1 Introduction
Permutahedra and associahedra are among the most classical polytopes in algebraic combinatorics. The
$(n-1)$
-dimensional permutahedron has a vertex for each permutation of 
and an edge for each pair of permutations related by the transposition of two adjacent entries. The
$(n-1)$
-dimensional associahedron has a vertex for each binary tree on n nodes and an edge for each pair of binary trees related by a rotation. These two families of polytopes admit common generalizations explaining their similar behavior, including the permutreehedra of [Reference Pilaud and PonsPP18], the quotientopes of [Reference Pilaud and SantosPS19], and the
$(m,n)$
-multiplihedra of [Reference Chapoton and PilaudCP22]. All these polytopes are actually deformed permutahedra (defined as generalized permutahedra in [Reference PostnikovPos09, Reference Postnikov, Reiner and WilliamsPRW08]), meaning that their normal fans all coarsen the braid fan. This article is devoted to another common generalization to the permutahedra and associahedra, which are not deformed permutahedra in general.
The combinatorics of this generalization is based on pebble trees. A 
is an ordered tree where each node receives some colored pebbles in such a way that each unary node receives at least one pebble, and each subtree has either one more or as many leaves as pebbles of each color (see Figure 1 and the more precise Definition 2.1). We consider the set of pebble trees with a fixed number of leaves and fixed sets of unbalanced and balanced colors (i.e., the colors for which the number of leaves equals or exceeds the number of pebbles). It is immediate from the definition that these pebble trees are closed by arc contractions, and our main result is that the contraction poset is the face poset of a simple convex polytope, that we call the 
(see Figure 14). For this, we first construct the 
, associating to each pebble tree a polyhedral cone in a natural but subtle way, and then prove that this fan is the normal fan of a polytope by checking that its wall-crossing inequalities are satisfiable.
Our construction recovers the combinatorics and geometry of the permutahedra and associahedra in two degenerate situations. On the one hand, pebble trees with a single leaf can be seen as ordered partitions of their balanced colors, and the pebble tree polytope is the permutahedron. On the other hand, pebble trees with no pebbles are Schröder trees, and the pebble tree polytope is the associahedron. But the special situation which motivated this article is when there is exactly one color of pebbles. The pebble trees are then in bijection with some specific oriented planar trees considered by K. Poirier and T. Tradler in [Reference Poirier and TradlerPT18] for the combinatorics of algebraic structures endowed with a binary product and a co-inner product. These structures are closely connected to the
$V_\infty $
-algebras of T. Tradler and M. Zeinalian [Reference Tradler and ZeinalianTZ07] that arose in a tentative algebraic model for string topology operations defined by M. Chas and D. Sullivan [Reference Chas and SullivanCS99]. It is proved in [Reference Poirier and TradlerPT18] that the contraction posets on these oriented planar trees are face lattices of the 
, which are polytopal complexes refining the boundary complex of the Cartesian product of an associahedron with a simplex. We prove here that all assocoipahedra can actually be realized as convex polytopes using faces of pebble tree polytopes (see Figure 15).
The paper is organized as follows. Section 2 is devoted to the combinatorics of pebble trees. In Section 2.1, we provide more precise definitions and notations for pebble trees, we introduce some natural maps between families of pebble trees, and we give the precise bijection with the oriented planar trees of [Reference Poirier and TradlerPT18]. We introduce in Section 2.2 the pebble tree contraction poset, prove in Section 2.3 that it is the face poset of a pseudomanifold called the pebble tree complex, and discuss in Section 2.4 the adjacency graph of this complex called the pebble tree flip graph. Section 3 is devoted to the geometry of pebble trees. After quickly reminding the reader some geometric preliminaries in Section 3.1, we construct the pebble tree fan in Section 3.2 and the pebble tree polytope in Section 3.3. Finally, Section 4 is devoted to the numerology of pebble trees. We compute the generating functions of the maximal pebble trees in Section 4.1 and of all the pebble trees in Section 4.2, and gather explicit expansions of these generating functions in Section 4.3. While the methods are standard computations based on generatingfunctionology [Reference Flajolet and SedgewickFS09], the results reveal a few surprises.
2 Pebble tree combinatorics
In this section, we define pebble trees (Section 2.1) and study the pebble tree contraction poset (Section 2.2), the pebble tree complex (Section 2.3), and the pebble tree flip graph (Section 2.4).
2.1 Pebble trees
Recall that an 
is either a 
or a 
with an ordered list of subtrees. These subtrees are the 
of the node, and this node is the 
of these subtrees. As we only consider ordered trees, we omit the adjective ordered and only say tree. For a node n in a tree T, we denote by
$T_n$
the subtree of T rooted at n. For a subtree S, we denote by 
the set of leaves of S.
Definition 2.1 For a finite set of colors 
, a 
is a tree with pebbles colored by 
placed on its nodes such that
-
(1) each leaf receives no pebble, each node with a single child receives at least one pebble, and each node with at least two children receives arbitrary many pebbles (possibly none),
-
(2) for each subtree S and each color
, the number of leaves minus the number of pebbles of color in S is either
$0$
or
$1$
.
, the number of leaves minus the number of pebbles of color 
in S is either 
Example 2.1 Two classical combinatorial objects are extreme examples of pebble trees:
-
• pebble trees with a single leaf can be seen as ordered partitions of their pebble colors,
-
• pebble trees with no pebbles (meaning
) are Schröder trees (where each node has either none or at least two children).
) are Schröder trees (where each node has either none or at least two children).Some more generic examples of pebble trees are illustrated in Figure 1. Note that all our pictures of trees start with a vertical half-edge attached to the root, and end with half-edges representing the leaves.
Notation 2.2 We call 
the pebbles of color 
. We call 
of a subtree S the difference 
between the number of leaves and the number of 
-pebbles of S. We say that S is 
(resp. 
) if 
(resp. 
). We denote by 
(resp. 
) the set of colors 
for which S is 
-balanced (resp. 
-unbalanced).
Notation 2.3 We denote by 
the set of all pebble trees with leaves L and pebble colors 
, and by 
the subset of B-balanced and U-unbalanced pebble trees of 
for any 
. For
$\ell , b, u \in \mathbb {N}$
, we define 
as 
. In this case, we label the leaves by
$[\ell ]$
from left to right, the balanced pebble colors by
$[b]$
and the unbalanced pebble colors by
$[b+1,b+u]$
. Note that 
is isomorphic to 
for arbitrary
$L, B, U$
with
$|L| = \ell $
,
$|B| = b$
and
$|U| = u$
. It is however convenient to keep the notation 
to define certain operations on pebble trees (see Definitions 2.2–2.5) and for recursive decompositions of the pebble trees (see Section 4).
Remark 2.4 Some immediate consequences of Definition 2.1:
-
•
is finite for any L and , thus is finite for any
${\ell , b, u \in \mathbb {N}}$
. -
• The number of
-pebbles at a node p with children
$c_1, \dots , c_k$
is . Hence, the number of -unbalanced children of p is the number of -pebbles at p, plus
$1$
if p is -unbalanced. -
• Each unary node has at least one pebble, and at most one of each color.
