A Projection

Since the dimension of the _ν _ν-brick polyhedron is usually much higher than the dimension of the _ν _ν-associahedron, it is interesting to study suitable projections to obtain figures in the appropriate dimension. In this section, we provide an elegant projection in the case where _ν _ν has no two consecutive north steps.

For convenience, we consider paths of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =(NE^{k_n}) \cdot \cdot \cdot (NE^{k_1})$$\end{document}ν=(NEkn)···(NEk1), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_i\ge 1$$\end{document}ki≥1. Notice that we are adding a north step N at the beginning and some east steps at the end of the path, but this does not affect the combinatorics of the bounded components of the brick polyhedron. One can double check that the ν ν-brick polyhedron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}(Q\nu ,w\nu ) \subseteq \mathbb {R}^{n+2+\sum (k_i-1)}$$\end{document}B(Qν,wν)⊆Rn+2+∑(ki-1). Since the first and last coordinates of the brick vectors b(T) are constant for every ν ν-tree T, we omit them, and write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{b}(T)$$\end{document}b~(T) for the resulting vectors. Moreover, we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{\mathcal {B}}(Q\nu ,w\nu )$$\end{document}B~(Qν,wν) the result of omitting the first and last coordinates of the intersection of the brick polyhedron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}} (Q_\nu ,w_\nu )$$\end{document}B(Qν,wν) with the affine subspace defined by the first and the last coordinates being those constant numbers. In particular, the bounded components of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}} (Q_\nu ,w_\nu )$$\end{document}B(Qν,wν) are in correspondence with the bounded components of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{\mathcal {B}} (Q_\nu ,w_\nu )$$\end{document}B~(Qν,wν). After removing the first and the last coordinates, we havewhere \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N:=n+\sum (k_i-1)$$\end{document}N:=n+∑(ki-1). We denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_I:= \sum _{i\in I} x_i$$\end{document}xI:=∑i∈Ixi. Our projection uses the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widetilde{M}_1, \dots , \widetilde{M}_n $$\end{document}M1,⋯,Mn, which are defined recursively by setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widetilde{M}_j $$\end{document}Mj to be the last \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (k_i - 1) $$\end{document}(ki-1) elements of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [N] {\setminus } \bigcup _{i=1}^{j-1} \widetilde{M}_i $$\end{document}[N]\⋃i=1j-1Mi, and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M_j = \widetilde{M}_j \cup { j } $$\end{document}Mj=M~j∪{j} for _ j ∈ [n] _j∈[n].

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widetilde{\mathcal {B}} (Q_\nu ,w_\nu )\subseteq \mathbb {R}^{n+\sum (k_i-1)}=\mathbb {R}^N $$\end{document}B~(Qν,wν)⊆Rn+∑(ki-1)=RN

Definition 16

(Projection) We define the projectionNext, letFinally, we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi :\mathbb {R}^{N} \rightarrow \mathbb {R}^{n-1}$$\end{document}π:RN→Rn-1 as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi =\pi _2 \circ \pi _1$$\end{document}π=π2∘π1.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \pi _1: \mathbb {R}^{N} \longrightarrow \mathbb {R}^{n}, (x_1,...,x_{N})\mapsto (x_{M_1},...,x_{M_{n}})\in \mathbb {R}^{n} . \end{aligned}$$\end{document}π1:RN⟶Rn,(x1,...,xN)↦(xM1,...,xMn)∈Rn.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \pi _2: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n-1},(x_{M_1},...,x_{M_{n}}) \mapsto (x_{M_1},x_{M_1}+x_{M_2},...,x_{M_1}+...+x_{M_{n-1}}). \end{aligned}$$\end{document}π2:Rn→Rn-1,(xM1,...,xMn)↦(xM1,xM1+xM2,...,xM1+...+xMn-1).

Theorem 12

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =(NE^{k_n}) \cdot \cdot \cdot (NE^{k_1})$$\end{document}ν=(NEkn)···(NEk1) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_i\ge 1$$\end{document}ki≥1 (no consecutive north steps). Then the projection \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi : \mathbb {R}^N \rightarrow \mathbb {R}^{n-1}$$\end{document}π:RN→Rn-1 of the bounded components of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{\mathcal {B}} (Q_\nu ,w_\nu )$$\end{document}B~(Qν,wν) is a realization of the _ν _ν-associahedron of the desired dimension.

