ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.02 / 541–559
https://doi.org/10.26493/1855-3974.2044.fd1
(Also available at http://amc-journal.eu)
Multivariate polynomials for
generalized permutohedra*
Eric Katz †
Department of Mathematics, The Ohio State University,
Columbus, Ohio, United States
McCabe Olsen ‡
Department of Mathematics, Rose-Hulman Institute of Technology,
Terre Haute, Indiana, United States
Received 9 July 2019, accepted 8 September 2021, published online 3 August 2022
Abstract
Using the notion of a Mahonian statistic on acyclic posets, we introduce a q-analogue
of the h-polynomial of a simple generalized permutohedron. We focus primarily on the
case of nestohedra and on explicit computations for many interesting examples, such as
Sn-invariant nestohedra, graph associahedra, and Stanley-Pitman polytopes. For the usual
(Stasheff) associahedron, our generalization yields an alternative q-analogue to the well-
studied Narayana numbers.
Keywords: Generalized permutohedron, h-polynomial, q-analogues.
Math. Subj. Class. (2020): 52B12, 05A15, 06A07, 05C31
1 Introduction
Given any combinatorially defined polynomial, a common theme in enumerative combina-
torics is to consider multivariate analogues which further stratify and enrich the encoded
data by an additional combinatorial statistic. A notable example of is the Euler–Mahonian
polynomial
An(t, q) =
∑
π∈Sn
tdes(π)qmaj(π)
*The authors thank Vic Reiner for helpful comments at the beginning of this project. The authors also thank
the anonymous referee for helpful comments and suggestions.
†Partially supported by NSF DMS 1748837.
‡Corresponding author.
E-mail addresses: katz.60@osu.edu (Eric Katz), olsen@rose-hulman.edu (McCabe Olsen)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
542 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559
which is a bivariate generalization of the more foundational Eulerian polynomial
An(t) =
∑
π∈Sn
tdes(π),
both of which are specializations of the n− 1 variable polynomial
An(t1, t2, . . . , tn−1) =
∑
π∈Sn
∏
i∈Des(π)
ti.
In this case, we further stratify the descent statistic on permutations by the additional data
of the major index. Such a generalization is commonly referred to as a q-analogue in
reference to usual choice of added variable.
Given a convex polytope P ⊂ Rn, the h-polynomial is an encoding of the face numbers
of P obtained as a linear change of variables of the generating function for the face num-
bers. If P is simple or simplicial, then the Dehn–Sommerville equations for P are reflected
in the palindromicity of the h-polynomial. For simple rational polytopes, the h-polynomial
is the Poincaré polynomial of the cohomology groups of the toric variety attached to the
polytope. Moreover, for simplicial polytopes, the h-polynomial is the generating function
for facets of P according to the size of their restriction sets [25, Section 8.3].
Generalized permutohedra are a broad class of convex polytopes which exhibit many
nice properties. First introduced by Postnikov [20], these polytopes have been the subject
of much study and are of wide interest in many areas of algebraic and enumerative com-
binatorics, including the combinatorics of Coxeter groups, cluster algebras, combinatorial
Hopf algebras and monoids, and polyhedral geometry (see, e.g., [2, 4, 15, 16]).
Of particular interest for our purposes, Postikov, Reiner, and Williams [21] give a com-
binatorial description of the h-polynomial for any simple generalized permutohedron using
an Eulerian descent statistic on posets. Moreover, they provide a formula for well-behaved,
special cases of generalized permutohedra. We give a bivariate generalization of their de-
scription for any simple generalized permutohedron: for P a simple generalized permuta-
hedron and Qσ the cone poset for a full dimensional cone σ in the normal fan N (P ) (See
Definition 2.1), we define
hP (t, q) :=
∑
σ∈N (P )
tdes(Qσ)qmaj(Qσ)
where des and maj are statistics defined below. Furthermore, we are able to be more
explicit when restricting to particular classes of generalized permutahedra, specifically Sn-
invariant nestohedra, graph associahedra, and Stanley–Pitman polytopes.
Our definition of the bivariate h-polynomial, which specializes to the usual h-polynomial
is justified by analogy with the Euler–Mahonian polynomial. Other possible definitions ex-
ist. An inequivalent definition is the principal specialization of the Frobenius characteristic
of the permutohedral toric variety. This definition does not extend to generalized permuto-
hedra and is not discussed in the body of the paper. However, it does make use of the major
index.
