# Electrical networks and the Grove algebra

Yibo Gao  Department of Mathematics, University of Michigan, Ann Arbor, MI
48109<EMAIL_ADDRESS>,  Thomas Lam  Department of Mathematics, University
of Michigan, Ann Arbor, MI 48109<EMAIL_ADDRESS>and  Zixuan Xu  Department
of Mat7ematics, Massachusetts Institute of Technology, Cambridge, MA 02139
<EMAIL_ADDRESS>
###### Abstract.

We study the ring of regular functions on the space of planar electrical
networks, which we coin the _grove algebra_. This algebra is an electrical
analogue of the Plücker ring studied classically in invariant theory. We
develop the combinatorics of double groves to study the grove algebra, and
find a quadratic Gröbner basis for the grove ideal.

T.L. was supported by Grant No. DMS-1953852 from the National Science
Foundation.

##  1\. Introduction

Electrical resistor networks are modeled by undirected graphs whose edges are
given positive weights, and have been studied for two centuries. The physical
axioms governing such networks, Ohm’s Law and Kirchhoff’s Law, were discovered
in the 19th century. In the last few decades, planar resistor networks have
been studied from a modern perspective in the works of Curtis–Ingerman–Morrow
[CIM98], de Verdière–Gitler–Vertigan [CdVGV96], Kenyon–Wilson [KW11],
Lam–Pylyavskyy [LP15], Lam [Lam18], Chepuri–George–Speyer [CGS21], and
Bychkov–Gorbounov–Kazakov–Talalaev [BGKT21], and others.

###  1.1. Grove coordinates

In Kirchhoff’s classical work, he gave formulae for the effective resistance
of an electrical network by counting spanning trees. We formalize the tree
enumeration in terms of groves [KW11]. Let $\Gamma$ be a planar electrical
network with $n$ boundary nodes. A _grove_ in $\Gamma$ is a spanning forest
$F$ such that every connected component contains a boundary node. Each grove
has a noncrossing boundary partition $\sigma(F)$.

The grove coordinates $L_{\sigma}(\Gamma)$ count groves in $\Gamma$ with
chosen boundary partition $\sigma$ and determine $\Gamma$ up to electrical
equivalence (series-parallel and $Y-\Delta$ moves). In [Lam18], the second
author constructed a compactification of the space of planar electrical
networks with $n$ boundary vertices, using the grove coordinates
$(L_{\sigma})$. Furthermore, an embedding of this compactification into the
Grassmannian $\operatorname{Gr}(n-1,2n)$ is constructed; we denote the image
of this compactification by $\mathcal{X}_{n,\geq 0}$ and let
$\mathcal{X}_{n}\subset\operatorname{Gr}(n-1,2n)$ denote its Zariski-closure,
an irreducible subvariety of the Grassmannian.

The starting point of our work is the analogy between the Grassmannian
$\operatorname{Gr}(k,n)$ and the variety $\mathcal{X}_{n}$, and between the
Plücker coordinates $\Delta_{I}$ and the grove coordinates $L_{\sigma}$.

$\operatorname{Gr}(k,n)$ | $\mathcal{X}_{n}$
---|---
Totally nonnegative Grassmannian $\operatorname{Gr}(k,n)_{\geq 0}$ | Space of cactus networks $\mathcal{X}_{n,\geq 0}$
Dimer model or plabic graph | Cactus network $\Gamma$
$k$-element subsets of $[n]$ | noncrossing partitions on $[n]$
Plücker coordinates $\Delta_{I}$ | Grove coordinates $L_{\sigma}$
Perfect matchings | Spanning forests
Partial noncrossing matchings $A_{k,n}$ |  $3$-noncrossing matchings $\mathcal{TC}_{n}$
Temperley-Lieb immanants $F_{\tau,T}$ | Bush basis $B_{\xi}$
Double dimers | Double groves
Semistandard tableaux ${\mathbf{T}}$ | Nested Dyck paths ${\mathbf{P}}\in\mathcal{C}^{(d)}_{n}$
Standard monomial basis $\Delta_{{\mathbf{T}}}$ | Standard grove basis $L_{\mathbf{P}}$
Dual canonical basis $H({\mathbf{T}})$ | Electrical canonical basis $E_{\mathbf{P}}$ ??
Bounded affine permutations | Matchings on $[2n]$
Positroid variety $\Pi_{f}$ | Electroid variety $\mathcal{X}_{\tau}$
Figure 1. Parallels between $\mathcal{X}_{n}$ and $\operatorname{Gr}(k,n)$

###  1.2. Plücker algebra and Grove algebra

The Plücker algebra is the homogeneous coordinate ring
$R(k,n):={\mathbb{C}}[\operatorname{Gr}(k,n)]$ of the Grassmannian. It can be
identified with the quotient of the polynomial ring ${\mathbb{C}}[\Delta_{I}]$
generated by the Plücker coordinates $\Delta_{I}$, labeled by $k$-element
subsets $I\subset\binom{[n]}{k}$, modulo the Plücker ideal generated by the
Plücker relations. The Plücker algebra $R(k,n)$ is also isomorphic to the ring
of $\operatorname{SL}(k)$-invariants in the polynomial functions on a $k\times
n$ matrix. In this latter setting, the description of $R(k,n)$ in terms of
Plücker coordinates and Plücker relations are known as the first and second
fundamental theorems of invariant theory.

We recall the following classical theorem regarding the basis for the Plücker
ideal and the Plücker ring; see [SW89].

######  Theorem 1.1.

  1. (1)

The Plücker relations form a quadratic Gröbner basis for the Plücker ideal.

  2. (2)

The degree $d$ homogeneous piece $R(k,n)_{d}$ of the Plücker ring has basis
the set of standard monomials
$\Delta_{\mathbf{T}}=\Delta_{T_{1}}\Delta_{T_{2}}\cdots\Delta_{T_{d}}$ as
$(T_{1},\ldots,T_{d})$ varies over the columns of a semistandard Young
tableaux ${\mathbf{T}}$ with rectangular shape $k\times d$ and entries in
$\\{1,2,\ldots,n\\}$.

In analogy to the objects involved in Theorem 1.1, we define the _grove
algebra_ $G_{n}:={\mathbb{C}}[\mathcal{X}_{n}]$ to be homogeneous coordinate
ring of $\mathcal{X}_{n}$ and use $G_{d,n}$ denote the degree $d$ homogeneous
piece of $G_{n}$. There is a natural bijection $P\mapsto\sigma(P)$ (see
Section 2.4) from Dyck paths of semilength $2n$ to noncrossing partitions of
$[n]$. We abuse notation by writing $L_{P}$ for $L_{\sigma(P)}$. Let
$\mathcal{C}^{(d)}_{n}$ denote the set of $d$-tuples of nested Dyck paths (see
Section 2.4). For ${\mathbf{P}}=(P_{1},\ldots,P_{d})\in\mathcal{C}^{(d)}_{n}$,
we define the standard grove monomials

$L_{\mathbf{P}}:=L_{P_{1}}L_{P_{2}}\cdots L_{P_{d}}.$

In this work, we present a result analogous to Theorem 1.1 for the grove
algebra, stated as follows.

######  Theorem 1.2 (Theorem 6.12 and 6.4).

  1. (1)

The grove algebra $G_{n}$ is generated by the grove coordinates $L_{\sigma}$
modulo the ideal $\mathfrak{I}_{n}$ generated by the quadratic relations
$r_{P,Q}$ described in 6.9. The relations $r_{P,Q}$ give a Gröbner basis of
the ideal $\mathfrak{I}_{n}$, with respect to a term order described in
Section 6.

  2. (2)

We have $\dim(G_{d,n})=\prod_{1\leq i\leq j\leq n-1}\frac{i+j+2d}{i+j}$ and
$G_{d,n}$ has basis the set of standard grove monomials $L_{\mathbf{P}}$,
${\mathbf{P}}\in\mathcal{C}^{(d)}_{n}$.

Chepuri, George, and Speyer [CGS21] and Bychkov, Gorbounov, Kazakov, and
Talalaev [BGKT21] have shown that the variety $\mathcal{X}_{n}$ is isomorphic
to the Lagrangian Grassmannian $\operatorname{LG}(n-1,2n-2)$, extending
earlier work of Lam and Pylyavskyy [LP15] relating electrical networks to the
symplectic group. By Borel-Weil theory, each homogeneous piece $G_{d,n}$ of
$G_{n}$ is isomorphic to an irreducible representation of the symplectic group
$\operatorname{Sp}(2n-2)$; see Section 6.2. The relation between our
description of $G_{d,n}$, including the standard grove basis
$\\{L_{\mathbf{P}}\\}$, and the usual constructions from highest weight theory
of $\operatorname{Sp}(2n-2)$ is far from being clear. It is an interesting
problem to compare these approaches.

###  1.3. Electrical canonical basis

Recall that the standard monomial basis $\Delta_{{\mathbf{T}}}$ is not
compatible with the cyclic symmetry of $\operatorname{Gr}(k,n)$. Lusztig’s
dual canonical basis of $R(k,n)$ is compatible with cyclic symmetry and
exhibits remarkable positivity properties. See [Lam19] for the following
result.

######  Theorem 1.3.

The space $R(k,n)_{d}$ has a dual canonical basis $H({\mathbf{T}})$ with the
following properties:

  1. (1)

For $d=1$, we have $H({\mathbf{T}})=\Delta_{I}$, where $I$ is the set of
entries in the one-column tableau ${\mathbf{T}}$.

  2. (2)

For $d=2$, the set $\\{H({\mathbf{T}})\\}$ is exactly the set of Temperley-
Lieb immanants [Lam15] $\\{F_{\tau,T}\mid(\tau,T)\in A_{k,n}\\}$ which we
describe in Section 5.