Moreover, the coordinates of the projected vertices can be simply described as follows. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0$$\end{document}T0 be the minimal _ν _ν-tree of the _ν _ν-Tamari lattice. We denote byWe can think of these y-coordinates as the result of translating the projection of the bounded components of the _ν _ν-brick polyhedron by the constant vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- \pi (\widetilde{b}(T_0) )$$\end{document}-π(b~(T0)). In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(T_0)=(0,\dots ,0)$$\end{document}y(T0)=(0,⋯,0).

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ y(T)=(y_1,\dots ,y_{n-1}) := \pi (\widetilde{b}(T) ) - \pi (\widetilde{b}(T_0) ). $$\end{document}y(T)=(y1,⋯,yn-1):=π(b~(T))-π(b~(T0)).

It turns out that these new coordinates can be described in a very simple and elegant combinatorial way. First, we label the horizontal lines of the Ferrers diagram determined by the path _ν _ν from 1 to _n-1_n-1, from top to bottom, omitting the top most row. We denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_i=P_i(T)$$\end{document}Pi=Pi(T) the unique shortest path connecting the root to the left most node of T on the i-th horizontal line, and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\operatorname {area}( P_i )$$\end{document}area(Pi) be the number of boxes left to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_i$$\end{document}Pi. Two examples are illustrated in Fig. 19. Magically, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_i=\operatorname {area}(P_i)$$\end{document}yi=area(Pi).

Fig. 19: The canonical coordinates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(T_0)=(y_1^{0},y_2^{0},y_3^{0})=(0,0,0)$$\end{document}y(T0)=(y10,y20,y30)=(0,0,0), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(T)=(y_1,y_2,y_3)=(3,2,3)$$\end{document}y(T)=(y1,y2,y3)=(3,2,3) for the two νν-trees \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0$$\end{document}T0 and T. The entry \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_i(T)$$\end{document}yi(T) is the area (i.e. number of boxes to the left) of the path \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_i(T)$$\end{document}Pi(T) connecting the root to the leftmost node of T at level i (increasing from top to bottom)

Theorem 13

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =(NE^{k_n}) \cdot \cdot \cdot (NE^{k_1})$$\end{document}ν=(NEkn)···(NEk1) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_i\ge 1$$\end{document}ki≥1 (no consecutive north steps). For a _ν ν-tree T we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(T) =(y_1,...,y{n-1})$$\end{document}y(T)=(y1,...,yn-1), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_i = \operatorname {area} (P_i(T))$$\end{document}yi=area(Pi(T)). These coordinates determine a realization of the _ν _ν-associahedron.

Remark 6

As a consequence, we obtain that the projection \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi : \mathbb {R}^N \rightarrow \mathbb {R}^{n-1}$$\end{document}π:RN→Rn-1 of the bounded components of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{\mathcal {B}} (Q_\nu ,w_\nu )$$\end{document}B~(Qν,wν) is a translation of the canonical realization of the _ν _ν-associahedron described by the first author in [21]. This may be regarded as a Loday-like realization of the _ν _ν-associahedron, because in the classical case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =(NE)^n$$\end{document}ν=(NE)n both realizations are affinely equivalent.

Example 6

We continue Examples 1 and 2, but add a north step N at the beginning and an east step E at the end of the path for convenience, obtaining _ν =NENEENE_ν=NENEENE (instead of _ν =ENEEN_ν=ENEEN). The seven _ν _ν-trees are shown in Fig. 20.

Fig. 20: Projection of the bounded components of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{\mathcal {B}}(Q\nu , w_\nu )$$\end{document}B~(Qν,wν) for ν =NENEENEν=NENEENE_