The structure of this note is as follows. In Section 2, we provide a review of necessary
background and terminology on permutations, posets, polyhedral geometry, and general-
ized permutohedra. Section 3 defines and discusses the the q-analogue for the h-polynomial
of any simple generalized permutohedron. In Section 4, we focus on general results for a
E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 543
large class of simple generalized permutohedra called nestohedra, including a palidromic-
ity result for special cases. Section 5 is devoted to several explict examples, including
Sn-invariant nestohedra, graph associahedra, the classical associahedron, the stellohedron,
and the Stanley–Pitman polytope. These examples produce some alternative q-analogues
of some well-known combinatorial sequences, including the Narayana numbers.
2 Background
In this section, we provide a brief review of basic properties of permutations statistics,
posets, polytopes and normal fans, and generalized permutohedra.
2.1 Permutation statistics
Let A = {a1 < a2 < . . . < an} be a set of n elements. The symmetric group on A,
denoted SA, is the set of all permutations of the elements of A. In the case of A = [n], we
will simply write Sn. Given π = π1π2 · · ·πn ∈ SA, the descent set of π is
Des(π) = {i ∈ [n− 1] : πi > πi+1},
the descent number of π is des(π) = |Des(π)|, and the major index of π is
maj(π) =
∑
i∈Des(π)
i.
The descent statistic is commonly referred to as an Eulerian statistic, due to the con-
nection to polynomial first studied by Euler [14]. The Eulerian polynomial An(t) is the
unique polynomial which satisfies∑
k≥0
(k + 1)ntk =
An(t)
(1− t)n+1
However, this polynomial can be interpreted entirely combinatorially as
An(t) =
∑
π∈Sn
tdes(π).
The major index, on the other hand, is commonly known as a Mahonian statistic, as it was
introduced by MacMahon [18]. The descent statistic and major index statistic are naturally
linked as they both encode information regarding the descent set of a permutation. Thus,
it is fruitful to consider the joint distribution of these statistics, which motivates the Euler–
Mahonian polynomial
An(t, q) =
∑
π∈Sn
tdes(π)qmaj(π),
which specializes to the Eulerian polynomial under the substitution q = 1. This polynomial
and various generalizations are widely of interest (see, e.g., [1, 5, 7, 9]).
2.2 Posets
Let Q be a partially ordered set (poset) on [n] with relation <Q. Given x, y ∈ Q, let x⋖Q y
denote the covering relation. Two elements x, y ∈ Q are incomparable if we have neither
544 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559
x <Q y nor y <Q x. A chain in Q, is a collection of elements x1, x2, . . . , xk ∈ Q such
that x1 <Q · · · <Q xk. A chain x1, x2, . . . , xk ∈ Q is called saturated if x1⋖Q · · ·⋖Qxk.
The Hasse diagram of Q is the graph with an oriented upwards direction such that there is
an edge from x up to y if and only if x ⋖ y. We say that Q is acyclic if for all x, y ∈ [n]
with x <Q y there is a unique saturated chain from x to y.
Given two posets Q1 and Q2, the ordinal sum Q1⊕Q2 is the poset on the disjoint union
of the ground sets of Q1 and Q2 such that x < y if
(i) x, y ∈ Q1 and x <Q1 y,
(ii) x, y ∈ Q2 and x <Q2 y, or
(iii) x ∈ Q1 and y ∈ Q2.
The poset Q is called graded (or ranked) if there is a function ρ : Q → Z≥0 such that if
x⋖Q y, then ρ(y) = ρ(x) + 1. While there are infinitely many rank functions for a graded
Q, there is a unique minimal rank function ρ such that ρ(x)−1 is not a valid rank function.
Given a poset Q on [n], we can generalize the notion of the descent statistic for per-
mutations. The descent set of Q is
Des(Q) := {(i, j) : i⋖Q j and i >Z j}
and thus the descent number of Q is des(Q) := |Des(Q)|. If Q is a graded poset on [n]
with minimal rank function ρ, we further have a notion of major index of Q
maj(Q) :=
∑
(i,j)∈Des(Q)
ρ(j).
We note that if Q is a totally ordered set with labels π1 <Q π2 <Q · · · <Q πn, these
quantities are precisely des(π) and maj(π).
2.3 Polytopes, fans, and h-vectors
A (convex) polytope P is the convex hull of finitely many point x1, . . . ,xk ∈ Rd. The di-
mension of P , denoted dim(P ), is the dimension of the smallest affine subspace containing
P . A face F of P is the collection of points where a linear functional ℓ ∈ (Rn)∗ is maxi-
mized on P . Faces of dimension 0 are called vertices and faces with dim(F ) = dim(P )−1
are called facets. A polytope P is called simple if every vertex is contained in exactly
dim(P ) many facets. The set of all faces of P forms a poset L(P ) under inclusion of
faces, which we will the face lattice of P . We say that two polytopes P1 and P2 are com-
binatorially equivalent if L(P1) = L(P2).
