graded by the number of blocks in a partition.

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Conjecture

The lattice NCPpTq is graded by the number of blocks in a partition. How we want to prove this conjecture: Show that every covering relation in NCPpTq is given by merging two blocks of a partition (which is what happens with NCpnq). 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

• FGpTq. We know that B

corresponds to a facial interval in Ý Ý Ñ

• FGpTq. We want to show that it

is contained in a facial interval “one dimension lower” than the entire lattice.

B ÞÑ ψpxq „ ra, xs Ĺ ra1, x1s Ĺ Ý Ý Ñ FGpTq

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Corollaries of conjecture and further structure

Garver and McConville defined a bijection NCPpTq called the Kreweras Complement. The Kreweras complement sends a partition with m blocks to a partition with #V opTq ` 1 ´ m blocks. A corollary

• f this map and the previous conjecture is the following:

Corollary

The lattice NCPpTq is rank-symmetric. The above property is shared by NCpnq. A natural question to ask is: How many of the nice properties of NCpnq carry over to NCPpTq? We provide a partial answer here:

Theorem

In general, NCPpTq is not self-dual.

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We conclude our discussion of NCPpTq with a method of calculating the number of maximal chains, denoted mcpTq. We will exploit the following fact in order to obtain recursions: let taiun

i“1 be the set of coatoms of NCPpTq; then

mcpTq “

n

ÿ

i“1

mcprˆ 0, aisq. From here, we can note that rˆ 0, ais is isomorphic to the product of two noncrossing tree partitions of smaller trees, as shown by the following picture:

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2 3 4 5 6 1

ai =

We have that rˆ 0, ais – NCPpT1q ˆ NCPpT2q, where

T1 = T2 =

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Using this method, we can count the maximal chains of the kth star-graph, denoted Sk, which is the family of trees of the following form:

S5

k = # of edges attached to central vertex

We get that mcpSkq “ k!Fk`1

2

, where Fk`1 is the pk ` 1qth fibonacci number.

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Segments S1, S2 P SegpTq whose composition is also in SegpTq are composable.

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A subset B Ă SegpTq is closed if for any composable S1, S2 P B, we have S1 9 S2 P B.

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A subset B Ă SegpTq is closed if for any composable S1, S2 P B, we have S1 9 S2 P B. A subset B Ă SegpTq is biclosed if both B and BC are closed.

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A subset B Ă SegpTq is closed if for any composable S1, S2 P B, we have S1 9 S2 P B. A subset B Ă SegpTq is biclosed if both B and BC are closed. BicpTq is a poset who elements are biclosed sets B Ă SegpTq, partially

• rdered by inclusion.

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We will explicitly demonstrate the CU-labeling for BicpTq.

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For a segment ra, cs with vertex b in between, we say that ra, bs and rb, cs constitute a break of ra, cs

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For a segment ra, cs with vertex b in between, we say that ra, bs and rb, cs constitute a break of ra, cs Each of ra, bs and rb, cs is a split of ra, cs corresponding to that break.

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Recall that a CU-labeling is a map λ : tcovering relations of BicpTqu Ñ P for some poset P of labels.

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Recall that a CU-labeling is a map λ : tcovering relations of BicpTqu Ñ P for some poset P of labels. We choose P with elements of the form S∆ where S P SegpTq and ∆ is a set of splits of S. The partial ordering is given by S∆ ě Qµ if S contains Q.

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Covering relations in BicpTq look like:

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Covering relations in BicpTq look like:

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Example:

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Example:

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Example:

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Example:

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Example:

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Example: Covering relations look like

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Example: Covering relations look like

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Example: Covering relations look like

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Example: Covering relations look like

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ΨpBicpTqq has a maximum element. If we can show that for all C, D P BicpTq, there exists some B P BicpTq such that ψpCq X ψpDq “ ψpBq, then we can conclude that ΨpBicpTqq is a lattice.

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Elements of ψpBq are those of the form S∆ where S is a composition of some of S1, S2, . . . , Sm and ∆ is a set of splits of S with certain stipulations.