The brick vectors are:Note that these are just translations of the brick vectors we computed in Example 2 (because of the extra N and E). The vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{b}(T_i)\in \mathbb {R}^4$$\end{document}b~(Ti)∈R4 are obtained by removing the first and last coordinates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b(T_i)$$\end{document}b(Ti). Furthermore, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1={1}$$\end{document}M1={1}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_2={2,4}$$\end{document}M2={2,4} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_3={3}$$\end{document}M3={3}. The projected points are then:These coordinates coincide with the vertex coordinates of the canonical realization by [21], up to a translation by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (\widetilde{b}(T_0) )=-(11,21)$$\end{document}π(b~(T0))=-(11,21):The resulting projection is illustrated in Fig. 20.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b(T_0)&= -(12,11,8,7,2,0)&b(T_4)&= -(12,10,7,9,2,0) \\ b(T_1)&= -(12,9,10,7,2,0)&b(T_5)&= -(12,9,8,9,2,0) \\ b(T_2)&= -(12,8,10,7,3,0)&b(T_6)&= -(12,8,8,9,3,0) \\ b(T_3)&= -(12,11,7,8,2,0) \end{aligned}$$\end{document}b(T0)=-(12,11,8,7,2,0)b(T4)=-(12,10,7,9,2,0)b(T1)=-(12,9,10,7,2,0)b(T5)=-(12,9,8,9,2,0)b(T2)=-(12,8,10,7,3,0)b(T6)=-(12,8,8,9,3,0)b(T3)=-(12,11,7,8,2,0)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \pi (\widetilde{b}(T_0))&= -(11,21)&\pi (\widetilde{b}(T_4))&= -(10,19) \\ \pi (\widetilde{b}(T_1))&= -(9,21)&\pi (\widetilde{b}(T_5))&= -(9,19) \\ \pi (\widetilde{b}(T_2))&= -(8,21)&\pi (\widetilde{b}(T_6))&= -(8,19) \\ \pi (\widetilde{b}(T_3))&= -(11,20) \end{aligned}$$\end{document}π(b~(T0))=-(11,21)π(b~(T4))=-(10,19)π(b~(T1))=-(9,21)π(b~(T5))=-(9,19)π(b~(T2))=-(8,21)π(b~(T6))=-(8,19)π(b~(T3))=-(11,20)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} y(T_0)&= (0,0)&y(T_4)&= (1,2) \\ y(T_1)&= (2,0)&y(T_5)&= (2,2) \\ y(T_2)&= (3,0)&y(T_6)&= (3,2) \\ y(T_3)&= (0,1) \end{aligned}$$\end{document}y(T0)=(0,0)y(T4)=(1,2)y(T1)=(2,0)y(T5)=(2,2)y(T2)=(3,0)y(T6)=(3,2)y(T3)=(0,1)

Example 7

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =NENENEENE=(NE^1)(NE^1)(NE^2)(NE^1)$$\end{document}ν=NENENEENE=(NE1)(NE1)(NE2)(NE1), we obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=n+\sum (k_i-1)=4+(0+1+0+0)=5$$\end{document}N=n+∑(ki-1)=4+(0+1+0+0)=5. Furthermore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1={1}$$\end{document}M1={1}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_2={2,5}$$\end{document}M2={2,5}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_3={3}$$\end{document}M3={3}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_4={4}$$\end{document}M4={4}. So we group coordinates 2 and 5 of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{b}(T)$$\end{document}b~(T) together. In order to illustrate how the projection works, let us consider the two _ν _ν-trees T and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0$$\end{document}T0 in Fig. 19. We obtain:The difference between the corresponding projected brick vectors isAs we can see from Fig. 19, the entry \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_i=\operatorname {area}(P_i)$$\end{document}yi=area(Pi) counts the number of boxes left to the path \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_i$$\end{document}Pi connecting the root to the left most node of T at level i.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b(T)&= -(17, 13, 13, 9, 13, 2, 0)&b(T_0)&= -(17, 16, 12, 10, 10, 2, 0) \\ \widetilde{b}(T)&= -(13, 13, 9, 13, 2)&\widetilde{b}(T_0)&= -(16, 12, 10, 10, 2) \\ \pi _1(\widetilde{b}(T))&= -(13, 15, 9, 13)&\pi _1(\widetilde{b}(T_0))&= -(16, 14, 10, 10) \\ \pi (\widetilde{b}(T))&= -(13, 28, 37)&\pi (\widetilde{b}(T_0))&= -(16, 30, 40) \\ \end{aligned}$$\end{document}b(T)=-(17,13,13,9,13,2,0)b(T0)=-(17,16,12,10,10,2,0)b~(T)=-(13,13,9,13,2)b~(T0)=-(16,12,10,10,2)π1(b~(T))=-(13,15,9,13)π1(b~(T0))=-(16,14,10,10)π(b~(T))=-(13,28,37)π(b~(T0))=-(16,30,40)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ y(T)=(y_1,y_2,y_3)=\pi (\widetilde{b}(T)) - \pi (\widetilde{b}(T_0)) = (3,2,3). $$\end{document}y(T)=(y1,y2,y3)=π(b~(T))-π(b~(T0))=(3,2,3).

Although the ν ν-brick polyhedron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}} (Q\nu ,w\nu )\subseteq \mathbb R^7$$\end{document}B(Qν,wν)⊆R7, the projection of its bounded components lies in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^3$$\end{document}R3 and is illustrated in Fig. 21.