A polyhedral cone σ ⊂ Rn is solution set to the weak inequality Ax ≥ 0 for some real
matrix A. A cone σ is called pointed if σ contains no linear subspaces. The dimension of
σ, denoted dim(σ), is the dimension of the smallest affine subspace containing σ. A cone
is called simplical if it is defined by exactly dim(σ) many independent inequalities. A face
of σ is the subset obtained by replacing some of the defining equalities with equality. Two
cones σ1 and σ2 intersect properly if σ1 ∩ σ2 is a face of both σ1 and σ2. A collection of
cones F is called a fan if it is closed under taking faces, and any two cones σ1, σ2 ∈ F
intersect properly. We say F is a complete fan if F covers Rn.
E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 545
Let P be a polytope with face F . The normal cone of F in P , denoted NP (F ) is the
subset of linear functions ℓ ∈ (Rn)∗ whose maximum on P occurs at all points of F . That
is,
NP (F ) := {ℓ ∈ (Rn)∗ : ℓ(x) = max{ℓ(y) : y ∈ P} for all x ∈ F}
The normal fan of P , denote N (P ) is the complete fan formed by the normal cones of all
faces. Note that N (P ) is pointed if and only if dim(P ) = n. However, one can always
reduce N (P ) to a pointed fan in the space (Rn)∗/P⊥, where P⊥ ⊂ (Rn)∗ is the subset of
linear functionals constant on P .
Given a polytope P , the f -vector of P is the vector (f0(P ), f1(P ), . . . , fdim(P )(P ))
where fi(P ) is the number of i-dimensional faces of P . The f -polynomial of P is the
generating function fP (t) =
∑dim(P )
i=0 fi(P )t
i. Moreover, one can define f -vector and
f -polynomial of fan F in the obvious way. The f -vectors of a polytope P and its normal
fan N (P ) are related by fi(P ) = fdim(P )−i(N (P )).
Given P a simple polytope, or equivalently if F is a simplical fan, one can instead
consider a different vector. The h-vector of P is the vector (h0(P ), . . . , hdim(P )(P )) ∈
Zdim(P )+1≥0 and the h-polynomial is hP (t) =
∑dim(P )
i=0 hi(P )t
i defined uniquely by the
relation fP (t) = hP (t+ 1). Likewise, the h-polynomial of F , hF (t) =
∑dim(F)
i=0 hi(F)ti
is given by the relation tdim(F)fF (t−1) = hF (t + 1). Hence, the h-polynomial of a
polytope P and the h-polynomial of its normal fan N (P ) coincide. In this case, it happens
that the h-polynomial satisfies the Dehn-Sommerville relations hi(P ) = hdim(P )−i(P ) for
i = 0, 1, . . . ,dim(P ) (see, e.g., [25, Section 8.3]).
2.4 Generalized permutohedra
Given α = (α1, α2, . . . , αn) ∈ Rn such that 0 ≤ α1 < α2 < · · · < αn, the α-
permutohedron or usual permutohedron Παn ⊂ Rn is the convex hull of the Sn-orbit
of α. Note that this is an (n − 1)-dimensional polytope, as it lies in the hyperplane∑n
i=1 xi =
∑n
j=0 αj . Regardless of the choice of α, the normal fan of Π
α
n is the braid
fan is
Brn := {σ(π) : π ∈ Sn} ⊆ Rn/(1, 1, . . . , 1)
where the full dimensional cones σ(π) are
σ(π) = {x ∈ Rn/(1, 1, . . . , 1) : xπ1 ≤ xπ2 ≤ · · · ≤ xπn}.
See Figure 1 for the example of Br3. Given that any choice of α produces the normal fan
of Brn, we will usually consider usual permutohedron for α = (0, 1, 2, . . . , n− 1), which
we will simply denote Πn. It is a well-known result that the h-polynomial for Πn is given
by the Eulerian polynomial
hΠn(t) =
∑
π∈Sn
tdes(π).
Introduced by Postnikov in [20], a generalized permutohedron P ⊂ Rn is a convex
polytope whose normal cone N (P ) ⊂ Rn/(1, 1, . . . , 1) can be refined to Brn. We say that
N (P ) is a coarsening of Brn if there is a polytopal realization for N (P ) which can be
refined by Brn.
Definition 2.1. Suppose that F is a coarsening of Brn. Given a full-dimensional cone
σ ∈ F , the cone poset Qσ is a poset on [n] given by the relations i <Qσ j if xi ≤ xj for
all x ∈ σ.
546 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559
123213
312321
132231
x1 = x2
x2 = x3x1 = x3
Figure 1: Br3 in R3/(1, 1, 1).