-
• There is no consecutive chain formed by
unary nodes
is finite for any L and 
, thus 
is finite for any 
-pebbles at a node p with children 
. Hence, the number of 
-unbalanced children of p is the number of 
-pebbles at p, plus 
-unbalanced.
unary nodes We now define five natural maps between pebble trees (see Figures 2 and 3), that will induce isomorphisms in Propositions 2.10, 2.17, and 3.11. In Definitions 2.4 and 2.6, we call 
-leaf the only pebble tree of 
, i.e., whose root has a single pebble of color 
and a single child which is a leaf.
Figure 2: A
$\circ $
-balanced and
$\bullet $
-unbalanced
$\{\circ , \bullet \}$
-pebble tree (left), the
$\circ $
-balanced and
$\bullet $
-unbalanced
$\{\circ , \bullet \}$
-pebble tree obtained by mirroring it (middle left), the
$\{\circ ,\bullet \}$
-balanced
$\{\circ , \bullet \}$
-pebble tree obtained by
$\bullet $
-balancing it (middle right), and the
$\{\circ ,\star \}$
-balanced and
$\bullet $
-unbalanced
$\{\circ , \bullet , \star \}$
-pebble tree obtained by
$\star $
-inserting it (right).
Figure 3: A fully balanced
$\{\circ , \bullet \}$
-pebble tree and the fully balanced
$\{\circ , \bullet \}$
-pebble trees obtained by rerooting it at leaves x, y and z respectively (left). A fully unbalanced
$\{\circ , \bullet \}$
-pebble tree and the fully balanced
$\{\circ ,\bullet ,\star \}$
-pebble tree obtained by
$\star $
-uprooting it (right).
Definition 2.2 The 
sends a pebble tree T of 
to the pebble tree
$\mu (T)$
of 
obtained by a vertical symmetry of the tree, meaning that
$\mu (T)$
is defined inductively by
-
• if T is just a leaf, then
$\mu (T)$
is just a leaf, -
• if T is a node with pebbles P and children
$C_1, \dots , C_j$
, then
$\mu (T)$
is a node with pebbles P and children
$\mu (C_j), \dots , \mu (C_1)$
.
Definition 2.3 If 
, the 
sends a pebble tree T of 
to the pebble tree 
of 
whose root has a single pebble of color 
and a single child T.
Definition 2.4 If 
, the 
sends a pebble tree T of 
to the pebble tree 
of 
obtained from T by replacing each leaf by a 
-leaf.
Definition 2.5 If
$x \in L$
and
$U = \varnothing $
, the 
sends a pebble tree T of 
to the pebble tree 
obtained by hanging T from its leaf x. In other words,
$\rho _x(T)$
is the tree on the nodes of T where the children of each node p of T are the neighbors c of p in T so that x is in the connected component of p in the tree T were we have deleted the edge joining p to c.
Definition 2.6 If
$\ell> 1$
,
$B = \varnothing $
and 
, the 
sends a pebble tree T of 
to the pebble tree 
of 
obtained from T by first hanging T from its rightmost leaf, then deleting the leftmost leaf and placing a 
-pebble at its parent, and finally replacing all remaining leaves except the first by a 
-leaf.
Finally, our next three remarks connect pebble trees with other relevant families of trees.
Remark 2.5 Consider a word 
starting with 
(here, 
and 
stand for incoming and outgoing). An 
is a rooted oriented planar tree such that
-
• labeling the external arrows counterclockwise starting from the root, the ith arrow is incoming if
and outgoing if , -
• each internal node has at least one outgoing arrow,
-
• there is no node with precisely one incoming and one outgoing arrow.
and outgoing if 
,These trees arise in the combinatorics of algebras endowed with a binary product and a co-inner product. They are studied in details in [Reference Poirier and TradlerPT18]. It turns out that they can be understood from pebble trees.
First, as illustrated in Figure 4, there are simple bijections between the pebble trees of 
and the 
-trees (meaning the trees where all leaves are outgoing):
-
• Starting from a pebble tree
, orient each arc
$(p,c)$
joining a parent p to one of its children c in T from c to p if c is balanced, and from p to c if c is unbalanced, orient the root and leaves outward, and forget all pebbles. -
• Starting from a
-tree, place at each node one less pebbles than its outdegree, and forget the orientations.
, orient each arc 
-tree, place at each node one less pebbles than its outdegree, and forget the orientations.In the present paper, we prefer our interpretation as pebble trees as it enables us to consider several pebble colors simultaneously.
Figure 4: Examples of the bijection between the pebble trees of 
and the 
-trees.
Consider now an arbitrary signature
$\alpha $
. Although not explicit in [Reference Poirier and TradlerPT18], there is a clear map from
$\alpha $
-trees to 
-trees, which consists in replacing each incoming external arrow (like 
) by a node with a pair of outgoing arrows (like 
). This leads to a bijection between the
$\alpha $
-trees and the pebble trees of 
where the parent of the ith leaf is a unary node marked with a pebble.
Remark 2.6 The reader familiar with the bijective combinatorics of planar maps might also see some connections with the
$\beta $
-trees of [Reference Jacquard and SchaefferJS98, Reference Cori and SchaefferCS03]. Indeed, labeling each node p of a pebble tree by the pebble default 
, we obtain a
$\beta (1,1)$
-tree. However, this map is clearly injective but not surjective, and the additional condition given by the pebble trees is unclear on
$\beta (1,1)$
-trees.
Remark 2.7 There is also a natural map from the pebble trees of 
to the painted trees corresponding to the faces of the multiplihedron [Reference StasheffSta70, Reference ForceyFor08, Reference Chapoton and PilaudCP22]. Namely, we can just forget all pebbles which have a pebble on the path to the root of the pebble tree to obtain a painted tree.
2.2 Pebble tree contraction poset
We now define the contraction poset on pebble trees, and will see that it is the face poset of a simplicial complex in Section 2.3 and of a polytope in Section 3.3.
Definition 2.7 For any node c (not a leaf, nor the root) with parent p in a pebble tree T, the 
of c in T is the pebble tree
$T/c$
obtained by replacing c by its children in the list of children of p and adding to p the pebbles of c. The 
is the poset of contractions on pebble trees of 
.
Example 2.8 In the extreme situations of Example 2.1:
-
• the pebble tree contraction poset
is the refinement poset on ordered partitions of
$[b]$
, -
• the pebble tree contraction poset
is the contraction poset on Schröder trees with
$\ell $
leaves.
is the refinement poset on ordered partitions of 
is the contraction poset on Schröder trees with 
The pebble tree contraction posets 
and 
are illustrated in Figures 5 and 6. The fact that 
and 
are isomorphic can be seen applying successively Points (5), (4), and (2) of Proposition 2.10 below.
Figure 5: The pebble tree contraction poset 
.
Figure 6: The pebble tree contraction poset 
.
Remark 2.9 Observe that:
-
• The set of pebble trees
is clearly closed under contraction. Hence, the pebble tree contraction poset is a simplicial poset (a poset where each interval is a boolean algebra). -
• The pebble tree contraction poset is ranked: the rank of a pebble tree is its number of nodes.