  3. (3)

For any semistandard tableau ${\mathbf{T}}$, the function $H({\mathbf{T}})$ is
a nonnegative function on the totally nonnegative Grassmannian
$\operatorname{Gr}(k,n)_{\geq 0}$.

  4. (4)

For any $T$, we have $\chi^{*}(H({\mathbf{T}}))=H(\chi({\mathbf{T}}))$ where
$\chi^{*}$ denotes the pullback map induced by the signed cyclic symmetry
$\chi:\operatorname{Gr}(k,n)\to\operatorname{Gr}(k,n)$, and
$\chi({\mathbf{T}})$ is the _promotion_ of ${\mathbf{T}}$.

In [Lam19] (see also [Lam16]) it is further shown that the dual canonical
basis $H({\mathbf{T}})$ is compatible with restrictions to the homogeneous
coordinate ring of positroid varieties $\Pi_{f}$ [KLS13].

Similar to the standard monomial basis for $\operatorname{Gr}(k,n)$, the
standard grove basis $L_{\mathbf{P}}$ is not compatible with the cyclic
symmetry of planar electrical networks (or that of $\mathcal{X}_{n}$).
Therefore we conjecture that there exists a canonical basis for the space
$G_{d,n}$ analogous to the dual canonical basis $H({\mathbf{T}})$ for
$R(k_{n})_{d}$ in Theorem 1.3.

######  Conjecture 1.4.

The space $G_{d,n}$ has an electrical canonical basis $E_{\mathbf{P}}$,
${\mathbf{P}}\in\mathcal{C}^{(d)}_{n}$ with the following properties:

  1. (1)

For $d=1$, we have $E_{P}=L_{P}$.

  2. (2)

The electrical canonical basis takes nonnegative values on the
compactification of the space of electrical networks $\mathcal{X}_{n,\geq 0}$.

  3. (3)

Any monomial in the grove coordinates $L_{\sigma}$ expands positively into the
electrical canonical basis $E_{P}$.

  4. (4)

There are actions of the cyclic group
$\mathbb{Z}/2n\mathbb{Z}=\langle\chi\rangle$ on $\mathcal{C}^{(d)}_{n}$ and on
$\mathcal{X}_{n}$, preserving $\mathcal{X}_{n,\geq 0}$, such that
$\chi^{*}(E_{\mathbf{P}})=E_{\chi({\mathbf{P}})}$, where $\chi^{*}$ denotes
the pullback on $\mathcal{X}_{n}$ induced by $\chi$.

The action of $\chi$ on $\mathcal{C}^{(d)}_{n}$ is an electrical analogue of
promotion on tableaux. For $d=1$, it corresponds to rotation on noncrossing
matchings, under the bijection between Dyck paths and noncrossing matchings
(Section 2.4).

Lusztig has defined a totally nonnegative part
$\operatorname{LG}(n-1,2n-2)_{\geq 0}$ of $\operatorname{LG}(n-1,2n-2)$; see
[Kar18]. The spaces $\mathcal{X}_{n,\geq 0}$ and
$\operatorname{LG}(n-1,2n-2)_{\geq 0}$ are both “totally positive” versions of
the Lagrangian Grassmannian. However, they are quite different. For example,
they are cell complexes with a different number of cells. We expect the
analogy between $\operatorname{LG}(n-1,2n-2)_{\geq 0}$ and
$\mathcal{X}_{n,\geq 0}$ to parallel the analogy between Lusztig’s dual
canonical basis and the electrical canonical basis.

In [Lam18], a stratification of $\mathcal{X}_{n}$ by electroid varieties
$\mathcal{X}_{\tau}$ is constructed, indexed by matchings $\tau$ on $2n$
points. We conjecture the electrical canonical basis to have the following
properties with respect to $\mathcal{X}_{\tau}$.

######  Conjecture 1.5.

  1. (1)

For a matching $\tau$, the electrical canonical basis elements
$E_{\mathbf{P}}$ that do not restrict to 0 on $\mathcal{X}_{\tau}$ form a
basis of the homogeneous coordinate ring ${\mathbb{C}}[\mathcal{X}_{\tau}]$.

  2. (2)

For a matching $\tau$, if $E_{\mathbf{P}}$ is not identically 0 on
$\mathcal{X}_{\tau}$, then it takes strictly positive values on
$\mathcal{X}_{\tau,>0}=\mathcal{X}_{\tau}\cap\mathcal{X}_{n,\geq 0}$.

###  1.4. Electrical canonical basis in degree two

Using the combinatorics of electrical networks, we construct the electrical
canonical basis in degree two, which we call the _Bush basis_. As a
consequence, we discover some remarkable combinatorial properties of _double
groves_.

There is a bijection between the set $\mathcal{C}^{(2)}_{n}$ of pairs of
nested Dyck paths of semilength $2n$, and the set of $3$-noncrossing matchings
on $2n$ points, defined in Section 3. We construct the Bush basis $B_{\xi}$ of
$G_{2,n}$ labeled by $3$-noncrossing matchings $\xi$, defined by (4.4)

$B_{\xi}(\Gamma):=\sum_{H\subset 2\Gamma}\alpha(H)_{\xi}\operatorname{wt}(H),$

where the summation is over double groves in $\Gamma$, and $\alpha(H)_{\xi}$
are certain nonnegative integers. This definition is compatible with the
natural rotation action on 3-noncrossing matchings. In Theorem 4.8 we give a
positive expansion of the quadratic monomials $L_{\sigma}L_{\sigma^{\prime}}$
into the Bush basis $B_{\xi}$.

The relation between $L_{\sigma}$ and $\Delta_{I}$ is more than an analogy. In
[Lam18], it is shown that the embedding
$\iota:\mathcal{X}_{n}\hookrightarrow\operatorname{Gr}(n-1,2n)$ is given by
the formula

$\iota^{*}(\Delta_{I})=\sum_{I}M_{I\sigma}L_{\sigma}$

where $M_{I\sigma}$ is the concordance matrix defined in [Lam18] which we
recall in Section 6.1. Likewise, we give in Section 5 a positive formula
expanding the Temperley-Lieb immanants $F_{\tau,T}$ in terms of the Bush basis
$B_{\xi}$. This produces a commutative square of positive expansions:

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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
It would also be interesting to carry out our program in the setting of the
Ising model and orthogonal Grassmannian [GP20].

##  2\. Background

In this section, we introduce relevant background, following conventions in
[Lam18].

###  2.1. Electrical networks and cactus networks

A _circular planar electrical network_ , is a finite weighted undirected graph
$\Gamma$ embedded into a disk, with boundary vertices labeled
$\bar{1},\bar{2},\ldots,\bar{n}$ in clockwise order. From now on, we will
simply refer to circular planar electrical networks as _electrical networks_.

_Cactus networks_ are introduced by the second author [Lam18] in order to
study the compactification of the space of electrical networks. Let $S$ be a
circle with boundary vertices $\bar{1},\ldots,\bar{n}$, and let $\zeta$ be a
noncrossing partition of $[\bar{n}]$. Identifying the boundary vertices
according to the parts of $\zeta$ gives a _hollow cactus_ $S_{\zeta}$, which
can be viewed as a union of circles glued at some points. A _cactus_ is
$S_{\zeta}$ together with its interior and a _cactus network_ $\Gamma$ is a
weighted graph embedded into a cactus, which can be intuitively thought of as
an electrical network where boundary vertices are identified in a manner given
by $\zeta$. In particular, an electrical network is a cactus network with
$\zeta=(\bar{1}|\bar{2}|\cdots|\bar{n})$. See Fig. 2 for an example.

$\bar{2}$$\bar{1}$$\bar{7}$$\bar{6}$$\bar{3}$$\bar{5}$$\bar{4}$ Figure 2. An
example of a cactus network for
$\zeta=(\bar{1}|\bar{2}|\bar{3},\bar{5}|\bar{4}|\bar{6}|\bar{7})$

We can define a weight function $\operatorname{wt}$ on the edges in a network
$\Gamma$ where we typically treat each value $\operatorname{wt}(e)$ as an
indeterminate. For a subgraph $F\subset\Gamma$, define

$\operatorname{wt}(F)=\prod_{e\in F}\operatorname{wt}(e).$

We call the underlying unweighted graph of a cactus network a _cactus graph_.

###  2.2. Groves

A _grove_ $F$ on a cactus network $\Gamma$ is a spanning forest, such that
each component is connected to the boundary. The _boundary partition_
$\sigma(F)$ is a set partition of $[\bar{n}]$ that records which boundary
vertices are in the same connected components of $F$. Note that since our
network is planar, $\sigma(F)$ must be a noncrossing partition of $[\bar{n}]$
that coarsens $\zeta$, if $\Gamma$ lies in the cactus $S_{\zeta}$. For a
noncrossing partition $\sigma$, we define the _grove measurement_ as

$L_{\sigma}(\Gamma):=\sum_{F\>|\>\sigma(F)=\sigma}\operatorname{wt}(F)$

where the summation is over all groves with boundary partition $\sigma$.