It follows immediately from the definition that the cone poset Qσ is connected and
acyclic if and only if σ is a full-dimensional simplical cone. For additional exposition and
details on this correspondence, the reader should consult [21, Section 3]. Moreover, we
make the following observation.
Proposition 2.2. Given σ ∈ N (P ) be a full dimensional cone for a simple generalized
permutahedron P . Then the poset Qσ is an connected, acyclic, graded poset with a unique
minimal rank function ρ : Qσ → Z≥0.
Proof. Since P is simple, this implies that σ is simplicial. As noted above, it follows
directly from Definition 2.1 that the poset Qσ is connecteded and acyclic. This implies
that if x <Qσ y, there is a unique saturated chain from x to y. Hence, we can define a
rank function ρ such that ρ(x) ≥ 0 for all x ∈ Qσ and if x ⋖Qσ y then ρ(y) = ρ(x) + 1.
To obtain the unique minimal rank function, let ρ be any valid function above and define
ρ̃(x) = ρ(x)− α, where α = miny∈Qσ ρ(y).
Remark 2.3. In [21], the authors use the alternative language of a tree-poset, which is
poset whose Hasse diagram in a spanning tree on [n]. This is equivalent to a poset which
is acyclic and connected.
In the case of a simple generalized permutohedron, or rather a simplicial coarsening of
Brn, one can give a combinatorial formula for the h-polynomial in terms of descents on
acyclic posets, which is a natural generalization of the result for the usual permutohedron.
Theorem 2.4 ([21, Theorem 4.2]). Let P be a simple generalized permutahedron and let
{Qσ}σ∈N (P ) be the cone posets for full dimensional cones in the normal fan N (P ) as in
Definition 2.1. Then
hP (t) =
∑
σ∈N (P )
tdes(Qσ).
One should note that it is straightforward to verify that
hΠn(t) =
∑
π∈Sn
tdes(π).
using Theorem 2.4, as the posets for the full dimensional cones are simply linear orderings
of [n].
E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 547
3 Simple generalized permutohedra
In this section, we will introduce a bivariate generalization for the h-polynomial of any
simple generalized permutohedron. Particularly, we will give a formula for a q-analogue of
Theorem 2.4, which gives us the expected bivariate polynomial in the case of Πn. Unfor-
tunately, our generalization does not produce a polynomial invariant for the combinatorial
type of N (P ). Rather, the polynomials will vary based upon the particular choice of coars-
ening of Brn, and thus one may have combinatorially equivalent generalized permutohedra
with different polynomials.
Based on the obserations of Proposition 2.2, we can now give a q-analogue of the h-
polynomial for a simple generalized permutohedron.
Definition 3.1. Let P be a simple generalized permutahedron and let {Qσ}σ∈N (P ) be the
posets for full dimensional cones in the normal fan N (P ). Then, the q-h-polynomial is
given by
hP (t, q) :=
∑
σ∈N (P )
tdes(Qσ)qmaj(Qσ).
In the case of the usual permutohedron Πn, this q-analogue gives us the expected gen-
eralization. The full dimensional cones of the braid fan correspond to permutations π ∈ Sn
giving the total order Qπ which is π1 < π2 < · · · < πn. By definition, des(Qπ) = des(π)
and maj(Qπ) = maj(π). Thus we have
hΠn(t, q) =
∑
π∈Sn
tdes(π)qmaj(π)
which is the Euler–Mahonian polynomial, an expected q-analogue of the Eulerian polyno-
mial.
Unfortunately, this construction is not invariant under reordering of the ground set.
That is, the q-analogue depends on the choice of embedding or (equivalently) the choice of
coarsening of the braid fan, as demonstrated by the following example.
Example 3.2. Consider the associahedron A(3) ⊂ R3 which is the polytope whose normal
fan is obtained by merging exactly 2 full-dimensional cones that intersect in an edge in Br3
(see Section 5.3 for an in depth discussion of A(n)). Two different choices of coarsening
will produce combinatorially equivalent fans (resp. polytopes), but different multivariate
polynomials. If one coarsens the braid fan by merging the cones corresponding to the
permutations 132 and 312, the obtained q-analogue is hF1(t, q) = 1 + tq + 2tq
2 + t2q3.
Alternatively, if one instead coarsens the braid fan by merging the cones corresponding to
231 and 321, the obtained q-analogue is hF2(t, q) = 1+2tq+ tq
2 + t2q2. Of course when
q = 1 in either case we have hF (t) = 1 + 3t + t2 as expected. These two choices of
coarsening are depicted in Figure 2.