-
• The
pebble trees of are the pebble trees with only unary nodes containing a single pebble and binary nodes containing no pebble (hence, they have
$\ell (b+u)-u$
unary nodes and
$\ell -1$
binary nodes, thus rank
$\ell (1 + b + u) - u - 1$
). -
• The
pebble tree of is the corolla with
$\ell $
leaves and
$\ell (b+u)-u$
pebbles at the root (hence it has rank
$1$
).
is clearly closed under contraction. Hence, the pebble tree contraction poset is a simplicial poset (a poset where each interval is a boolean algebra).
pebble trees of 
are the pebble trees with only unary nodes containing a single pebble and binary nodes containing no pebble (hence, they have 
pebble tree of 
is the corolla with 
Observe now that the mirroring, balancing, inserting, rerooting and uprooting maps of Definitions 2.2–2.6 obviously commute with contractions. This implies the following statement.
Proposition 2.10 Consider the operations of Definitions 2.2–2.6.
-
(1) The mirroring map of Definition 2.2 defines a poset automorphism of
. -
(2) If
$u> 1$
, the balancing map of Definition 2.3 defines a poset isomorphism from the pebble tree contraction poset to a principal upper set of the pebble tree contraction poset . Hence, is isomorphic to a principal upper set of . -
(3) The inserting map of Definition 2.4 defines a poset isomorphism from the pebble tree contraction poset
to a principal upper set of the pebble tree contraction poset . -
(4) The rerooting maps of Definition 2.5 define poset automorphisms of
. -
(5) If
$\ell> 1$
, the uprooting map of Definition 2.6 defines a poset isomorphism from the pebble tree contraction poset to a principal upper set of the pebble tree contraction poset .
.
to a principal upper set of the pebble tree contraction poset 
. Hence, 
is isomorphic to a principal upper set of 
.
to a principal upper set of the pebble tree contraction poset 
.
.
to a principal upper set of the pebble tree contraction poset 
. Finally, we connect the
$\alpha $
-trees of [Reference Poirier and TradlerPT18] to an upper set of the pebble tree contraction poset.
Remark 2.11 Following Remark 2.5, observe that for any signature 
, the
$\alpha $
-tree contraction poset is isomorphic to the principal upper set of the pebble tree contraction poset 
generated by the pebble tree whose root has 
pebbles and
$\ell $
children, and whose ith children is a leaf if 
and a unary node with one pebble and a leaf if 
.
Remark 2.12 The following observations are consequences of Remarks 2.5 and 2.11 and Proposition 2.10:
-
• the
-tree contraction poset is isomorphic to the pebble tree contraction poset , -
• for any
with a single occurrence of , the
$\alpha $
-tree contraction poset is isomorphic to the pebble tree contraction poset , -
• for
or , the
$\alpha $
-tree contraction poset is isomorphic to the pebble tree contraction poset (i.e., the contraction poset on Schröder trees).
-tree contraction poset is isomorphic to the pebble tree contraction poset 
,
with a single occurrence of 
, the 
,
or 
, the 
(i.e., the contraction poset on Schröder trees).2.3 Pebble tree complex
As mentioned in Remark 2.9, the pebble tree contraction poset 
is a simplicial poset. We now construct the corresponding simplicial complex. For a pebble tree in 
, recall that we label the leaves by
$[\ell ]$
from left to right, the balanced pebble colors by
$[b]$
, and the unbalanced pebble colors by
$[b+1,b+u]$
. Recall also that we denote by 
and 
the sets of leaves and of balanced colors in a subtree S. We will moreover need the following notation and definition, which might be mysterious at first sight, but aims at generalizing Example 2.14 and will be crucial in Section 3.2.
Notation 2.13 For an interval 
and a subset
$B \subseteq [b+u]$
, we define the sets
Definition 2.8 The 
is the simplicial complex whose simplices are the sets 
for all pebble trees 
, where 
.
Example 2.14 In the extreme situations of Example 2.1:
-
• the simplices of the pebble tree complex
are the flags
$B_1 \subsetneq B_2 \subsetneq \dots \subsetneq B_k \subseteq [b]$
, -
• the simplices of the pebble tree complex
are the collections of pairwise nested or non-adjacents intervals of
$[\ell -1]$
.
are the flags 
are the collections of pairwise nested or non-adjacents intervals of 
Figure 7 illustrates some more generic examples of simplices
$\Lambda (T)$
. Note that in these trees, the white color
$\circ $
is labeled
$1$
while the black color
$\bullet $
is labeled
$2$
.
Figure 7: Some
$\circ $
-balanced and
$\bullet $
-unbalanced
$\{\circ , \bullet \}$
-pebble trees T and the associated simplices
$\Lambda (T)$
. Each node n of T is labeled by the concatenation of the elements of the set
$\lambda (T_n)$
.
Proposition 2.15 The pebble tree complex 
is a pseudomanifold, whose face poset is isomorphic to the pebble tree poset 
.
Proof Observe first that
$\Lambda (T/n) = \Lambda (T) \smallsetminus \{\lambda (T_n)\}$
for any node n (not a leaf, nor the root) in a pebble tree T. Hence, the face poset of 
is indeed isomorphic to 
. We thus obtain that 
is a pure simplicial complex since 
is a ranked simplicial poset. It remains to prove that 
is a pseudomanifold, meaning that any ridge (i.e., codimension
$1$
face) is contained in precisely two facets (i.e., maximal dimensional faces). Consider thus a pebble tree of corank
$1$
, and let p be the only node which is neither unary with a pebble, nor binary with no pebble. We want to prove that there are two ways to open p, meaning to replace p by an edge whose contraction gives back p. We distinguish three cases:
-
• If p has three children and no pebble, then there are two ways to open p as usual (see Figure 8 (left)) and it does not matter whether p and its children are balanced or not for each pebble color.
Figure 8: The pebble tree complex is a pseudomanifold. All possible corank
$1$
pebble trees are obtained by contracting precisely two maximal pebble trees. In the second (resp. third) picture, we mark Y (resp. X) with a
$\bullet $
to indicate that it is
$\bullet $
-balanced. Neither X nor Y are
$\bullet $
-balanced in the fourth picture. -
• If p has two children and a
-pebble, then p is -balanced, and there are still two ways to open p depending on whether its children are -balanced or not (see Figure 8 (middle)). -
• If p has one child and two pebbles of different colors, then there are still two ways to open p choosing which pebble goes in the parent and which pebble goes in the child (see Figure 8 (right)).
-pebble, then p is 
-balanced, and there are still two ways to open p depending on whether its children are 
-balanced or not (see Figure Remark 2.16 In contrast to the special situations of Example 2.1, the pebble tree complex is not flag in general. For instance, 
and 
are not flag.
Finally, we translate Proposition 2.10 and Remark 2.11 to the pebble tree complex.
Proposition 2.17 Consider the operations of Definitions 2.2–2.6.