###  2.3. Medial graph and medial pairing

Given a cactus network $\Gamma$, we define its _medial graph_ $G(\Gamma)$ as
follows. The vertices of $G(\Gamma)$ consist of vertices
$t_{1},t_{2},\ldots,t_{2n}$ on the boundary in clockwise order such that the
orginal boundary vertex $\bar{i}$ lies between $t_{2i-1}$ and $t_{2i}$, and
vertices $t_{e}$’s for each edge $e$ in the network. The edges of $G(\Gamma)$
are joined as follows: we first join $t_{e}$ with $t_{e^{\prime}}$ if $e$ and
$e^{\prime}$ share a vertex of $\Gamma$ and are incident to the same face.
Then for each boundary vertex $\bar{i}$ of $\Gamma$, let the edges incident to
$\bar{i}$ be $e_{1},\ldots,e_{k}$ in counterclockwise order, i.e. $t_{2i-1}$
is closest to $e_{1}$ while $t_{2i}$ is closest to $e_{k}$, and join
$t_{2i-1}$ with $t_{e_{1}}$ and $t_{2i}$ with $t_{e_{k}}$; in the case of
$k=0$, i.e. there are no edges incident to $\bar{i}$ in $\Gamma$, we join
$t_{2i-1}$ with $t_{2i}$ instead. Note that in $G(\Gamma)$, each $t_{e}$ has
degree $4$ for $e\in E(\Gamma)$, and each $t_{i}$ has degree $1$ for
$i\in[2n]$.

We then define the _medial pairing_ $\tau(\Gamma)$ on a medial graph
$G(\Gamma)$. From a boundary vertex $t_{i}$, which has degree $1$ of
$G(\Gamma)$, we trace through edges such that whenever we encounter a degree
$4$ vertex $t_{e}$, we go straight through, ending at another boundary vertex
$t_{j}$. This procedure forms $n$ _strands_ or _wires_ which naturally results
in a matching $\tau(\Gamma)$ of $[2n]$. See Fig. 3 for an example.

$t_{5}$$t_{4}$$\bar{2}$$t_{3}$$t_{2}$$\bar{1}$$t_{1}$$t_{14}$$\bar{7}$$t_{13}$$t_{12}$$\bar{6}$$t_{11}$$t_{10}$$\bar{3}$$\bar{5}$$t_{6}$$t_{9}$$\bar{4}$$t_{7}$$t_{8}$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bar{4}$$\bar{3}$$\bar{2}$$\bar{1}$$\bar{7}$$\bar{6}$$\bar{5}$$t_{8}$$t_{7}$$t_{6}$$t_{5}$$t_{4}$$t_{3}$$t_{2}$$t_{1}$$t_{14}$$t_{13}$$t_{12}$$t_{11}$$t_{10}$$t_{9}$

Figure 3. A cactus network $\Gamma$ with its medial graph $G(\Gamma)$ (left)
and medial pairing
$\tau(\Gamma)=\\{(1,2),(3,11),(4,13),(5,12),(6,8),(7,9),(10,14)\\}$ (right)

For later sections, it might be more convenient to think about medial graphs
and medial pairings to be on a circle instead of on a cactus. To reconstruct
$\Gamma$ from its medial graph $G(\Gamma)$, either drawn on a cactus or drawn
on a circle, we first place a boundary vertex $\bar{i}$ between $t_{2i-1}$ and
$t_{2i}$ for $i\in[n]$. Note that the graph $G$ divides the interior of the
disk or the cactus into regions. We color the regions with black and white so
that adjacent regions have different colors, and that the regions with
boundary vertices $\bar{i}$’s are colored white. Contract the boundary
vertices inside the same regions to form a cactus. Place a vertex in each
white interior region and connect two vertices if the two corresponding
regions in $G$ share a vertex.

A medial graph is _lensless_ if no wire has a self intersection, and no two
wires cross twice, i.e. there are no lens that look like . A cactus network
$\Gamma$ is _critical_ , or _reduced_ , if its medial graph is lensless. In
this case, the number of crossings in the medial pairing $\tau(\Gamma)$ equals
the number of edges in $\Gamma$. We note that for any matching $\tau$ on
$[2n]$, there exists a reduced cactus network $\Gamma$ such that
$\tau(\Gamma)=\tau$, by drawing $\tau$ in a reduced way on a disk, viewing it
as a medial graph and applying the procedure above. More specifically, we
recall the following 2.1 from [Lam18].

######  Proposition 2.1 (Proposition 2.12 of [Lam18]).

We have:

  1. (1)

any matching on $[2n]$ can be realized as a medial pairing of some reduced
cactus graph;

  2. (2)

any two reduced cactus graphs with the same medial pairing can be obtained
from each other via $Y-\Delta$ moves.

Given a cactus network $\Gamma$ with its medial graph $G(\Gamma)$, we define
its _dual network_ $\Gamma^{\vee}$ as follows: we keep only the information of
$G(\Gamma)$, put $\bar{i}$ between $t_{2i}$ and $t_{2i+1}$ in clockwise order
where the indices are taken modulo $2n$, and recover a cactus network
$\Gamma^{\vee}$ via the procedure described above, called the _dual network_
of $\Gamma$. See Fig. 4 for an example. If we take the dual twice,
$(\Gamma^{\vee})^{\vee}$ will be $\Gamma$ with indices shifted by $1$ in the
clockwise order.

$\bullet$$\bullet$$\bullet$$\bullet$$\bar{2}$$\bar{4}$$\bar{1}$$\bar{3}$$t_{3}$$t_{2}$$t_{1}$$t_{8}$$t_{7}$$t_{6}$$t_{5}$$t_{4}$$\bullet$$\bullet$$\bullet$$\bullet$$\bar{1}$$\bar{2}$$\bar{4}$$\bar{3}$
Figure 4. A network $\Gamma$ and its dual $\Gamma^{\vee}$

The dual graphs $\Gamma$ and $\Gamma^{\vee}$ have the same number of edges.
Furthermore, each edge of $\Gamma$ intersects a unique edge of $\Gamma^{\vee}$
(and vice-versa), inducing a natural bijection between the edges of $\Gamma$
and of $\Gamma^{\vee}$. Each pair of intersecting edges corresponds to an
intersection in the medial graph $G(\Gamma)$.

###  2.4. Catalan objects and their bijections

We need to use all of the following objects: Dyck paths, noncrossing matchings
and noncrossing partitions, all of which are enumerated by the Catalan numbers
$C_{n}$. For easier reference, we summarize the notations we use in the
following:

  * -

$\mathcal{C}_{n}$:= the set of Dyck paths of semilength $n$.

  * -

$\mathcal{M}_{n}$:= the set of matchings on $1,2,\ldots,2n$.

  * -

$\mathcal{NM}_{n}$:= the set of noncrossing matchings on $1,2,\ldots,2n$.

  * -

$\mathcal{TC}_{n}$:= the set of $3$-noncrossing matchings on $1,2,\ldots,2n$.

  * -

$\mathcal{NP}_{n}$:= the set of noncrossing partitions of
$[\bar{n}]=\\{\bar{1},\bar{2},\dots,\bar{n}\\}$.

Now we start our discussion on these Catalan objects.

######  Definition 2.2.

A Dyck path of semilength $n$ is a walk from $(0,0)$ to $(2n,0)$ consisting of
$n$ upsteps $(1,1)$ and $n$ downsteps $(1,-1)$ that stays above the $x$-axis.

We denote the set of Dyck paths of semilength $n$ by $\mathcal{C}_{n}$. There
is a natural poset structure on Dyck paths: for two Dyck paths of semilength
$n$, we say that $P\leq Q$ if the path $P$ stays weakly below $Q$. We can also
represent a Dyck path $P$ by a vector $v(P)\in\\{0,1\\}^{2n}$ such that
$v(P)_{j}=1$ if the $j$th step of $P$ is an upstep and $v(P)_{j}=0$ if the
$j$th step of $P$ is a downstep. Then there are always as many $1$’s as $0$’s
within $v(P)_{1},\ldots,v(P)_{j}$ for all $j=1,\ldots,2n$. We write
$P\preccurlyeq Q$ if $v(P)$ is lexicographically (weakly) smaller than $v(Q)$.
Note that $\preccurlyeq$ is a total order and that if $P\leq Q$ then
$P\preccurlyeq Q$.

For a positive integer $k$, let

$\mathcal{C}_{n}^{(k)}=\\{P_{1}\leq P_{2}\leq\cdots\leq
P_{k}\\}\subset\mathcal{C}_{n}^{k}$

be the set of chains of length $k$ in the poset on Dyck paths of semilength
$n$. In particular, $\mathcal{C}_{n}^{(1)}=\mathcal{C}_{n}$ and
$\mathcal{C}_{n}^{(2)}$ is the set of pairs of comparable Dyck paths $(P\leq
Q)$.

Another important family of combinatorial objects that we need is _matchings_.
Denote the set of matchings on $1,2,\ldots,2n$ by $\mathcal{M}_{n}$. For a
pair $(i,j)$ in a matching $M$ with $i<j$, we say that $i$ is the _left
endpoint_ and $j$ is the _right endpoint_. Let $\mathrm{cr}(M)$ denote the set
of crossings of $M$, given by

$\mathrm{cr}(M)=\\{(a,b,c,d)\>|\>a<b<c<d,\ (a,c),(b,d)\in M\\}.$

A _strand diagram_ of a matching $M\in\mathcal{M}_{n}$ is a drawing of $M$ by
$n$ arcs, called _strands_ or _wires_ , either on a disk or a straight line
such that no two strands cross twice. We consider the strand diagrams up to
homotopy so for example, if $M=\\{(1,2),(3,4),\ldots,(2n-1,2n)\\}$, then there
is a unique strand diagram of $M$.

######  Definition 2.3.

A matching $M$ of $1,2,\ldots,2n$ is called $k$-noncrossing if there do not
exist $k$ pairs in $M$ that cross pairwise. Denote the set of $2$-noncrossing
matchings, i.e. noncrossing matchings by $\mathcal{NM}_{n}$ and the set of
$3$-noncrossing matchings by $\mathcal{TC}_{n}$.