4 Nestohedra
In this section, we focus on a broad class of simple generalized permutohedra known as
nestohedra, for which one can be more explicit in producing combinatorial definitions for
these q-analogues. The nestohedra were first introduced by Postnikov [20]. To construct a
nestohedron, we need the notion of a building set.
548 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559
123213
321
231
x1 = x2
x2 = x3x1 = x3
1 3
2
123213
312
132
x1 = x2
x2 = x3x1 = x3
23
1
Figure 2: Two coarsenings of Br3 which are combinatorially equivalent but produce differ-
ent q-h-polynomials in Example 3.2.
Definition 4.1 ([20, Definition 7.1]). A collection B of nonempty subsets of [n] is called a
building set if it satisfies the following conditions:
1. If I, J ∈ B and I ∩ J ̸= ∅, then I ∪ J ∈ B.
2. B contains all singletons {i}, such that i ∈ [n].
A building set B is connected if [n] ∈ B. For any building set, one can define a
nestohedron. For any subset I ⊆ [n], let ∆I := conv{ei : i ∈ I}. The following
definition appears implicitly in the results of [20, Section 7], but stated explicitly in this
form in [21, Definition 6.3].
Definition 4.2 (see [21, Definition 6.3], [20, Section 7]). Given a building set B on [n].
The nestohedron PB on the building set B is the polytope obtained from the Minkowski
sum
PB :=
∑
I∈B
yI∆I
for some strictly positive parameters yI .
One can see that PB is a generalized permutohedron because N (∆I) is refined by Brn
and thus N (
∑
I∈B yI∆I) must also be refined by Brn [25, Proposition 7.12]. For our
purposes, we are primarily interested in the explicit cones and associated posets in N (PB).
These can be described through combinatorial means. Given a rooted tree T on [n] which
is directed such that all edges are oriented away from the root and a vertex i in T , let T≤i
be the tree of descendants of i. That is, j ∈ T≤i if there is a directed path from i to j in T .
We define B-trees for a connected building set B.
Definition 4.3 ([20, Definition 7.7]). For a connected building set B on [n], a B-tree is a
rooted tree T on the set [n] such that
1. For any i ∈ [n], one has T≤i ∈ B
2. For any k ≥ 2 incomparable nodes i1, . . . , ik ∈ [n], one has
⋃k
j=1 T≤ij ̸∈ B.
One can algorithmically construct all of these B-trees using the following proposition.
E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 549
Proposition 4.4 ([21, Proposition 8.5], [20, Proposition 7.8]). Let B be a connected build-
ing set on [n] and let i ∈ [n]. Let B1, . . . ,Br be the connected components of of the
restriction B|[n]\{i}. Then all B-trees with root at i are obtained by picking a Bj-tree Tj ,
for each component Bj , j = 1, . . . , r, and connecting the roots of T1, . . . , Tr to the node i
by edges.
For a building set B, a B-tree T has the structure of a poset by x⋖ y provided that the
there is an edge (x, y) and y is closer to the root. For ease of notation, we will write x⋖T y
to denote an edge (x, y) in T and to indicate which element is closer to the root. So,
Des(T ) = {(i, j) : i⋖T j and i >N j}
and des(T ) = |Des(T )|. Given x ∈ T , we say that the depth of x, denoted dp(x),
is the length of the unique path from x to the root. The depth of T is depth(T ) :=
maxx∈T dp(x). The major index of T is
maj(T ) :=
∑
(i,j)∈Des(T )
(depth(T )− dp(j))
Remark 4.5. Note that for any x ∈ T , the quantity depth(T ) − dp(x) is precisely ρ(x)
where ρ is the minimal rank function on the poset representation of T .
Proposition 4.6 ([21, Corollary 8.4]). For any connected building set B on [n], the h-
polynomial of the generalized permutohedron PB is
hB(t) =
∑
T
tdes(T )
where the sum is over B-trees T .
Given connected building sets B1, . . . ,Br on pairwise disjoint sets S1, . . . , Sr, we can
form the combined connected building set B on S =
⋃r
i=1 Si by B = (
⊔r
i=1 Bi) ⊔ {S}.
We will now give a formula for the h-polynomial of such a building set.
Proposition 4.7. Let B1, . . . ,Br be connected building sets on the pairwise disjoint sets
S1, . . . , Sr, and let B be the combined connected building set on S =
⋃r
i=1 Si. Then
hB(t) = (1 + t+ · · ·+ tr−1)
r∏
i=1
hBi(t).