-
(1) The map defined by
$\ell j + i - \delta _{j \ne 0} \mapsto \ell (j + 1) - i$
for any
${(i,j) \in ([\ell ] \times [0,b+u]) \smallsetminus \{(\ell ,0)\}}$
induces an automorphism of the pebble tree complex . -
(2) If
$u> 1$
, the pebble tree complex is isomorphic to the link of the face
${[\ell ] \boxtimes [b]}$
in the pebble tree complex . Hence, is isomorphic to a link of . -
(3) The pebble tree complex
is isomorphic to the link of the face
$\left \{ \{i\} \boxtimes [1] \;\middle |\; i \in [\ell ] \right \}$
in the pebble tree complex . -
(4) The rerooting maps of Definition 2.5 induce automorphisms of the pebble tree complex
. -
(5) If
$\ell> 1$
, the pebble tree complex is isomorphic to the link of the face
$\left \{ \{i\} \boxtimes [1] \;\middle |\; i \in [2,\ell -1] \right \}$
in the pebble tree complex .
.
is isomorphic to the link of the face 
. Hence, 
is isomorphic to a link of 
.
is isomorphic to the link of the face 
.
.
is isomorphic to the link of the face 
.
, the 
in the pebble tree complex 
.2.4 Pebble tree flip graph
As the pebble tree complex 
is a pseudomanifold by Proposition 2.15, it is natural to consider its dual graph.
Definition 2.9 Two maximal pebble trees T and
$T'$
of 
are related by a 
if there are nodes n of T and
$n'$
of
$T'$
such that the following equivalent conditions hold:
-
• the contraction
$T/n$
coincides with the contraction
$T'/n'$
, -
•
$\Lambda (T) \smallsetminus \{\lambda (T_n)\} = \Lambda (T') \smallsetminus \{\lambda (T^{\prime }_{n'})\}$
.
All possible types of flips are illustrated in Figure 9. The 
is the graph whose vertices are the maximal pebble trees of 
and whose edges are the flips between them.
Figure 9: All possible flips in a maximal pebble tree. In the second (resp. third) picture, we mark Y (resp. X) with a
$\bullet $
to indicate that it is
$\bullet $
-balanced. Neither X nor Y are
$\bullet $
-balanced in the fourth picture.
Example 2.19 In the extreme situations of Example 2.1:
-
• the flip graph on
is the graph of adjacent transpositions on permutations of
$[b]$
, -
• the flip graph on
is the rotation graph on binary trees with
$\ell $
leaves.
is the graph of adjacent transpositions on permutations of 
is the rotation graph on binary trees with 
These extreme situations correspond to the right and left cases of Figure 9 respectively. Figure 10 illustrates a sequence of flips in maximal pebble trees of 
. Figures 11 and 12 illustrate the flip graphs on maximal pebble trees of 
and 
(which are isomorphic by Proposition 2.10 as already mentioned before).
Figure 10: A sequence of flips in maximal
$\circ $
-balanced and
$\bullet $
-unbalanced
$\{\circ , \bullet \}$
-pebble trees.
Figure 11: The flip graph on pebble trees of 
.
Figure 12: The flip graph on pebble trees of 
.
As the dual graph of a pure simplicial pseudomanifold, the pebble tree flip graph is regular. Its degree is
$\ell (1 + b + u) - u - 2$
. As we will see in Theorem 3.8 that it is the graph of a simple polytope, it has the connectivity of its degree. Among various further properties of this graph that would require more investigations, we mention the following problem in connection to [Reference Sleator, Tarjan and ThurstonSTT88, Reference PourninPou14].
Problem 2.20 Evaluate the diameter of the flip graph on maximal pebble trees of 
.
Finally, note that Proposition 2.10 and Remark 2.11 directly translate to morphisms between the flip graphs on the corresponding trees.
3 Pebble tree geometry
This section is devoted to the geometry of pebble trees. After quickly reminding the reader some geometric preliminaries (Section 3.1), we construct the pebble tree fan (Section 3.2) and the pebble tree polytope (Section 3.3).
3.1 Geometric preliminaries
We refer to [Reference ZieglerZie98] for a reference on polyhedral geometry, and only remind the basic notions needed later in the paper.
A (polyhedral) 
is the positive span 
of a finite set 
of vectors of
$\mathbb {R}^d$
or equivalently, the intersection of finitely many closed linear half-spaces of
$\mathbb {R}^d.$
The 
of a cone are its intersections with its supporting hyperplanes. The 
(resp. 
) are the faces of dimension
$1$
(resp. codimension
$1$
). A cone is 
if its rays are linearly independent. A (polyhedral) 
$\mathcal{F} $
is a set of cones such that any face of a cone of
$\mathcal{F} $
belongs to
$\mathcal{F} $
, and any two cones of
$\mathcal{F} $
intersect along a face of both. A fan is 
if the intersection of its cones is the origin, 
if the union of its cones covers
$\mathbb {R}^d$
, and 
if all its cones are simplicial.
Note that a simplicial fan defines a simplicial complex on its rays (the simplices of the simplicial complex are the subsets of rays which span a cone of the fan). Conversely, given a simplicial complex
$\Delta $
with ground set V, one can try to realize it geometrically by associating a ray 
of
$\mathbb {R}^d$
to each
$v \in V$
, and the cone 
generated by the set 
to each
$\triangle \in \Delta $
. To show that the resulting cones indeed form a fan, we will need the following statement, which can be seen as a reformulation of [Reference De Loera, Rambau and SantosDRS10, Corollary 4.5.20].
Proposition 3.1 Consider a closed simplicial pseudomanifold
$\Delta $
with ground set V and a set of vectors 
of
$\mathbb {R}^d$
, and define 
for any
$\triangle \in \Delta $
. Then the collection of cones 
forms a complete simplicial fan of
$\mathbb {R}^d$
if and only if
-
• there exists a vector
${\boldsymbol {v}}$
of
$\mathbb {R}^d$
contained in only one of the open cones for
$\triangle \in \Delta $
, -
• for any two adjacent facets
$\triangle , \triangle '$
of
$\Delta $
with
$\triangle \smallsetminus \{v\} = \triangle ' \smallsetminus \{v'\}$
, we have
$\alpha _v \alpha _{v'}> 0$
where denotes the unique (up to rescaling) linear dependence on.
for 
.
A 
is the convex hull of finitely many points of
$\mathbb {R}^d$
or equivalently, a bounded intersection of finitely many closed affine half-spaces of
$\mathbb {R}^d$
. The 
of a polytope are its intersections with its supporting hyperplanes. The 
(resp. 
, resp. 
) are the faces of dimension
$0$
(resp. dimension
$1$
, resp. codimension
$1$
).
The 
of a face 
of a polytope 
is the cone generated by the normal vectors to the supporting hyperplanes of 
containing 
. Said differently, it is the cone of vectors
${\boldsymbol {c}}$
of
$\mathbb {R}^d$
such that the linear form
${\boldsymbol {x}} \mapsto \left \langle \, {\boldsymbol {c}} \; \middle | \; {\boldsymbol {x}} \, \right \rangle $
on 
is maximized by all points of the face 
. The 
of 
is the set of normal cones of all its faces.