We now describe a bijection between Dyck paths $\mathcal{C}_{n}$ and
noncrossing matchings $\mathcal{NM}_{n}$. For a Dyck path
$P\in\mathcal{C}_{n}$, label its steps (edges) by $1,2,\ldots,2n$ from left to
right and pair up the upsteps with downsteps of the same height with no steps
in between to obtain a noncrossing matching $M\in\mathcal{NM}_{n}$. For the
inverse, given a noncrossing matching $M$, construct a Dyck path $P$ so that
the $i$-th step is a upstep if $i$ is a left endpoint in $M$, and a downstep
if $i$ is a right endpoint in $M$. See Fig. 5 for an example.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$

Figure 5. Example of a Dyck path of semilength 5 and its corresponding
noncrossing matching under the bijection described above

Th cover relations in the poset structure on Dyck paths can be described as
follows: $P\lessdot P^{\prime}$ if $P^{\prime}$ can be obtained from $P$ by
changing an upstep followed by a downstep, to a downstep followed by an
upstep. The poset structure on noncrossing matchings is inherited from that on
Dyck paths. And the cover relations can be described similarly: $M\lessdot
M^{\prime}$ if we can change two pairs $(a,b)$ and $(b+1,c)$ in $M$ to $(a,c)$
and $(b,b+1)$ in $M^{\prime}$, where $a<b<b+1<c$.

Our third Catalan object is the set of noncrossing partitions.

######  Definition 2.4.

A partition $\sigma$ of $[\bar{n}]=\\{\bar{1},\bar{2},\ldots,\bar{n}\\}$ is
called _noncrossing_ if there do not exist $a<b<c<d$ such that $\bar{a}$ and
$\bar{c}$ are in one part of $\sigma$ while $\bar{b}$ and $\bar{d}$ are in
another. Denote the set of noncrossing partitions as $\mathcal{NP}_{n}$.

The natural bijections between $\mathcal{NP}_{n}$ and $\mathcal{NM}_{n}$ and
$\mathcal{C}_{n}$ are also easy to describe.

We first start by describing the bijection between $\mathcal{NM}_{n}$ and
$\mathcal{NP}_{n}$. Let $M\in\mathcal{NM}_{n}$ be a noncrossing matching on
$[2n]$. We draw the strand diagram $M$ either on a circle or on a straight
line, dividing into smaller regions, put $\bar{i}$ between $2i-1$ and $2i$,
for $i=1,\ldots,n$. We obtain a noncrossing partition $\sigma$ of $[\bar{n}]$
whose parts consist of the points $\bar{i}$ in the same region. The inverse
bijection from $\mathcal{NP}_{n}$ to $\mathcal{NM}_{n}$ is more conveniently
described inductively. Let $\sigma$ be a noncrossing partition, then $\sigma$
must contain a part of the form $(a+1,\ldots,a+k)$ for some $a\geq 0$ and
$k\geq 1$. We remove this part and add a partial matching by connecting $2a+1$
with $2a+2k$, $2a+2$ with $2a+3$, $2a+4$ with $2a+5$ and so on up to $2a+2k-2$
with $2a+2k-1$. We repeat this process until we obtain a noncrossing matching.

Now we describe the bijection between $\mathcal{C}_{n}$ and
$\mathcal{NP}_{n}$. Let $P\in\mathcal{C}_{n}$ be a Dyck path. There are $2n+1$
lattice points in $P$, including $(0,0)$ and $(2n,0)$. Label the $2i$-th
lattice point in $P$ by $\bar{i}$, for $i=1,\ldots,n$, so that $(1,1)$ is
labeled by $\bar{1}$. Then we identify $\bar{i}$ and $\bar{j}$ to be in the
same part of the noncrossing partition $\sigma$ if they are on the same
horizontal line, not separated by the Dyck path $P$. To go the other way, we
apply the following recursive procedure. Let $\sigma$ be a noncrossing
partition and let the part containing $1$ be
$\\{1=a_{0}<a_{1}<\cdots<a_{k}=m\\}$ with $k\geq 0$. This means that each part
of $\sigma$ either has all entries at most $m$, or has all entries at least
$m+1$. We deal with these two kinds separately and attach the two resulting
Dyck paths at coordinate $(2m,0)$. Thus assume $m=n$. Similarly, all other
parts of $\sigma$ have their entries strictly between $a_{i}$ and $a_{i+1}$,
and let $P^{(i)}$ be the Dyck path obtained from such $i$, after adjusting the
entries from $\\{a_{i}+1,\ldots,a_{i+1}-1\\}$ to
$\\{1,\ldots,a_{i+1}-a_{i}-1\\}$. Attach each $P^{(i)}$ from $(2a_{i}+2,2)$ to
$(2a_{i+1}-2,2)$ and we have the desired Dyck path $P\in\mathcal{C}_{n}$.

See Fig. 6 for an illustration of the bijections between
$\mathcal{NP}_{n},\mathcal{NM}_{n}$ and $\mathcal{C}_{n}$.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bar{1}$$\bar{2}$$\bar{3}$$\bar{4}$$\bar{5}$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bar{1}$$\bar{2}$$\bar{3}$$\bar{4}$$\bar{5}$

Figure 6. The noncrossing matching (left) and the Dyck path (right) in
bijection with the noncrossing partition
$\sigma=(\bar{1}\bar{2}|\bar{3}\bar{4}\bar{5})$

For a noncrossing partition $\sigma$ on $[\bar{n}]$, one has a _dual
noncrossing_ partition $\tilde{\sigma}$ on $[\tilde{n}]$ obtained by drawing
$\sigma$ on a disk, connecting boundary vertices in the same parts, putting
$\tilde{i}$ between $\bar{i}$ and $\overline{i{+}1}$ and identifying the
partition $\tilde{\sigma}$ as the regions. See Fig. 7. We also often denote
$\sigma$ as a partition on $1,3,\ldots,2n-1$ and $\tilde{\sigma}$ as a
partition on $2,4,\ldots,2n$.

$\bar{1}$$\bar{2}$$\bar{3}$$\bar{4}$$\bar{5}$$\bar{6}$$\tilde{1}$$\tilde{2}$$\tilde{3}$$\tilde{4}$$\tilde{5}$$\tilde{6}$
Figure 7. A noncrossing partition
$\sigma=(\bar{1}|\bar{2}\bar{3}\bar{6}|\bar{4}\bar{5})$ with its dual
$\tilde{\sigma}=(\tilde{1}\tilde{6}|\tilde{2}|\tilde{3}\tilde{5}|\tilde{4})$

It is easy to obtain that $|\sigma|+|\tilde{\sigma}|=n+1$, where $|\sigma|$ is
the number of parts of $\sigma$ (Lemma 2.2 of [Lam18]).

As for notations, we typically use $P,Q$ for Dyck paths, $\tau,M,\xi$ for
matchings, and $\sigma$ for noncrossing partitions. Also, denote these
bijections by $\sigma:\mathcal{C}_{n}\rightarrow\mathcal{NP}_{n}$,
$\sigma:\mathcal{NM}_{n}\rightarrow\mathcal{NP}_{n}$,
$\tau:\mathcal{C}_{n}\rightarrow\mathcal{NM}_{n}$,
$\tau:\mathcal{NP}_{n}\rightarrow\mathcal{NM}_{n}$.

##  3\. Combinatorics of $3$-noncrossing matchings

###  3.1. 3-noncrossing matchings and pairs of comparable Dyck paths

In this section, we describe a bijection between $3$-noncrossing matchings
$\mathcal{TC}_{n}$ of $n$ and pairs of comparable Dyck paths
$\mathcal{C}_{n}^{(2)}$ in two languages, one formulated as in [CDD+07] and
the other one to be used later.

We first formulate the construction in Chen et al. [CDD+07]. For a matching
$M\in\mathcal{M}_{n}$, associate a sequence of standard Young tableaux
$T_{0},T_{1},\ldots,T_{2n}$, starting with $T_{2n}=\emptyset$, and for $j<2n$,

  * •

if $j$ is the right end point of a pair $(i,j)$ in $M$, insert $i$ into
$T_{j+1}$ to obtain $T_{j}$, in the sense of row insertion in RSK (see for
example [Sta12]);

  * •

if $j$ is the left end point of a pair $(j,k)$ in $M$, delete $j$ from
$T_{j+1}$ to obtain $T_{j}$.

For a $k$-noncrossing matching $M$, $k\geq 2$, Chen et al. [CDD+07] showed
that the tableaux $T_{0},\ldots,T_{2n}$ contain at most $k-1$ rows. Thus, for
$M\in\mathcal{TC}_{n}$, let the shape of the corresponding standard Young
tableau $T_{j}$ be $(x_{j}\geq y_{j})$. Note that the $y_{j}$’s can be $0$.
Define the map $\varphi:\mathcal{TC}_{n}\to\mathcal{C}_{n}^{(2)}$ as
$\varphi(M):=(P_{1}\leq P_{2})$, where $P_{1}$ is the Dyck path that passes
through the coordinates $\\{(j,x_{j}-y_{j})\\}_{j=0}^{2n}$ and $P_{2}$ is the
Dyck path that passes through the coordinates
$\\{(j,x_{j}+y_{j})\\}_{j=0}^{2n}$. See Fig. 8 for an example.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

$\emptyset$$1$$12$$12$$3$$1$$3$$15$$3$$156$$3$$15$$3$$35$$3$$\emptyset$

Figure 8. A bijection between $\mathcal{TC}_{n}$ and $\mathcal{C}^{(2)}_{n}$
by Chen et al. [CDD+07], with the sequence of standard Young tableau
$T_{0},T_{1},\ldots,T_{2n}$ shown below.

######  Theorem 3.1 (Corollary 5.4 of [CDD+07]).

The map $\varphi$ defined above is a bijection between $\mathcal{TC}_{n}$ and
$\mathcal{C}^{(2)}_{n}$.