Proof. Without loss of generality, let S = [n] and let the sets S1, . . . , Sr partition [n] such
that if x ∈ Si and y ∈ Sj , x < y if and only if i < j for every
1 ≤ i, j ≤ r. Let T be a B-tree with vertex i as the root. Suppose that i ∈ Sj for
some j. By Proposition 4.4, T is formed by connecting the root i to the roots of trees on
the connected components of B|[n]\{i}. Note that the connected components are precisesly
Bk where k ̸= j and the connected components of Bj |Sj\{i}. Therefore, T is formed by
Bk-trees T1, T2, . . . , Tr such that for all k ̸= j, the root of Tk is connected to the root of Tj
for some j = 1, 2, . . . , r. Additionally, given any collection of Bk-trees, we can form a B-
tree by simply choosing one of the trees Tj to contain the root. Therefore, we will consider
T as being partitioned into Bk-trees T1, T2, . . . , Tr with root in Tj in this way. Now, it is a
550 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559
straightforward computations to note that des(T ) = r−j+
∑r
k=1 des(Tk) as the construc-
tion preserves all existing descents in each tree Tk and introduces exactly one new descent
between Tj and Tk where k > j. Since we the choices of trees for each k are independent,
the contribution of all trees where Tj has the root to the h-polynomial is tr−j
∏r
k=1 hBk(t).
Thus, summing over all choices of j gives us the desired expression.
Now we give a different characterization of the q-h-polynomial of the generalized per-
mutohedron. This description comes from specializing Definition 3.1 to the case of nesto-
hedra, making use of alternative descriptions of the descent set and major index.
Proposition 4.8. For any connected building set B on [n], the q-h-polynomial of the gen-
eralized permutohedron PB is
hB(t, q) =
∑
T
tdes(T )qmaj(T )
where the sum is over B-trees T .
Define the statistic µ(T ) :=
∑
(i,j)∈T (depth(T )− dp(j)). Note that this statistic
depends only on the isomorphism type of the rooted tree T not on the labeling. With
this, we introduce a trivariate analogue of the h-polynomial of a nestohedron on connected
building set
hB(t, q, u) :=
∑
T
tdes(T )qmaj(T )uµ(T )
By the Dehn-Sommerville relations, we have that the h-polynomial is palindromic. In
certain cases, we can provide a multivariate analogue of palindromicity.
Theorem 4.9. Let B be a connected building set on [n] which is invariant under the invo-
lution ω : [n] → [n] such that ω(i) = n− i+1. Then the h-polynomial for the nestohedron
PB is
hB(t, q, u) = t
n−1hB(t
−1, q−1, qu)
Proof. Let B be a building set such that ω(B) = B. Suppose that T is a B-tree. By
Proposition 4.4, there exists a B-tree T̃ such that T and T̃ such that T̃ = ω(T ). That is, the
trees are isomorphic as unlabeled rooted trees, and one can obtain the appropriate labels of
one tree by applying the involution. It is clear that Des(T̃ ) = {(i, j) : (i, j) ̸∈ Des(T )}.
Hence des(T̃ ) = n− 1− des(T ) and maj(T̃ ) = µ(T )−maj(T ). This gives the equality
above.
5 Examples
We conclude with a section computing explicit examples of q-h-polynomials for nestohe-
dra of interest. Included in the list are Sn-invariant nestohedra, graph associahedra, the
associahedron, the stellahedron, and the Stanley–Pitman polytope.
5.1 Sn-invariant nestohedra
We will now specialize to the case of building sets which are invariant under the action of
Sn on the ground set [n]. Note that a connected building set B on [n] is Sn-invariant if and
only if
B =
{
{1}, . . . , {n},
(
[n]
j
)
, j = k, . . . , n
}
E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 551
for some 2 ≤ k ≤ n. Therefore, for a fixed n and fixed 2 ≤ k ≤ n, we will denote this
building set Bkn.
Proposition 5.1. Let Bkn be the Sn-invariant connected building set of [n] with minimal
nonsingleton set of cardinality k. Suppose that T1 and T2 are any two B-trees. Then
T1 and T2 are isomorphic as unlabeled rooted trees. Moreover, for any B-tree T , T ∼=
Ak−1⊕Cn−k+1 as a poset, where Ai is an antichain on i elements, Cj is a totally ordered
chain on j elements, and ⊕ is ordinal sum.
Proof. This follows from Proposition 4.4 with the observation that Bkn|[n]\{i} ∼= Bk−1n−1
which is a connected building set. Continuing in this fashion, repeated restictions will result
in connected building sets until we arrive at Bkn|[n]\W where W ⊂ [n] with |W | = n−k+1,
which consists only of singleton elements.