Fig. 21: Left: Projection of the bounded components of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{\mathcal {B}}(Q\nu , w_\nu )$$\end{document}B~(Qν,wν), Right: νν-associahedron_

Example 8

Figure 22 shows the projections of _ν _ν-brick polyhedra to the canonical realization of _ν _ν-associahedra for three different lattice paths:_ν = (NE)(NE)(NEE)(NE)_ν=(NE)(NE)(NEE)(NE)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = (NE^2)^4$$\end{document}ν=(NE2)4 (Fuss-Catalan case)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = (NE^3)^4$$\end{document}ν=(NE3)4 (Fuss-Catalan case)Animations of these figures can be found in [22].

Fig. 22: Some νν-associahedra obtained as projections of νν-brick polyhedra

Proof of Theorems 12 and 13

Since Theorem 13 provides the explicit coordinates, which, up to translation, coincide with the known canonical realization of the _ν _ν-associahedron in [21], it is sufficient to prove Theorem 13.

Proof of Theorem 13

Observe that the dimension of the maximal polytopal part and the projected points align precisely. Let T be a _ν _ν-tree and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T'=T\setminus {b} \cup {b'}$$\end{document}T′=T{b}∪{b′} be a rotation as in Fig. 23.

Fig. 23: Structure of νν-trees T and T'T′

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(T)=(c_1,...,c_d)$$\end{document}c(T)=(c1,...,cd) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(T')=(c'_1,...,c'_d)$$\end{document}c(T′)=(c1′,...,cd′) be the coordinates of the _ν _ν-trees T and _T'_T′ in the canonical realization. We need to showConsider the pipe dream representation of the ν ν-trees T and T'T′. The translation vector from the canonical realization to the projection is then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (b(T_0))$$\end{document}π(b(T0)), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0$$\end{document}T0 is the minimal ν ν-tree. Suppose that node b∈ T_b∈T touches pipes i and j, and let k be the lower horizontal level of a descendant node of b in the tree T, see Fig. 23. Denoting the number of boxes covered by a rotation as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {area}{i,j}$$\end{document}areai,j we obtain the following in the canonical realization:We need to show that this formula holds for the projected points. LetThen \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i'=x_i+\text {area}{i,j}$$\end{document}xi′=xi+areai,j, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_j'=x_j-\text {area}{i,j}$$\end{document}xj′=xj-areai,j and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x'\ell = x\ell $$\end{document}xℓ′=xℓ otherwise. This is because, by Corollary 2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i$$\end{document}xi is the negative of the number of lattice points in the Ferrers diagram that are below pipe i. When we make the rotation, this number of points is reduced by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {area}{i,j}$$\end{document}areai,j. Similarly, the number of points below pipe j increases by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {area}{i,j}$$\end{document}areai,j. The number of points below the other pipes remains constant.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c(T)-c(T') = \pi (b(T')) - \pi (b(T)). \end{aligned}$$\end{document}c(T)-c(T′)=π(b(T′))-π(b(T)).

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c'_\ell = {\left\{ \begin{array}{ll} c_\ell +\text {area}_{i,j} & \text { for }i \le \ell \le k \\ c_\ell & \text {otherwise} \end{array}\right. }. \end{aligned}$$\end{document}cℓ′=cℓ+areai,jfori≤ℓ≤kcℓotherwise.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&y = (y_1,...,y_d) = \pi (b(T))\text {, }y' = (y_{1'},...,y_{d'}) = \pi (b(T'))\text {, }\\&x = (x_1,...,x_{N-2}) = b(T)\text {, }x' = (x_{1'},...,x_{N-2}') = b(T'). \end{aligned}$$\end{document}y=(y1,...,yd)=π(b(T)),y′=(y1′,...,yd′)=π(b(T′)),x=(x1,...,xN-2)=b(T),x′=(x1′,...,xN-2′)=b(T′).

Now it is not hard to see that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in M_{k+1}$$\end{document}j∈Mk+1. If there is a node below b in T then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=k+1 \in M_{k+1}$$\end{document}j=k+1∈Mk+1, if not then j is the diagonal level plus 1 of the point s∈ S_s∈S that is in the k-th horizontal line, so \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in M{k+1}$$\end{document}j∈Mk+1. Therefore, sincethenThis finishes the proof. _□ _□

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} y_\ell =(x_{M_1}+...+x_{M_\ell })\text { and }y_\ell '=(x'_{M_1}+...+x'_{M_\ell }) \end{aligned}$$\end{document}yℓ=(xM1+...+xMℓ)andyℓ′=(x1′+...+xℓ′)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} y'_\ell = {\left\{ \begin{array}{ll} y_\ell +\text {area}_{i,j} & \text { for }i \le \ell \le k \\ y_\ell & \text {otherwise} \end{array}\right. }. \end{aligned}$$\end{document}yℓ′=yℓ+areai,jfori≤ℓ≤kyℓotherwise.