We give another description of the bijection $\varphi$ via resolutions of
crossings. Recall that the set of crossings of $M\in\mathcal{M}_{n}$ is
denoted $\mathrm{cr}(M)$. For a vector $v\in\\{0,1\\}^{\mathrm{cr}(M)}$,
define the resolution of crossings of $M$ with respect to $v$ to be the
noncrossing matching obtained from a planar drawing of $M$ by resolving each
crossing $x=(a,b,c,d)\in\mathrm{cr}(M)$ locally by connecting the strand $a$
to $b$ and $c$ to $d$ if $v_{x}=0$, or connecting $a$ to $d$ and $b$ to $c$ if
$v_{x}=1$. Denote this resolution by $M(v)$, which is a noncrossing matching.
We note that in the planar drawing of such a resolution, it is possible for a
connected component of strands that do not connect to the vertices
$1,2,\ldots,2n$ to appear, and we temporarily allow such resolutions. Details
regarding this issue will be discussed in Section 4. Let $\mathbf{0}$ denote
the all $0$ vector and $\mathbf{1}$ denote the all $1$ vector. We are
particularly interested in the resolution of crossings $M(\mathbf{0})$ and
$M(\mathbf{1})$.

Examples of resolutions of crossings are shown in Fig. 9.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$M(\mathbf{0})$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$M(\mathbf{1})$

Figure 9. Examples for resolution of crossings with $M$ shown in Fig. 9.

######  Theorem 3.2.

Let $M\in\mathcal{TC}_{n}$. Then we have
$\varphi(M)=(M(\mathbf{0}),M(\mathbf{1}))$, where $M(\mathbf{0})$ and
$M(\mathbf{1})$ are identified with Dyck paths via the bijection
$\tau^{-1}:\mathcal{NM}_{n}\to\mathcal{C}_{n}$.

###### Proof.

We first recall the bijection between noncrossing matchings $\mathcal{NM}_{n}$
and Dyck paths $\mathcal{C}_{n}$. A noncrossing matching
$M\in\mathcal{NM}_{n}$ is in bijection with a Dyck path $P\in\mathcal{C}_{n}$
if for all $j=1,\ldots,2n$, $j$ is a left endpoint in $M$ if and only if the
$j$-th step is an upstep in $P$.

For a $3$-noncrossing matching $M\in\mathcal{TC}_{n}$, write
$\varphi(M)=(P_{0}\leq P_{1})$ for notation purposes of this proof. Let
$T_{0},\ldots,T_{2n}$ be the sequence of tableaux corresponding to $M$ as
defined above in this section. By construction, $T_{j}$ has one more entry
than $T_{j-1}$ if and only if $j$ is a left endpoint in $M$. At the same time,
we see from the construction of $M(\mathbf{1})$ that $M(\mathbf{1})$ and $M$
have the same set of left endpoints and right endpoints (see Fig. 9 for a
reference). Recall that the height of $P_{1}$ is given by the sizes of
$T_{j}$’s, and by the bijection between noncrossing matchings and Dyck paths,
we directly see that $P_{1}=M(\mathbf{1})$.

We use induction on $n$ and on $|\mathrm{cr}(M)|$ to show that $P_{0}$ and
$M(\mathbf{0})$ are in bijection under $\tau$. The base case $n=1$ is trivial.
The other base case is when $\mathrm{cr}(M)=\emptyset$, i.e. $M$ is
noncrossing. Then the tableau $T_{0},\ldots,T_{2n}$ all consist of a single
row, so we can see that $P_{0}=P_{1}$ is indeed in bijection with
$M(\mathbf{0})=M(\mathbf{1})=M$ under $\tau$.

For the inductive step, as $1$ is a left endpoint in $M$ and $2n$ is a right
endpoint in $M$, there must exist a $j\in\\{1,\ldots,2n-1\\}$ such that $j$ is
a left endpoint and $j+1$ is a right endpoint in $M$. There are two possible
cases: either $(j,j+1)$ are paired in the matching $M$, or they are not paired
in $M$. In the first case where $(j,j+1)$ are paired $M$, let
$M^{\prime}\in\mathcal{TC}_{n-1}$ be obtained from $M$ by deleting the pair
$(j,j+1)$ from $M$ and flattening other entries. By the induction hypothesis,
we have $\varphi(M^{\prime})=(P_{0}^{\prime}\leq P_{1}^{\prime})$ where
$P_{0}^{\prime}$ is in bijection with $M^{\prime}(\mathbf{0})$. We know that
$M^{\prime}(\mathbf{0})$ is obtained from $M(\mathbf{0})$ by removing the pair
$(j,j+1)$, so to show that $P_{0}$ and $M(\mathbf{0})$ are in bijection given
that $P_{0}^{\prime}$ and $M^{\prime}(\mathbf{0})$ are in bijection, it
suffices to show that $P_{0}^{\prime}$ is obtained from $P_{0}$ by deleting an
upstep at $j$ and a downstep at $j+1$. By definition of $T_{0},\ldots,T_{2n}$,
$T_{j+1}$ is obtained from $T_{j+2}$ by inserting $j$, which is strictly
larger than other values in $T_{j+2}$ so $j$ gets inserted as the rightmost
entry of the first row in $T_{j+1}$. Then $T_{j}$ is obtained from $T_{j+1}$
by deleting $j$ from $T_{j+1}$, so $T_{j}=T_{j+2}$. This means that step $j$
of $P_{0}$ is an upstep and step $j+1$ is a downstep. Moreover, the sequence
of tableaux for $M^{\prime}$ is
$T_{0},\ldots,T_{j}=T_{j+2},T_{j+3},\ldots,T_{2n}$ so we conclude that
$P_{0}^{\prime}$ is obtained from $P_{0}$ by deleting step $j$ and $j+1$ and
we are done.

In the second case where $(j,j+1)$ are not paired in the matching $M$. Suppose
that $j$ is paired with $b$ and $j+1$ is paired with $a$ in $M$, where $b>j+1$
and $a<j$. Let $M^{\prime}$ be obtained from $M$ by swapping the roles of $j$
and $j+1$, i.e. $M^{\prime}$ is obtained from $M$ by replacing the pair
$(a,j+1)$ and $(j,b)$ with $(a,j)$ and $(j+1,b)$. We show that
$M^{\prime}\in\mathcal{TC}_{n}$. If there exist three strands $A,B,C$ in
$M^{\prime}$ that intersect pairwise, then at least one of them is either
$(a,j)$ or $(j+1,b)$. Say $A=(a,j)$. Then as $(j+1,b)$ does not intersect with
$(a,j)$, $(j+1,b)$ does not belong to one of these strands. This means that
$(a,j+1)$, $B$ and $C$ also intersect pairwise in $M$, contradicting
$M\in\mathcal{TC}_{n}$. Let $T_{0}^{\prime},\ldots,T_{2n}^{\prime}$ be the
sequence of tableau associated to $M^{\prime}$ and let
$\varphi(M^{\prime})=(P_{0}^{\prime}\leq P_{1}^{\prime})$. Let the shape of
$T_{i}$ be $(x_{i},y_{i})$ and the shape of $T_{i}^{\prime}$ be
$(x_{i}^{\prime},y_{i}^{\prime})$ for $i=1,\ldots,2n$. The definition of
$M(\mathbf{0})$ gives $M(\mathbf{0})=M^{\prime}(\mathbf{0})$, as $M^{\prime}$
can be viewed as resolving one crossing from $M$. As
$|cr(M^{\prime})|<|cr(M)|$, by induction hypothesis, $P_{0}^{\prime}$ and
$M^{\prime}(\mathbf{0})$ are in bijection. To show that $P_{0}$ and
$M(\mathbf{0})$ are in bijection, it suffices to show that
$P_{0}=P_{0}^{\prime}$, i.e. $x_{i}-y_{i}=x_{i}^{\prime}-y_{i}^{\prime}$ for
$i=1,\ldots,2n.$

It is clear that $T_{i}=T_{i}^{\prime}$ if $i>b$, and that $T_{i}$ has the
same shape as $T_{i}^{\prime}$ if $j+2\leq i\leq b$, while replacing the value
$j+1$ in $T_{i}^{\prime}$ by $j$. For simplicity, we use $\overline{j}$ to
mean $j$ in the tableaux $T_{i}$, and $j+1$ in $T_{i}^{\prime}$, for
$i=1,\ldots,2n$. We see that $\overline{j}$ is the largest value in $T_{j+2}$
and $T_{j+2}^{\prime}$. Now, $T_{j+1}$ is obtained from $T_{j+2}$ by inserting
$a$ and $T_{j}$ is obtained from $T_{j+1}$ by removing $\overline{j}$, while
$T_{j+1}^{\prime}$ is obtained from $T_{j+2}^{\prime}$ by removing
$\overline{j}$ and $T_{j}^{\prime}$ is obtained from $T_{j+1}^{\prime}$ by
inserting $a$. If $\overline{j}$ is in the second row of $T_{j+2}$ and
$T_{j+2}^{\prime}$, then $a$ must be inserted in the first row of $T_{j+2}$,
since otherwise, insertion of $a$ will bump some value $c<\overline{j}$ to the
second row of $T_{j+2}$, resulting in another bump that creates a third row,
contradicting $T_{j+1}$ having at most $2$ rows. In this case,
$(x_{j+1},y_{j+1})=(x_{j+2}+1,y_{j+2})$ while
$(x_{j+1}^{\prime},y_{j+1}^{\prime})=(x_{j+2},y_{j+2}-1)$, satisfying
$x_{j+1}-y_{j+1}=x_{j+1}^{\prime}-y_{j+1}^{\prime}$. Moreover, removing
$\overline{j}$ and inserting $a$ commute on $T_{j+2}$ and $T_{j+2}^{\prime}$,
so $T_{i}=T_{i}^{\prime}$ for $i\leq j$. If $\overline{j}$ is in the first row
of $T_{j+2}$ and $T_{j+2}^{\prime}$, inserting $a$ will increase the length of
the second row, so $(x_{j+1},y_{j+1})=(x_{j+2},y_{j+2}+1)$ while
$(x_{j+1}^{\prime},y_{j+1}^{\prime})=(x_{j+2}-1,y_{j+2})$, satisfying
$x_{j+1}-y_{j+1}=x_{j+1}^{\prime}-y_{j+1}^{\prime}$. Moreover, whether $a$
bumps $\overline{j}$ to the second row or $a$ bumps some other value to the
second row, we direcly see that removing $\overline{j}$ and inserting $a$
commute. As a result, $x_{i}-y_{i}=x_{i}^{\prime}-y_{i}^{\prime}$ in all
cases, concluding our proof. ∎

######  Proposition 3.3.

For $M\in\mathcal{TC}_{n}$ and $v\in\\{0,1\\}^{\mathrm{cr}(M)}$, $M(v)\leq
M(\mathbf{1})$, with equality if and only if $v=\mathbf{1}$.