Consider now a complete simplicial fan
$\mathcal{F} $
of
$\mathbb {R}^d$
with rays 
and cones 
for
${\triangle \in \Delta }$
, where 
as in Proposition 3.1. To realize the fan
$\mathcal{F} $
, one can try to pick a height vector 
and consider the polytope ℙ
h
: = {
x
∈ℝ
d
| 〈
r
v |
x
〉≤h
v
for all v∈V}. The following classical statement characterizes the height vectors
${\boldsymbol {h}}$
for which the fan
$\mathcal{F} $
is the normal fan of this polytope 
. We borrow the formulation from [Reference Chapoton, Fomin and ZelevinskyCFZ02, Lemma 2.1].
Proposition 3.2 Let
$\mathcal{F} $
be an essential complete simplicial fan in
$\mathbb {R}^n$
with rays 
and cones 
for
$\triangle \in \Delta $
. Then the following are equivalent for any height vector
${\boldsymbol {h}} \in \mathbb {R}^V$
:
-
• The fan
$\mathcal{F} $
is the normal fan of the polytope -
• For two adjacent facets
$\triangle , \triangle '$
of
$\Delta $
with
$\triangle \smallsetminus \{v\} = \triangle ' \smallsetminus \{v'\}$
, the height vector
${\boldsymbol {h}}$
satisfies the where
$$\begin{align*}\alpha_v \, h_v + \alpha_{v'} \, h_{v'} + \sum_{w \in \triangle \cap \triangle'} \alpha_w \, h_w> 0 \end{align*}$$
denotes the unique linear dependence on such that
$\alpha _v + \alpha _{v'} = 2$
.
such that 
3.2 Pebble tree fan
Fix
$\ell , b, u \in \mathbb {N}$
and consider the intervals
whose union is the interval
We work in the Euclidean space
$\mathbb {R}^I$
with canonical basis
$({\boldsymbol {e}}_i)_{i \in I}$
. We denote by 
the characteristic vector of a subset
$J \subseteq I$
. As our constructions actually live in the linear subspace
we define the vector
for each subset
$J \subseteq I$
. It is immediate to check that these vectors satisfy the linear dependences
for any
$J, K \subseteq I$
. Finally, we associate to any pebble subtree S the vector 
where 
and 
denote the sets of leaves and of balanced colors in S, and the operation
$\boxtimes $
was defined in Definition 2.13. Note that 
when S is the entire tree T (because 
and 
so that
$\lambda (T) = I_0$
) or when S is a leaf i (because 
and 
so that
$\lambda (S) = \varnothing $
). We now use these vectors 
to construct the pebble tree fan.
Definition 3.1 The 
is the collection of cones
for all pebble trees 
, where 
.
Example 3.3 In the extreme situations of Example 2.1:
-
• the pebble tree fan
is the braid fan, with a ray for each proper subset
${\varnothing \neq J \subsetneq [b]}$
and a maximal cone for each permutation
$\sigma $
of
$[b]$
, defined by the inequalities
${x_{\sigma (1)} \le \dots \le x_{\sigma (b)}}$
, -
• the pebble tree fan
is the sylvester fan, with a ray for each proper interval J of
$[\ell ]$
and a maximal cone for each binary tree T, defined by the inequalities
$x_i \le x_j$
whenever there is a path from i to j in the tree T labeled in inorder and oriented toward its root.
is the braid fan, with a ray 
for each proper subset 
for each permutation 
is the sylvester fan, with a ray 
for each proper interval J of 
for each binary tree T, defined by the inequalities 
Note that the sylvester fan coarsens the braid fan: the cone 
of the sylvester fan can also be obtained by glueing the cones 
of the braid fan corresponding to the linear extensions
$\sigma $
of T.
Theorem 3.4 The pebble tree fan 
is an essential complete simplicial fan in
$\mathbb {H}_\ell ^{b,u}$
, whose face lattice is the pebble tree contraction poset 
.
The proof of Theorem 3.4 relies on the description of the linear dependences among adjacent maximal cones described in Lemma 3.6. To obtain these dependences, we need the following preliminary statement, where we use the operation
$\otimes $
defined in Definition 2.13.
Lemma 3.5 For any maximal pebble tree S and any 
, there are in S some distinct unary subtrees
$U_1, \dots , U_k$
with children
$V_1, \dots , V_k$
respectively such that 
.
Proof If a subtree U has a 
-pebble and a unique child V, then we have 
because 
and 
. Hence, for any 
, if we denote by
$U_1, \dots , U_k$
the closest descendants of S with a 
-pebble and by
$V_1, \dots , V_k$
their respective children, then we have 
because 
. The result follows since 
.
Lemma 3.6 Let T and
$T'$
be two adjacent maximal pebble trees and let S and
$S'$
be the subtrees of T and
$T'$
such that
$\Lambda (T) \smallsetminus \{\lambda (S)\} = \Lambda (T') \smallsetminus \{\lambda (S')\}$
. Then there is a linear dependence among the rays 
associated with the subtrees R of T and
$T'$
where the rays 
and 
both have coefficient
$1$
.
Proof We analyse the five possible types of flips described in Figure 9. In all cases, we denote by R the parent of S and
$S'$
. See Figure 13 here.
Figure 13: The four cases of the proof of 3.6.
Since 
(resp. 
, resp. 
) is a linear combination of the rays 
for some subtrees P of S distinct from S (resp. of
$S'$
distinct from
$S'$
, resp. of Y) by Lemma 3.5, this is indeed a linear dependence among the rays 
associated with the subtrees Q of T and
$T'$
where the rays 
and 
both have coefficient
$1$
.
Again, we can develop 
using Lemma 3.5, so that we indeed obtained a linear dependence among the rays 
for the subtrees Q of T and
$T'$
where the rays 
and 
both have coefficient
$1$
.
Case 3. The case of the third flip of Figure 13, is symmetric to Case 2.
We have
and moreover 
. Using Lemma 3.5, we get
or, written differently
Now if R has no ancestor with a
$\bullet $
-pebble, then using that 
, we obtain similarly
where
-
•
$U_1, \dots , U_k$
are the closest descendants of the root of T but not descendants of R which have a
$\bullet $
-pebble, -
•
$V_1, \dots , V_k$
are the (unique) children of
$U_1, \dots , U_k$
respectively.
Proof of Theorem 3.4
Note that 
is included in
$\mathbb {H}_\ell ^{b,u}$
since all rays 
are. To prove that it is a complete simplicial fan, we just check the two criteria of Proposition 3.1. The second criterion is guaranteed by the description of the linear dependences in Lemma 3.6. For the first criterion, consider the vector
and a pebble tree T such that
${\boldsymbol {v}}$
is contained in the interior of 
. As the last
$\ell -1$
coordinates of
${\boldsymbol {v}}$
are strictly larger than all other coordinates, each of the last
$\ell -1$
leaves of T is preceded by a unary node with pebble colored by
$b+u$
. Repeating the argument, we obtain that the first leaf of T is preceded by a chain of unary nodes with pebbles colored
$1, \dots , b$
while each of the last
$\ell -1$
leaves if T is preceded by a chain of unary nodes with pebbles colored
$1, \dots , b+u$
. Finally, we obtain that the rest of the tree T is the left comb since it is the only Schröder tree whose cone in the sylvester fan contains the vector 
. Finally, 
is essential as the dimension of its cones matches the dimension of
$\mathbb {H}_\ell ^{b,u}$
.