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Vertices within S which are endpoints of some Si correspond to faultline breaks. Other vertices correspond to non-faultline breaks.

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Vertices within S which are endpoints of some Si correspond to faultline breaks. Other vertices correspond to non-faultline breaks.

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What splits of S are in ∆?

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What splits of S are in ∆? For non-faultline breaks, these are predetermined by ∆1, ∆2, . . . , ∆m.

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What splits of S are in ∆? For non-faultline breaks, these are predetermined by ∆1, ∆2, . . . , ∆m. For each faultline break, there is an independent choice.

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Labels in ψpCq X ψpDq are of the form S∆ where:

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Labels in ψpCq X ψpDq are of the form S∆ where: S must simultaneously be a composition of Si’s and Qi’s.

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Labels in ψpCq X ψpDq are of the form S∆ where: S must simultaneously be a composition of Si’s and Qi’s. Furthermore, the splits determined by the corresponding ∆i’s and µi’s must be compatible.

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S must simultaneously be a composition of Si’s and Qi’s:

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The only breaks for which there is a choice of what split of S to include in ∆ are when the break is a faultline for S viewed as a composition of Si’s and S viewed as a composition of Qi’s.

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The only breaks for which there is a choice of what split of S to include in ∆ are when the break is a faultline for S viewed as a composition of Si’s and S viewed as a composition of Qi’s.

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The only breaks for which there is a choice of what split of S to include in ∆ are when the break is a faultline for S viewed as a composition of Si’s and S viewed as a composition of Qi’s.

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Call an element of ψpCq X ψpDq pseudominimal if it does not contain any double faultlines in its composition pair.

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Call an element of ψpCq X ψpDq pseudominimal if it does not contain any double faultlines in its composition pair. Pseudominimal elements generate all of ψpCq X ψpDq in the sense that any S∆ P ψpCq X ψpDq satisfies

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Call an element of ψpCq X ψpDq pseudominimal if it does not contain any double faultlines in its composition pair. Pseudominimal elements generate all of ψpCq X ψpDq in the sense that any S∆ P ψpCq X ψpDq satisfies

1 S is a composition of the segment parts of pseudominimal labels. Alexander Clifton and Peter Dillery Shards and noncrossing tree partitions August 4, 2016 43 / 48

Call an element of ψpCq X ψpDq pseudominimal if it does not contain any double faultlines in its composition pair. Pseudominimal elements generate all of ψpCq X ψpDq in the sense that any S∆ P ψpCq X ψpDq satisfies

1 S is a composition of the segment parts of pseudominimal labels. 2 The only choices for which splits of S to include in ∆ occur at

breaks where two such pseudominimal lables are joined together.

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Pseudominimal elements of ψpCq X ψpDq generate ψpCq X ψpDq the same way ψpBq is generated by the labels on its covering relations. We can conceivably take B “ _t pseudominimal elements of ψpCq X ψpDqu to obtain ψpBq “ ψpCq X ψpDq.

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For the star graph Sk: |NCPpSkq| “

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For the star graph Sk: |NCPpSkq| “2|NCPpSk´1q| ` |NCPpSk´2q|

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For the star graph Sk: |NCPpSkq| “2|NCPpSk´1q| ` |NCPpSk´2q| with |NCPpS3q| “ 14, |NCPpS4q| “ 34

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Straight trees like are analagous to classical non-crossing partitions.

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Thank you to project mentor Al Garver, project TA Craig Corsi, Thomas McConville, Vic Reiner, the University of Minnesota, and the National Science Foundation.

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References

• A. Garver and T. McConville. Oriented flip graphs and noncrossing

tree partitions. arXiv: 1604.06009, 2016.

• N. Reading. Noncrossing partitions and the shard intersection
• rder. Journal of Algebraic Combinatorics, 2011.
• R. Simion and D. Ullman. On the structure of the lattice of

noncrossing partitions. Discrete Mathematics, 1989.

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