###### Proof.

For a matching $M\in\mathcal{TC}_{n}$ and $k=1,\ldots,2n$, let $l_{k}(M)$
denote the number of left endpoints among $1,2,\ldots,k$ in $M$, and let
$r_{k}(M)$ denote the number of right endpoints among $1,2,\ldots,k$. Recall
that the corresponding Dyck path for $M$ is the path that passes through
$(k,l_{k}(M)-r_{k}(M))$ for $k=0,\ldots,2n$. Thus, to show $M^{\prime}\leq M$,
it suffices to show that $l_{k}(M^{\prime})\leq l_{k}(M)$ for all $k$.

Consider a planar drawing of $M$ inside the rectangle $[1,2n]\times[0,1]$,
such that for a pair $(a,b)$ in $M$ where $a<b$, we connect the coordinate
$(a,0)$ to $(b,0)$ via a curve $\gamma_{a,b}:[0,1]\rightarrow[a,b]\times[0,1]$
such that $\gamma_{a,b}(0)=(a,0)$, $\gamma_{a,b}(1)=(b,0)$ and that the
projection of $\gamma_{a,b}$ onto the first coordinate is bijective, i.e. we
can view this curve as going from left to right. The matching shown in Fig. 8
is such an example. We view $M(v)$ as modifying and regrouping the curves in a
small neighborhood around each crossing in $M$ (see the top of Fig. 9). Note
that the crossings of $M$ decompose the curves into segments. For each
crossing $z$, there are four segments of curves touching $z$ in the planar
drawing of $M$, from the direction of NW, SW, NE and SE.

For $k=1,\ldots,2n$, the vertical line $x=k+0.5$ intersects the strands in the
drawing of $M$ with a left endpoint in $1,\ldots,k$ and a right endpoint in
$k+1,\ldots,n$. There are $l_{k}(M)-r_{k}(M)$ number of such strands. Let
$\xi_{1},\ldots,\xi_{h}$ be the segments that intersect $x=k+0.5$, where
$h=l_{k}(M)-r_{k}(M)$. Now, in any resolution of crossings $M(v)$, there are
$l_{k}(M(v))-r_{k}(M(v))$ strands with a left endpoint in $1,\ldots,k$ and a
right endpoint in $k+1,\ldots,n$. Each such strand needs to contain at least
one segment of $\xi_{1},\ldots,\xi_{h}$, meaning that
$l_{k}(M(v))-r_{k}(M(v))\leq h$ so $l_{k}(M(v))\leq
l_{k}(M)=l_{k}(M(\mathbf{1}))$. This means $M(v)\leq M(\mathbf{1})$ as
desired. ∎

###  3.2. $3$-noncrossing matchings and cactus networks

######  Proposition 3.4.

Let $\tau\in\mathcal{M}_{n}$ be a matching. The followings are equivalent:

  1. (1)

$\tau\in\mathcal{TC}_{n}$ is a $3$-noncrossing matching;

  2. (2)

all interior regions of (any) strand diagram of $\tau$ have at least $4$
sides;

  3. (3)

$\tau$ has a unique strand diagram;

  4. (4)

there is a reduced cactus network $\Gamma$ whose medial pairing is $\tau$ such
that $\Gamma$ does not have any triangular faces and all interior vertices of
$\Gamma$ have degree at least $4$.

###### Proof.

We prove (1)$\Rightarrow$(4)$\Rightarrow$(3)$\Rightarrow$(2)$\Rightarrow$(1).

(1)$\Rightarrow$(4). Assume $\tau$ is a $3$-noncrossing matching. Take any
strand diagram / medial graph of $\tau$ and recover a reduced cactus network
$\Gamma$ as described in Section 2. As $\Gamma$ is reduced, no interior vertex
has degree $2$. If $\Gamma$ has a triangular face with edges
$e_{1},e_{2},e_{3}$, then this medial graph $G(\tau)$ has a triangular face
given by $t_{e_{1}},t_{e_{2}},t_{e_{3}}$; if $\Gamma$ has an interior vertex
of degree $3$, then it comes from a triangular face in the strand diagram.
Both cases imply that there exists a triangular face in the strand diagram of
$\tau$, which corresponds to three strands that cross pairwise, contradicting
$\tau$ being $3$-noncrossing.

(4)$\Rightarrow$(3). If $\tau$ has two (or more) strand diagrams, then any
corresponding reduced cactus network admits a $Y-\Delta$ move, by 2.1. But
$\Gamma$ does not have any interior vertex of degree $3$ or a triangular face.
This is a contradiction.

(3)$\Rightarrow$(2). If any strand diagram of $\tau$ has a region with $3$
sides, i.e. a triangular face, then we can apply a local move of the form:
$\leftrightarrow$ . This implies that $\tau$ has more than one strand diagram.

(2)$\Rightarrow$(1). If $\tau$ is not $3$-noncrossing, then in a strand
diagram of $\tau$, there are three strands that cross pairwise, forming an
interior triangle. This triangle may not be a face, but any other strand that
passes through this triangle creates a smaller triangle. Thus any strand
diagram of $\tau$ contains an interior triangular face. ∎

For a $3$-noncrossing matching $\xi\in\mathcal{TC}$, let $\Gamma(\xi)$ be the
corresponding reduced cactus network, which is unique by 3.4.

##  4\. The Grove algebra $G_{n}$ and the Bush basis $\\{B_{\xi}\\}$ for
$G_{n,2}$

Let
$\mathfrak{I}_{n}\subset\mathbb{Z}[L_{\sigma}\>|\>\sigma\in\mathcal{NP}_{n}]$
be the ideal of polynomials in the variables
$\\{L_{\sigma}\\}_{\sigma\in\mathcal{NP}_{n}}$ that vanish on all electrical
networks $\Gamma$, or equivalently, on all cactus networks $\Gamma$. Define
the grove algebra (over $\mathbb{Z}$) to be

$G_{n}:=\mathbb{Z}[L_{\sigma}\>|\>\sigma\in\mathcal{NP}_{n}]/\mathfrak{I}_{n}$

which is a graded algebra by the degree of the polynomials, where we assign
each variable $L_{\sigma}$ to have degree $1$. Let the $d$-th graded piece of
$G_{n}$ be $G_{n,d}$. We will describe $G_{n}\otimes{\mathbb{C}}$ in more
algebro-geometric terms in Section 6.1.

In this section, we introduce a basis
$\\{B_{\xi}\>|\>\xi\in\mathcal{TC}_{n}\\}$ of $G_{n,2}$ indexed by
3-noncrossing matchings that exhibits cyclic symmetry. We first define
$B_{\xi}(\Gamma)$ for any cactus network $\Gamma$ and then lift them to
$G_{n,2}$.

###  4.1. Definition of the Bush basis $\\{B_{\xi}\\}$

######  Definition 4.1.

A double grove $H$ on boundary vertices $\bar{1},\bar{2},\ldots,\bar{n}$ is a
cactus graph where each edge has multiplicity $1$ or $2$ (which we call a
double edge).

Let $\mathbb{Z}\mathcal{TC}_{n}$ denote the free module with basis elements
$\mathcal{TC}_{n}$. For a double grove $H$, we define an invariant
$\alpha(H)\in\mathbb{Z}\mathcal{TC}_{n}$ by the following procedure.

######  Definition 4.2.

For a cactus graph $H$ with multiplicity (i.e. edges of $H$ are allowed to
have multiplicities being positive integers), $\alpha(H)$ is defined
recursively as follows:

  1. (0)

If some edge of $H$ has multiplicity $\geq 3$, then $\alpha(H)=0$.

  2. (1)

If all edges of $H$ have multiplicity $1$ and the medial pairing $\tau(H)$ is
a $3$-noncrossing matching $\xi$, i.e. $H$ is reduced and does not have any
interior vertex of degree equal to $3$ or any triangular faces (see 3.4), then
$\alpha(H)=\xi$.

  3. (2)

If $H$ has loops, then $\alpha(H)=0$.

  4. (3)

If $H$ has a double edge $e$, then $\alpha(H)=\alpha(H^{\prime})$, where
$H^{\prime}$ is obtained from $H$ by contracting $e$.

  5. (4)

If $H$ has edges $e_{1},\ldots,e_{r}$ of multiplicity $1$ between the same
pair of vertices, then $\alpha(H)=0$ if $r\geq 3$, and
$\alpha(H)=2\alpha(H^{\prime})$ where $H^{\prime}$ is obtained from $H$ by
contracting $e_{1}$ (and removing $e_{2}$) if $r=2$.

  6. (5)

If $H$ has an interior leaf or an interior vertex with degree $0$, then
$\alpha(H)=0$.