Theorem 5.2. Let Bkn be the Sn-invariant connected building set on [n] with minimal
nonsingleton set of cardinality k. The q-h-polynomial for the nestohedron PBkn is
hBkn(t, q) =
∑
A∈( [n]n−k+1)
∑
π∈SA
tdes(π)+|{j∈[n]\A : j>π1}|qmaj(π)+des(π)+|{j∈[n]\A : j>π1}|
Moreover, this polynomial satisfies
hBkn(t, q) = t
n−1q
k2−2kn−k+n2+3n−2
2 hBkn(t
−1, q−1).
Prior to giving the proof of this formula, it is instructive to give concrete example of
enumerating the descents in Bkn-trees.
Example 5.3. Consider the B58-tree T given in Figure 3. The descents which occur along
the chain are precisely the descents of the permutation π = 5481 ∈ S{1,4,5,8} which
has Des(5418) = {1, 3} and des(5418) = 2. Moreover, there are descents which oc-
cur between the antichain and the chain itself. The number of such descents is precisely
the number of elements of [8] \ {1, 4, 5, 8} which are larger than 5. There are precisely
2, and hence yielding des(T ) = des(5481) + |{j ∈ [8] \ {1, 4, 5, 8} : j > 5}| =
4. When computing the major index, we note that the contributions of π = 5418 is∑
i∈Des(5418)(i + 1) = maj(5418) + des(5418) = 4 + 2 = 6, to account for the cor-
rect rank. Moreover, every descent between the antichain and the chain has rank 1, so this
contributes a total of 2. Thus, maj(T ) = maj(5418)+des(5418)+|{j ∈ [8]\{1, 4, 5, 8} :
j > 5}| = 8.
Proof. By Proposition 5.1, we know that any T has the poset structure of Ak−1⊕Cn−k+1.
So any labeled tree is described by an n− k+1-element subset A of [n] and a permutation
π ∈ SA. The permutation labels Cn−k+1, and the remaining elements of [n] \ A label
the antichain Ak−1. There are two types of descents in the labeling: descents in Cn−k+1
which are enumerated by des(π), and descents where a label on the antichain Ak−1 is
greater than π1 which is enumerated by |{j ∈ [n] \ A : j > π1}|. To compute maj(T ),
note that if i ∈ Des(π) this corresponds to (j, ℓ) ∈ Des(T ) such that ρ(ℓ) = i + 1. So
the contribution from descents of this form is qmaj(π)+des(π). The other descents are of the
form (i, π1) ∈ Des(T ) and since ρ(π1) = 1, this contributes q|{j∈[n]\A : j>π1}|.
To see the palidromicity statement, note that since Bkn is Sn-invariant, then it is invariant
under the involution ω(i) = n − i + 1. It is clear that µ(T ) = k − 2 +
∑n−k+1
i=1 i =
552 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559
k2−2kn−k+n2+3n−2
2 for any B
k
n-tree T . Subsequently, applying the result of Theorem 4.9
and setting u = 1 yields the desired statement.
1
8
4
5
2 3 6 7
Figure 3: An example of a B58-tree T as appears in Example 5.3. By directly applying
the definitions of descent and major index statistics, we can see that des(T ) = 4 and
maj(T ) = 8.
5.2 Graph associahedra
We now consider a large family of examples of nestohedra arising from graphs. Given a
graph G = ([n], E), a tube of G is a proper, nonempty subset I ⊂ [n] such that the induced
subgraph G|I is connected. A k-tubing of G, χ, is a a collection of k distinct tubes subject
to:
1. For all incomparable A1, A2 ∈ χ, A1 ∪A2 ̸∈ χ (non-adjacency);
2. For all incomparable A1, A2 ∈ χ, A1 ∩A2 = ∅ (non-intersecting).
We do, however, allow for A1 ⊂ A2, which is called a nesting. We say that a tubing
χ is maximal if it cannot add any additional tubes to χ, or equivalently, if |χ| = n − 1.
Given a graph G, the graph associahedron of G is the polytope PG whose face lattice is
given by the set of all tubings of G where χ < χ′ if χ is obtained from χ′ by adding
tubes. Subsequently, the vertices of PG correspond to maximal tubings. This notion of
graph associahedra originates with Carr and Devadoss [12, 13] and has been a well-studied
family of examples of simple generalized permutohedra (see, e.g., [3, 6, 10, 11, 19]).
Remark 5.4. Given a simple graph G = ([n], E), the graph associahedron PG is an ex-
ample of nestohedron on a connected building set, even when G is not a connected graph.
The graphical building set of G, B(G) is the collection of nonempty J ⊆ [n] such that the
induced subgraph G|J is connected. While the building set B(G) is connected if and only
if G is connected (c.f. [21, Example 6.2]), the graph associahedra PG using the notions of
Carr and Devadoss [12, 13] is the nestahedron with building set B̂(G) = B(G)∪ [n] which
is always connected and B̂(G) = B(G) if G connected.