Remark 3.7 A few observations on the pebble tree fan:
-
• The simple descriptions of Example 3.3 for the braid fan and for the sylvester fan unfortunately fail for arbitrary
$b,u \ge 0$
. Indeed, there is a natural way to label the nodes of a maximal pebble tree: label the binary nodes in inorder by
${[\ell -1]}$
and the unary nodes by the only leaf first covered by this pebble. This labeling yields a cone for each maximal pebble tree T, defined by
$x_i \le x_j$
whenever there is a directed path from i to j in the tree T oriented toward its root. However, the cones for all maximal pebble trees T do not define a complete simplicial fan (check out the case
$\ell = 2$
,
$b = 1$
and
$u = 0$
). In fact, our pebble tree fan is not refined by the braid fan in general -
• Our definition of the pebble tree fan
respects some symmetries of the pebble tree complex but not all. See Proposition 3.11 for a precise statement directly on polytopes. -
• Lemma 3.6 actually proves that the pebble tree fan
is smooth, meaning that the principal vectors spanning the rays of any maximal cone of form an integral basis of the space (in other words, the corresponding toric variety is smooth).
for each maximal pebble tree T, defined by 
for all maximal pebble trees T do not define a complete simplicial fan (check out the case 
is not refined by the braid fan in general
respects some symmetries of the pebble tree complex 
but not all. See Proposition 
is smooth, meaning that the principal vectors spanning the rays of any maximal cone of 
form an integral basis of the space (in other words, the corresponding toric variety is smooth).3.3 Pebble tree polytope
Our next step is to construct a polytope whose normal fan is the pebble tree fan, using the criterion of Proposition 3.2.
Definition 3.2 A 
on n is a map f from the subsets of
$[n]$
to
$\mathbb {R}_{\ge 0}$
such that
$f_\varnothing = 0$
and
$$\begin{align*}f_{A \cup B} + f_{A \cap B} \le f_A + f_B \end{align*}$$
for any subsets A and B of
$[n]$
. We then define
where the minimum ranges over all subsets A and B of
$[n]$
such that
$A \not \subseteq B$
and
$A \not \supseteq B$
. Note that
-
•
$\sum _{i \in [k]} f_{A_i} - f_A \ge (k-1) \cdot \Delta f$
for any
$A = \bigcup _{i \in [k]} A_i$
where
$A_1, \dots , A_k$
are pairwise disjoint, -
•
$\Delta \lambda f = \lambda \Delta f$
for any scalar factor
$\lambda $
.
We say that f is 
when
$\Delta f> 0$
.
Theorem 3.8 Pick three strictly submodular functions f on
$\ell $
, g on
$\ell $
, and h on
$b+u$
such that
$$\begin{align*}\Delta f> 4 (\ell b + \ell u - u) \cdot (\max g + \max h) \qquad\text{and}\qquad \Delta g > (\ell b + \ell u - u + 1) \cdot \max h. \end{align*}$$
Then the pebble tree fan 
is the normal fan of the 
, the
$(\ell + \ell b + \ell u - u - 2)$
-dimensional polytope defined in the subspace
$\mathbb {H}_\ell ^{b,u}$
by the inequalities
for all
$1 \le s \le t \le \ell $
and all
$B \subseteq [b+u]$
.
Proof To shorten notations in this proof, we define for a maximal pebble subtree S
Observe that if V is the unique child of U, then
$\varphi _U - \varphi _V = g_U + h_U - h_V$
because 
and 
. We just need to prove that the function
$\varphi $
satisfies the wall-crossing inequalities of Proposition 3.2 for each of the linear dependences boxed in the proof of Lemma 3.6.
Case 1. By Lemmas 3.5 and 3.6, we have
for distinct unary subtrees
$U_1, \dots , U_k$
of S (resp.
$U^{\prime }_1, \dots , U^{\prime }_{k'}$
of
$S'$
, resp.
$U^{\prime \prime }_1, \dots , U^{\prime \prime }_{k"}$
of Y) with respective children
$V_1, \dots , V_k$
(resp.
$V^{\prime }_1, \dots , V^{\prime }_{k'}$
, resp.
$V^{\prime \prime }_1, \dots , V^{\prime \prime }_{k"}$
). Since 
and 
, we have
$$ \begin{align*} & f_S + f_{S'} - f_R - f_Y \ge \Delta f> 4 (\ell b + \ell u - u) (\max g + \max h) \\ &\hspace{6pt} \ge ( g_R + h_R ) + ( g_Y + h_Y ) + \sum_{i \in [k]} ( g_{U_i} + h_{U_i} ) + \sum_{i \in [k']} ( g_{U^{\prime}_i} + h_{U^{\prime}_i} ) + \sum_{i \in [k"]} ( g_{V^{\prime\prime}_i} + h_{V^{\prime\prime}_i} ), \end{align*} $$
where the last inequality holds since
$U_i \ne U_j$
(resp.
$U^{\prime }_i \ne U^{\prime }_j$
, resp.
$V^{\prime \prime }_i \ne V^{\prime \prime }_j$
) for
$i \ne j$
, and the pebble tree T has
$\ell b + \ell u - u$
unary subtrees. Since
$f, g, h$
take non-negative values and
$\varphi _U - \varphi _V = g_U + h_U - h_V$
when V is the unique child of U, we obtain that
$\varphi $
satisfies the wall-crossing inequality
$$\begin{align*}\varphi_S + \varphi_{S'} - \varphi_R - \varphi_Y - \sum_{i \in [k]} ( \varphi_{U_i} - \varphi_{V_i} ) - \sum_{i \in [k']} ( \varphi_{U^{\prime}_i} - \varphi_{V^{\prime}_i} ) + \sum_{i \in [k"]} ( \varphi_{U^{\prime\prime}_i} - \varphi_{V^{\prime\prime}_i} )> 0. \end{align*}$$
Case 2. By Lemma 3.6, we have
for distinct unary subtrees
$U_1, \dots , U_k$
of Y with children
$V_1, \dots , V_k$
, such that 
. Since 
, we obtain that
$$ \begin{align*} g_R - g_S - \sum_{i \in [k]} g_{U_i} &\ge (k-1) \cdot \Delta g \ge \Delta g> (\ell b + \ell u - u + 1) \cdot \max h \\ &\ge h_{S'} + h_S + \sum_{i \in [k]} h_{U_i}, \end{align*} $$
where the last inequality holds since
$S \ne U_i \ne U_j$
for
$i \ne j$
, and the subtree R has at most
$\ell b + \ell u - u$
unary subtrees. Since
$f, g, h$
take non-negative values and
${\varphi _U - \varphi _V = g_U + h_U - h_V}$
when V is the unique child of U, we obtain that
$\varphi $
satisfies the wall-crossing inequality
$$\begin{align*}( \varphi_R - \varphi_{S'} ) - ( \varphi_S - \varphi_X ) - \sum_{i \in [k]} ( \varphi_{U_i} - \varphi_{V_i} )> 0. \end{align*}$$
Case 3. Symmetric to Case 2.