  7. (6)

If $H$ has an interior vertex $v$ of degree $2$ with distinct neighbors, then
$\alpha(H)=2\alpha(H^{\prime})$ where $H^{\prime}$ is obtained from $H$ by
removing $v$.

  8. (7)

If $H$ has an interior vertex $v$ of degree $3$ with distinct neighbors
$v_{1},v_{2},v_{3}$, then
$\alpha(H)=\alpha(H_{1})+\alpha(H_{2})+\alpha(H_{3})$ where $H_{i}$ is
obtained from $H$ by removing $v$ together with the edges incident to it and
adding the edge $(v_{i+1},v_{i+2})$ where the indices are taken modulo $3$.

  9. (8)

If $H$ has a triangular face (not necessarily in the interior) with vertices
$v_{1},v_{2},v_{3}$, then
$\alpha(H)=\alpha(H_{1})+\alpha(H_{2})+\alpha(H_{3})$ where $H_{i}$ is
obtained from $H$ by identifying $v_{i}$ with $v_{i+1}$ and replacing the
three edges by a single edge between $v_{i}=v_{i+1}$ and $v_{i+2}$.

For $\xi\in\mathcal{TC}_{n}$, let $\alpha(H)_{\xi}\in\mathbb{Z}$ denote the
coefficient of $\xi$ in $\alpha(H)$.

A visualization of some moves in 4.2 is shown in Fig. 10.

$\bullet$$\bullet$$\bullet$$\rightarrow\ 2\cdot$Move (4)

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\rightarrow\ 2\cdot$Move (6)

$\bullet$$v_{1}$$\bullet$$\bullet$$v_{3}$$\bullet$$v_{2}$$v_{1}$$\bullet$$\bullet$$v_{3}$$\bullet$$v_{2}$$v_{1}$$\bullet$$\bullet$$v_{3}$$\bullet$$v_{2}$$v_{1}$$\bullet$$\bullet$$v_{3}$$\bullet$$v_{2}$$\rightarrow$$+$$+$Move
(7)

$v_{1}$$\bullet$$\bullet$$v_{3}$$\bullet$$v_{2}$$\bullet$$v_{1}{=}v_{2}$$\bullet$$v_{3}$$\bullet$$v_{1}{=}v_{3}$$\bullet$$v_{2}$$\bullet$$v_{1}$$\bullet$$v_{2}{=}v_{3}$$\rightarrow$$+$$+$Move
(8)

Figure 10. Moves (4), (6), (7) and (8) of 4.2

It is clear that the procedure in 4.2 terminates, as either the number of
vertices or the number of edges strictly decreases after step. However, it is
not immediately clear that $\alpha(H)$ is well-defined, i.e. the result of the
recursion is independent of choices of moves. we show in Lemma 4.3 that indeed
$\alpha(H)$ is well-defined.

######  Lemma 4.3.

For a double grove $H$, $\alpha(H)$ is well-defined. In other words,
$\alpha(H)$ does not depend on the order that we apply the moves in 4.2.

###### Proof.

By Newman’s lemma (also commonly known as the diamond lemma) [New42], it
suffices to show that for a cactus graph $H$ with multiplicities, if there are
two reduction moves $m$ and $m^{\prime}$ that can be applied to $H$ resulting
in $\alpha(H)$ being assigned to $A=\sum c_{i}\alpha(H_{i})$ and
$A^{\prime}=\sum c_{i}^{\prime}\alpha(H_{i}^{\prime})$, then there are
sequences of reduction moves that can be applied to $A$ and $A^{\prime}$
respectively, such that the end results are the same. Depending on which types
of moves $m$ and $m^{\prime}$ are and which edges are involved in these moves,
the number of cases to check is humongous, but most of them are very
straightforward, including the situations where $m$ and $m^{\prime}$ involve
disjoint sets of edges and any combinations of moves (0) - (6). We check a few
critical cases in this proof.

Case 1: $m$ and $m^{\prime}$ are moves (7) applied to vertices $v$ and
$v^{\prime}$ respectively. Notice that if $v$ and $v^{\prime}$ are not
adjacent, then moves (7) at $v$ and $v^{\prime}$ commute. So it suffices to
look at the following local configuration in $H$, and apply the calculation in
Fig. 11, where we apply moves (7), (7), (6) in 4.2 after move $m$. We see that
the same result can be achieved if we apply move $m^{\prime}$ first, as the
end result in Fig. 11 is symmetric horizontally.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$v$$v^{\prime}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$m$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$+$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$+$$\bullet$$\bullet$$\bullet$$\bullet$(7),(7),(6)$\bullet$$\bullet$$\bullet$$\bullet$$+$$\bullet$$\bullet$$\bullet$$\bullet$$+$$\bullet$$\bullet$$\bullet$$\bullet$$+$$\bullet$$\bullet$$\bullet$$\bullet$$+$$\bullet$$\bullet$$\bullet$$\bullet$$+$$\bullet$$\bullet$$\bullet$$\bullet$$+\
2\cdot$$\bullet$$\bullet$$\bullet$$\bullet$$=$$\bullet$$\bullet$$\bullet$$\bullet$$+$$\bullet$$\bullet$$\bullet$$\bullet$$+$$\bullet$$\bullet$$\bullet$$\bullet$$+$$\bullet$$\bullet$$\bullet$$\bullet$$+\
2\cdot$$\bullet$$\bullet$$\bullet$$\bullet$$+\ 2\cdot$ Figure 11. Case 1 of
Lemma 4.3

Case 2: $m$ and $m^{\prime}$ are moves (8) applied to the faces $v,x,y$ and
faces $v^{\prime},x,y$ respectively. We have a similar calculation shown in
Fig. 12. Note that $v,x,y,v^{\prime}$ may be incident to other edges and we
will not draw these edges for simplicity of the visualization. We conclude
that applying $m$ first or $m^{\prime}$ first can result in the same
configuration via symmetry.

$\bullet$$\bullet$$\bullet$$\bullet$$x$$v^{\prime}$$y$$v$$m$$\bullet$$\bullet$$\bullet$$v{=}x$$v^{\prime}$$y$$\bullet$$\bullet$$\bullet$$v{=}y$$v^{\prime}$$x$$+$$\bullet$$\bullet$$\bullet$$v$$x{=}y$$v^{\prime}$$+$(7),(7),(4)$\bullet$$\bullet$$v,x,y$$v^{\prime}$$\bullet$$\bullet$$v,x$$v^{\prime},y$$+$$\bullet$$\bullet$$v,x,v^{\prime}$$y$$+$$\bullet$$\bullet$$v,x,y$$v^{\prime}$$+$$\bullet$$\bullet$$v,y$$v^{\prime},x$$+$$\bullet$$\bullet$$x$$v,y,v^{\prime}$$+$$\bullet$$\bullet$$v$$v^{\prime},x,y$$+\
2\cdot$$\bullet$$\bullet$$v,x$$v^{\prime},y$$=$$\bullet$$\bullet$$v,y$$v^{\prime},x$$+$$\bullet$$\bullet$$v,x,v^{\prime}$$y$$+$$\bullet$$\bullet$$x$$v,y,v^{\prime}$$+$$\bullet$$\bullet$$v^{\prime}$$v,x,y$$+\
2\cdot$$\bullet$$\bullet$$v^{\prime},x,y$$v$$+\ 2\cdot$ Figure 12. Case 2 of
Lemma 4.3

Case 3: $m$ is a move of type (7) applied to a vertex $v$ and $m^{\prime}$ is
a move of type (8) applied to a triangular face $v,x,y$. We consider the
reductions separately for $m$ and $m^{\prime}$ in Fig. 13 and observe the
local confluence as desired. Notice that $v$ has degree $3$, which is adjacent
to the vertices $x,y,z$, while the vertices $x,y,z$ can have more edges
incident to them.

$\bullet$$\bullet$$\bullet$$\bullet$$x$$y$$v$$z$$m$$\bullet$$\bullet$$\bullet$$x$$y$$z$$\bullet$$\bullet$$\bullet$$x$$y$$z$$+$$\bullet$$\bullet$$\bullet$$x$$y$$z$$+$$\bullet$$\bullet$$\bullet$$x$$y$$z$(4)$\bullet$$\bullet$$\bullet$$x$$y$$z$$+$$\bullet$$\bullet$$x,y$$z$$+\
2\cdot$

$\bullet$$\bullet$$\bullet$$\bullet$$x$$y$$v$$z$$m^{\prime}$$\bullet$$\bullet$$\bullet$$x,v$$y$$z$$\bullet$$\bullet$$\bullet$$x$$y,v$$z$$+$$\bullet$$\bullet$$\bullet$$x,y$$v$$z$$+$$\bullet$$\bullet$$\bullet$$x$$y$$z$(6)$\bullet$$\bullet$$\bullet$$x$$y$$z$$+$$\bullet$$\bullet$$x,y$$z$$+\
2\cdot$

Figure 13. Case 3 of Lemma 4.3

The other cases involving some combinations of moves (0) to (6) of 4.2 are
tedious and straightforward to check so we omit them here. ∎

Recall that for an edge-weighted graph $H$ whose edges have multiplicities,
its weight $\operatorname{wt}(H)$ is given by the product of weights on the
edges with multiplicity. For example, if some edge $e$ has multiplicity $2$,
then it contributed $\operatorname{wt}(e)^{2}$ to $\operatorname{wt}(H)$.

######  Definition 4.4.

For $\xi\in\mathcal{TC}_{n}$, define

$B_{\xi}(\Gamma):=\sum_{H\subset 2\Gamma}\alpha(H)_{\xi}\operatorname{wt}(H)$

for any cactus network $\Gamma$, where the sum is over double groves $H$ whose
edges set is contained in that of $\Gamma$ and $\alpha(H)_{\xi}$ is defined in
4.2.