Case 4. Assume first that R has an ancestor with a
$\bullet $
-pebble. Then by Lemma 3.6, we have
Since 
, we obtain that
$$\begin{align*} & g_{U_0} - g_S - g_{S'} - \sum_{i \in [k]} g_{U_i} \\ &\quad \ge (k+1) \cdot \Delta g \ge \Delta g> (\ell b + \ell u - u + 1) \cdot \max h > h_{V_0} + h_S + h_{S'} + \sum_{i \in [k]} h_{U_i}, \end{align*}$$
where the last inequality holds since
$S \ne U_i \ne U_j \ne S'$
for
$i \ne j$
, and the subtree
$U_0$
has at most
$\ell b + \ell u - u$
unary subtrees. Since
$f, g, h$
take non-negative values and
${\varphi _U - \varphi _V = g_U + h_U - h_V}$
when V is the unique child of U, we obtain that
$\varphi $
satisfies the wall-crossing inequality
$$\begin{align*}( \varphi_{U_0} - \varphi_{V_0} ) - ( \varphi_S - \varphi_X ) - ( \varphi_{S'} - \varphi_Y ) - \sum_{i \in [k]} ( \varphi_{U_i} - \varphi_{V_i} )> 0. \end{align*}$$
Assume now that R has no ancestor with a
$\bullet $
-pebble. Then we have
The wall-crossing inequality is thus even easier to satisfy since
$\varphi _{U_0} - \varphi _{V_0}$
does not appear.
Case 5. By Lemma 3.6, we have
Since 
and 
, we have
$$\begin{align*}\varphi_S + \varphi_{S'} - \varphi_R - \varphi_X = h_S + h_{S'} - h_R - h_R> 0.\\[-34pt] \end{align*}$$
Remark 3.9 Note that the conditions of Theorem 3.8 are just sufficient conditions to ensure the wall-crossing inequalities. To find functions satisfying these conditions, pick three arbitrary strictly submodular functions
$f, g, h$
and rescale first g by a factor
$4 (\ell b + \ell u - u + 1) \cdot \max h / \Delta g$
, and then f by a factor
${(\ell b + \ell u - u) \cdot (\max g + \max h) / \Delta f}$
. We just write 
if we want to consider 
for arbitrary
$f,g,h$
satisfying the conditions of Theorem 3.8.
Example 3.10 In the extreme situations of Example 2.1:
-
• the pebble tree fan
(a.k.a. braid fan) is the normal fan of the classical permutahedron, which can be obtained for
$h_B = \binom {b+1}{2} - \binom {|B|+1}{2}$
(the functions f and g are irrelevant here, since
$[s,t]$
is constant to
$[1]$
), -
• the pebble tree fan
(a.k.a. sylvester fan) is the normal fan of the classical associahedron [Reference Shnider and SternbergSS93, Reference LodayLod04], which can be obtained for
$f_X = \binom {\ell +1}{2} - \binom {|X|+1}{2}$
(the functions g and h are irrelevant here, since B is constant to
$\varnothing $
).
(a.k.a. braid fan) is the normal fan of the classical permutahedron, which can be obtained for 
(a.k.a. sylvester fan) is the normal fan of the classical associahedron [
Figure 14 illustrates polytopal realizations of the pebble tree fans 
and 
. Note that, while they have the same combinatorics by Proposition 2.10, their geometric realizations differ.
Figure 14: The pebble tree polytopes 
(left) and 
(right).
Finally, we translate the first three points of Proposition 2.10 and Remark 2.11 to pebble tree polytopes. Note that the last two transformations of Proposition 2.10 do not respect the geometry of the pebble tree polytopes.
Proposition 3.11 Consider the operations of Definitions 2.2–2.4.
-
(1) The map defined by
${\boldsymbol {e}}_{\ell j + i - \delta _{j \ne 0}} \mapsto {\boldsymbol {e}}_{\ell (j + 1) - i}$
for any
$(i,j) \in ([\ell ] \times [0,b+u]) \smallsetminus \{(\ell ,0)\}$
induces an isometry of the pebble tree polytope . -
(2) If
$u> 1$
, the pebble tree polytope is a facet of the pebble tree polytope . Hence, is a codimension u face of . -
(3) The pebble tree polytope
is a codimension
$\ell $
face of the pebble tree polytope where
$h(X) = h'(\left \{ x+1 \;\middle |\; x \in X \right \})$
for
$X \subseteq [b+u]$
.
.
is a facet of the pebble tree polytope 
. Hence, 
is a codimension u face of 
.
is a codimension 
where 
Remark 3.12 Following Remarks 2.5, 2.11, and 2.18, observe that for any signature 
, the
$\alpha $
-assocoipahedron of [Reference Poirier and TradlerPT18] is realized by a face of the pebble tree polytope 
. For instance, Figure 15 shows faces of the pebble tree polytopes 
and 
which realize the
$\alpha $
-assocoipahedra for 
and 
presented in [Reference Poirier and TradlerPT18, Figures 8, 9, 14, and 15]. Note that the combinatorics of the 
-assocoipahedra represented in Figure 15 (right) is also isomorphic to the pebble tree polytopes 
represented in Figure 14 by combining Points (2) and (4) of Proposition 2.17.
Figure 15: The
$\alpha $
-assocoipahedra for 
(left) and 
(right), realized as faces of the pebble tree polytopes 
and 
.
4 Pebble tree numerology
In this section, we compute the generating functions of the maximal pebble trees (Section 4.1) and of all the pebble trees (Section 4.2), and gather explicit expansions of these generating functions (Section 4.3). While the methods are standard computations based on generatingfunctionology [Reference Flajolet and SedgewickFS09], the results reveal a few surprises. All references like A000108 are entries of the Online Encyclopedia of Integer Sequences [OEI10].
4.1 Enumeration of maximal pebble trees
We start with the enumeration of the maximal pebble trees which is significantly simpler.
Definition 4.1 For
$\ell , u, b \in \mathbb {N}$
, we denote by
$m_\ell ^{b,u}$
the number of maximal pebble trees of 
(i.e., with
$\ell $
leaves, b balanced and u unbalanced colors). We consider the generating function
Proposition 4.1 The generating functions 
satisfy the functional equations
where
$\delta $
is the Kronecker delta. Hence, 
is algebraic for any
$b,u \in \mathbb {N}$
.
Proof A maximal pebble tree in 
is:
-
• either a leaf if
$s = t$
and
$B = \varnothing $
, -
• or a unary node with a single pebble colored by
and a child in , -
• or a binary node with no pebble and two children in
and for some
$r \in [s,t]$
and
$V \subseteq U$
.
and a child in 
,
and 
for some 
The functional equations for 
are immediate consequences of this structural decomposition by classical generatingfunctionology [Reference Flajolet and SedgewickFS09]. The algebraicity of 
follows as it belongs to a system of finitely many polynomial equations (all equations for a given sum
$b+u$
).
Example 4.2 When
$b = u = 0$
, we recover the functional equation
${\mathfrak{M}^{0,0}(x)=x+\mathfrak{M}^{0,0}(x)^{2}}$
which yields the classical Catalan generating function
Example 4.3 For
$b + u = 1$
, we obtain that
from which we can compute the expansions
These functions actually both satisfy a cubic equation, namely
Example 4.4 For
$b + u = 2$
, we obtain
From which we can compute the expansions:
The expansion of 
is quite surprising, but can be explained by a tiny functional miracle. Indeed, observe that we obtain that 
from Equation (4.1) and that 
from Equation (4.2). Replacing 
on both sides of Equation (4.3), we obtain
This shows that
$m_\ell ^{1,1} = 10^{\ell -1} C_\ell $
, where
$C_\ell = \frac {1}{\ell +1} \binom {2\ell }{\ell }$
is the Catalan number.
Consider the map sending a maximal pebble tree of 
to its underlying binary tree. In view of the formula
${m_\ell ^{1,1} = 10^{\ell -1} C_\ell }$
, it is natural to expect that its fibers all have size
$10^{\ell -1}$
. However, while the size of the fiber of a binary tree T is clearly invariant under reordering the children of T, it is not constant on all binary trees already for
$\ell = 4$
. Namely, the fiber of the binary tree whose children are both the tree on
$2$
leaves contains
$968$
pebble trees of 
, while the fiber of each of the remaining
$4$
binary trees on
$4$
leaves contains
$1008$
pebble trees of 
.
Despite this disappointing observation, one can provide a bijective explanation for the appearance of the Catalan numbers in
$m_\ell ^{1,1}$
. It requires the following observation.
We now proceed to define, for
$\ell , r \ge 1$
, an explicit bijection
$\psi $
sending a triple
$(T, L, R)$
of maximal pebble trees of 
, 
, and 
respectively to a maximal pebble tree of 
. The image
$\psi (T, L, R)$
is described in Figure 16. Note that
$\psi (T, L, R)$
sometimes depends on the type of L or R in the sense of the decomposition of Remark 4.5. In this description, we denote by 
the
$\bullet $
-balanced
$\circ $
-unbalanced pebble tree obtained by exchanging the
$\circ $
and
$\bullet $
pebbles in a
$\circ $
-balanced
$\bullet $
-unbalanced pebble tree T. As the decomposition of Remark 4.5 is unique, the map
$\psi $
is well defined, and it is immediate to check that the resulting trees are maximal pebble trees of 
. Again by Remark 4.5, it is straightforward to check that
$\psi $
is bijective. Finally, as
$m_2^{1,1} = 10$
, the existence of the bijection
$\psi $
directly implies Equation (4.4), hence the fact that
${m_\ell ^{1,1} = 10^{\ell -1} C_\ell }$
.
Figure 16: Bijection
$\psi $
sending a triple
$(T, L, R)$
of maximal pebble trees of 
, 
and 
respectively to a maximal pebble tree of 
.
4.2 Enumeration of all pebble trees
We now consider all pebble trees.
Definition 4.2 For
$\ell , n, u, b \in \mathbb {N}$
, we denote by
$p_{\ell ,n}^{b,u}$
the number of pebble tree with
$\ell $
leaves, n nodes, b balanced and u unbalanced colors. We consider the generating function
Proposition 4.6 The generating functions 
satisfy the functional equations
where
$\delta $
is the Kronecker delta, where each
$X_i$
ranges among arbitrary subsets of
$[d]$
while each
$Y_j$
ranges among strict subsets of
$[d]$
, and where 
and 
for any
$k \in [d]$
. Hence, 
is algebraic for any
$b,u \in \mathbb {N}$
.
Proof A pebble tree of 
is:
-
• either a leaf if
$s = t$
and
$B = \varnothing $
, -
• or a unary node with some pebbles colored by a non-empty subset
$S \subseteq B$
(one pebble of each color in S) and a child in , -
• or a node with some pebbles and
$d \ge 2$
children in respectively, for some
$s = s_1 \le t_1 = s_2 \le \dots \le t_d = t$
and
$B_i \subseteq B$
and
$U_i \subseteq U$
for all
$i \in [d]$
, such that
$\bigcap _{i \in [d]} U_i = \varnothing $
.
,
respectively, for some 
The functional equations for 
are immediate consequences of this structural decomposition by classical generatingfunctionology [Reference Flajolet and SedgewickFS09]. The algebraicity of 
follows as it belongs to a system of finitely many polynomial equations (all equations for a given sum
$b+u$
).
Example 4.7 When
$b = u = 0$
, we recover the functional equation
which yield the classical Schröder generating function
The expansions of the generating functions 
and 
for
$b + u \le 2$
can be found in Section 4.3. Finally, we observe that the evaluations of 
at
$y = 1$
and
$y = -1$
have a geometric meaning.
Proposition 4.8 For any
$b,u \in \mathbb {N}$
, the evaluation 
is the generating function of the total number of faces of the pebble tree polytope 
, and
Proof By Theorem 3.8,
$p_{\ell ,n}^{b,u}$
is the number of
$(\ell + \ell b + \ell u - u - 2 - n)$
-dimensional faces of the
$(\ell + \ell b + \ell u - u - 2)$
-dimensional pebble tree polytope 
. This implies that
-
•
$\sum _n p_{\ell ,n}^{b,u}$
is the total number of faces of , -
•
$\sum _n p_{\ell ,n}^{b,u} (-1)^n = (-1)^{\ell + \ell b + \ell u - u - 2}$
by Euler’s formula.
,
This immediately implies the statement.
4.3 Expansions of generating functions
Below are the expansions of the generating functions 
, 
and 
of Definitions 4.1 and 4.2 for all
$b + u \le 2$
.
$\underline {b = 0 \text { and } u = 0}$
.
$\underline {b = 1 \text { and } u = 0.}$
$\underline {b = 0 \text { and } u = 1.}$
$\underline {b = 2 \text { and } u = 0.}$
$\underline {b = 1 \text { and } u = 1.}$
$\underline {b = 0 \text { and } u = 2.}$
Acknowledgements
I am grateful to Doriann Albertin for discussions around Example 4.4 and for a very constructive proofreading, to Wenjie Fang for pointing out Remark 2.6 and for discussions on Propositions 4.1 and 4.6, to Guillaume Laplante-Anfossi for motivating me to find polytopal realizations of the assocoipahedra, to Thibaut Mazuir for algebraic discussions on the assocoipahedra, and to Anna de Mier, Jordi Castellví Foguet and Clément Requilé for discussions on Example 4.4 during the 3rd Workshop on Geometric, Algebraic and Probabilistic Combinatorics in Monserrat on July 2022. Finally, I am grateful to an anonymous referee for useful suggestions on the presentation of this paper. This research benefited from computations and tests done using the open-source mathematical software Sage [Sd16] and its combinatorics features developed by the Sage-combinat community [SCc16], and from the Online Encyclopedia of Integer Sequences [OEI10].
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