######  Example 4.5.

Consider the electrical network $\Gamma$ shown here with $n=3$ that consists
of an interior vertex $x$ connected to $\bar{1},\bar{2},\bar{3}$, with weights
$a,b,c$ respectively:

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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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Consider $\xi=\\{(1,5),(2,6),(3,4)\\}$, for which we denote as $(15|26|34)$
for simplicity. In order to compute $B_{\xi}(\Gamma)$, we need to sum over all
$H\subset 2\Gamma$, where there are $27$ possibilities. However, notice that
if $H$ has $\leq 1$ edges, then (5) of 4.2 implies $\alpha(H)=0$; and if $H$
has $\geq 5$ edges counting with multiplicity, then after contracting once,
some edge will have multiplicity $\geq 3$, implying that $\alpha(H)=0$ as
well. Here, we list the calculation of $\alpha(H)$, for those $H\subset
2\Gamma$ such that $\alpha(H)_{\xi}\neq 0$.

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{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys<EMAIL_ADDRESS>
This calculation shows that $\alpha(\Gamma)=(15|26|34)+(13|24|56)+(12|35|46)$
so $\alpha(\Gamma)_{\xi}=1$. We also have

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One can check that $\alpha(H)_{\xi}=0$ if $H$ is not the above three choices.
As a result, we compute that $B_{\xi}(\Gamma)=abc+a^{2}c+ac^{2}$.

###  4.2. Expansion of $\\{L_{\sigma}L_{\sigma^{\prime}}\\}$ into
$\\{B_{\xi}\\}$

We now work towards lifting $B_{\xi}$’s to $G_{n,2}$ and showing that they
form a basis of $G_{n,2}$. Recall that for a matching $M\in\mathcal{TC}_{n}$
and a vector $v\in\\{0,1\\}^{\mathrm{cr}(M)}$, $M(v)\in\mathcal{NM}_{n}$ is
the noncrossing matching obtained from resolving the crossings in $M$ in the
way specified by $v$, called a resolution of crossings (see Section 3). We say
that such a resolution $M(v)$ is valid if no internal loops result from the
uncrossings, i.e. all segments are used.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$$M$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$Invalid$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$valid

Two loopless resolutions $M(v)$ and $M(v^{\prime})$ are called opposite if
$v+v^{\prime}=\mathbf{1}$. Note that $v+v^{\prime}=\mathbf{1}$ means that the
choices of uncrossings at every intersection in $M$ for $M(v)$ and
$M(v^{\prime})$ are different.

######  Definition 4.6.

For a $3$-noncrossing matching $\xi\in\mathcal{TC}_{n}$, and a pair of
noncrossing partitions
$(\sigma,\sigma^{\prime})\in\mathcal{NP}_{n}\times\mathcal{NP}_{n}$, define
$a_{\xi,(\sigma,\sigma^{\prime})}$ to be the number of valid opposite loopless
resolution of $\xi$ that results in $(\sigma,\sigma^{\prime})$. In other
words,

$a_{\xi,(\sigma,\sigma^{\prime})}:=\\#\\{v\in\\{0,1\\}^{\mathrm{cr}(\xi)}\>|\>\xi(v)=\tau(\sigma),\
\xi(\mathbf{1}{-}v)=\tau(\sigma^{\prime})\text{ are both valid}\\}$

where $\tau(\sigma)$ is the noncrossing matching that corresponds to the
noncrossing partition $\sigma$ as in Section 2.4.

######  Lemma 4.7.

For $\xi\in\mathcal{TC}_{n}$ and a pair of noncrossing partitions
$(\sigma,\sigma^{\prime})\in\mathcal{NP}_{n}\times\mathcal{NP}_{n}$,
$a_{\xi,(\sigma,\sigma^{\prime})}$ equals the number of ways of splitting
$\Gamma(\xi)$ into two groves $F$ and $F^{\prime}$ with boundary partition
$\sigma$ and $\sigma^{\prime}$ respectively.

###### Proof.

Consider the cactus network $\Gamma=\Gamma(\xi)$ and the medial graph
$G=G(\Gamma)$ of $\xi$ together. For each crossing $x\in\mathrm{cr}(\xi)$,
there is a corresponding edge $e_{x}$ in $\Gamma$ that passes through the this
crossing. Consider two opposite resolution $v$ and $v^{\prime}$ such that
$v+v^{\prime}=\mathbf{1}^{\mathrm{cr}(\xi)}$. Correspondingly, we partition
the edges of $\Gamma$ into $F\sqcup F^{\prime}$ such that at each crossing
$x\in\mathrm{cr}(\xi)$, if the resolution $\xi(v)$ does not intersect $e_{x}$,
which looks like  , then we assign $e_{x}$ to $F$, otherwise assign $e_{x}$ to
$F^{\prime}$. This procedure is reversible, as any partition of the edge set
of $\Gamma$ gives an opposite resolution.

We also see that if $F$ has a cycle, then $\xi(v)$ contains an interior loop
inside this cycle. And if there is a connected component $C$ of $F$ not
connected to the boundary, then $\xi(v)$ also contains an interior loop
following edges not assigned to $F$ around $C$. Both situations are shown in
FigureFig. 14. Conversely, if $\xi(v)$ contains an interior loop, by tracing
through the crossings in $\xi$ and the corresponding edges in $\Gamma$ around
this loop, we recover one of the two situations described previously. As a
result, the opposite resolutions are valid if and only if both $F$ and
$F^{\prime}$ are groves. So we obtain the desired statement.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

Figure 14. Invalid resolutions, where edges assigned to $F$ are solid, edges
assigned to $F^{\prime}$ are dashed, and internal loops in $\xi(v)$ are in
red.

∎

######  Theorem 4.8.

For $\sigma,\sigma^{\prime}\in\mathcal{NP}_{n}$ and any cactus network
$\Gamma$,

(1)
$L_{\sigma}(\Gamma)L_{\sigma^{\prime}}(\Gamma)=\sum_{\xi\in\mathcal{TC}_{n}}a_{\xi,(\sigma,\sigma^{\prime})}B_{\xi}(\Gamma).$

###### Proof.

By definition of $B_{\xi}(\Gamma)$, the right hand side of Eq. 1 equals

$\sum_{\xi\in\mathcal{TC}_{n}}a_{\xi,(\sigma,\sigma^{\prime})}\sum_{H\subset
2\Gamma}\alpha(H)_{\xi}\operatorname{wt}(H)=\sum_{H\subset
2\Gamma}\operatorname{wt}(H)\sum_{\xi\in\mathcal{TC}_{n}}a_{\xi,(\sigma,\sigma^{\prime})}\alpha(H)_{\xi}.$

Viewing every edge weight in $\Gamma$ as an indeterminate, we compare the
coefficient of $\operatorname{wt}(H)$ on both sides for every $H$. For a fixed
$H\subset 2\Gamma$, the coefficient of $\operatorname{wt}(H)$ on the left hand
side $L_{\sigma}(\Gamma)L_{\sigma^{\prime}}(\Gamma)$ is by definition the
number of ways to partition $H$ into $F\sqcup F^{\prime}$ such that $F$ is a
grove with boundary partition $\sigma$ and $F^{\prime}$ is a grove with
boundary partition $\sigma^{\prime}$. It suffices to show that this number
equals
$\sum_{\xi\in\mathcal{TC}_{n}}a_{\xi,(\sigma,\sigma^{\prime})}\alpha(H)_{\xi}$.
The proof also explains the origin of 4.2.

To split $H$ into two groves $F\sqcup F^{\prime}$, there are a few easy steps
of reduction. If some edge of $H$ has multiplicity $\geq 3$ (which cannot
happen at the very beginning) or $H$ has a loop or $H$ has an interior leaf,
then clearly no such splittings are possible. These situations correspond to
moves (0), (2) and (5) of 4.2. Additionally, we can do the following
reductions corresponding to moves (3), (4) and (6) of 4.2:

  * -

If $H$ has a double edge $e$, then both $F$ and $F^{\prime}$ need to contain
$e$, so we can simply assign $e$ to both $F$ and $F^{\prime}$ and then
contract $e$. (Corresponds to move (3).)

  * -

If $H$ has two edges $e_{1}$ and $e_{2}$ between the same pair of vertices,
then one needs to be assigned to $F$ and the other one to $F^{\prime}$, so we
can contract this edges and multiply the result by $2$. (Corresponds to move
(4).)

  * -

If $H$ has more than two edges between the same pair of vertices, then no
assignment is possible. (Corresponds to move (4).)

  * -

If $H$ has an interior vertex of degree $2$ with distinct neighbors, then
these two edges must be assigned to $F$ and $F^{\prime}$ respectively with
both choices being symmetric and allowed. Therefore we can remove this
interior vertex and multiply the result by $2$. (Corresponds to move (6).)

Next, if $H$ contains an interior vertex $v$ of degree $3$ connected to
$v_{1},v_{2},v_{3}$, then to assign these three edges to $F$ and $F^{\prime}$,
we need to partition these three edges into a group of one and a group of two.
There are three ways of doing so. If $(v,v_{1})$ is assigned to one of $F$ and
$F^{\prime}$ and $(v,v_{2})$ and $(v,v_{3})$ to the other, then effectively we
are removing $v$ (together with the three edges incident to it) from $H$, and
adding an edge $(v_{2},v_{3})$ to it, to obtain a smaller graph $H_{2}$. For a
partition of $H_{2}$ into two groves $F_{2}\sqcup F_{2}^{\prime}$, we can
recover a partition of $H$ into two groves $F\sqcup F^{\prime}$ by assigning
$(v,v_{2})$ and $(v,v_{3})$ to $F$ if $(v_{2},v_{3})$ in $H_{2}$ is assigned