In light of Remark 5.4, we can specialize Proposition 4.7 to determine the h-polynomial
of a disconnected graph.
E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 553
Corollary 5.5. Let G be a simple graph on [n] with connected components G1, G2, . . . , Gk.
Then
hG(t) = (1 + t+ · · ·+ tk−1)
k∏
i=1
hGi(t).
Let G = ([n], E) be a simple graph and let χ be a maximal tubing of G. Given i ∈ [n],
the nesting index of i, denoted νχ(i), is the number of tubes containing i. The nesting
number of χ is nest(χ) := maxi∈[n] νχ(i). Given any maximal χ, observe that for any
tube Aj ∈ χ, there exists a unique element αj ∈ Aj such that for any tube Ak ⊂ Aj , we
have αj ̸∈ Ak. For convenience, we will write Ak ⋖ Aj if Ak ⊂ Aj and there is no tube
Aℓ such that Ak ⊂ Aℓ ⊂ Aℓ. Let αn denote the unique element which is not contained in
any tube of χ.
The nesting descent set is
NestDes(χ) :={(αk, αj) : αk > αj and Ak ⋖Aj}
∪{(αℓ, αn) : αℓ > αn and Aℓ ̸⊂ Ap for any Ap}.
The nesting descent number is
nestDes(χ) := |NestDes(χ)|
and the nesting major index is
nestMaj(χ) :=
∑
(αk,αj)∈NestDes(χ)
(nest(χ)− νχ(αj))
We now state a formula for the q-h-polynomial of graph associahedra in terms of graph
tubings.
Proposition 5.6. Let G be a simple graph. The q-h-polynomial is
hG(t, q) =
∑
χ
tnestDes(χ)qnestMaj(χ)
where the sum is taken over all maximal tubings χ.
Proof. This follows by unpacking the definitions of B-trees in terms of graph tubings and
applying Proposition 4.8.
Remark 5.7. As was the case with nestohedra in general, we should note that this poly-
nomial is invariant only under labeled graph automorphisms. Under most circumstance, a
different choice of labeling of the vertices G will produce a different bivariate polynomial.
However, the specialization under q = 1 is invariant under permutation of the ground set.
Remark 5.8. As with nestohedra, we can similarly define a trivariant polynomial for graph
associahedra, namely
hG(t, q, u) =
∑
χ
tnestDes(χ)qnestMaj(χ)uµ(χ)
where the sum ranges over all maximal and µ(χ) =
∑
(αk,αj)
(nest(T ) − νT (αj)) where
this sum is over all pairs (αk, αj) such that Ak ⋖Aj , which is a direct translation of the µ
554 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559
statistic for nestohedra. If the involution ω : [n] → [n] such that ω(i) = n− i+1 produces
a labeled graph automorphism, then Theorem 4.9 gives us that palindromicity statement
hG(t, q, u) = t
n−1hG(t
−1, q−1, qu).
There are only two Sn-invariant graphs, namely the complete graph Kn and the null graph
Nn = Kn (i.e. the edgeless graph), which produce only the simplest examples of gen-
eralized permutohedra. PKn is the usual permutohedron Πn, and hence hKn(t, q) is the
usual Euler–Mahonian polynomial. PNn is simply an n− 1 dimensional simplex and thus
hNn(t, q) =
∑n−1
i=0 (tq)
i.
5.3 The associahedron and a new q-analogue of Narayana numbers
The associahedron A(n), which first appeared in the work of Stasheff [24], as well as the
notable work of Lee [17], is the graph associahedron for G = Path(n), where the vertices
are labeled linearly. It is well-known that
hPath(n)(t) =
n∑
k=1
N(n, k)tk−1
where N(n, k) = 1n
(
n
k
)(
n
k−1
)
is the Narayana number, which refine the Catalan numbers.
That is, hPath(n)(1) = Cn. To verify this formula, one should note that B-trees, or graph
tubings on Path(n), are in bijection with binary trees on n vertices (See [20, Section 8.2]).
The bijection sends descents in a B-tree to right edges in an unlabeled binary tree and
N(n, k) is known to enumerate the number of unlabeled binary trees on n vertices with
k − 1 right edges. Subsequently, we will phrase all formulae in terms of binary trees.
Let T be a binary tree. Given an edge e ∈ T , let dp(e) be the length of the path from
the root vertex to the closest vertex incident with e. Let depth(T ) = maxe∈T dp(e). The
right multiset of T is the multiset
R(T ) := {dp(e) : e is a right edge of T} .
The right number of T is r(T ) = |R(T )| and the right index of T is
rindex(T ) := depth(T )r(T )−
∑
j∈R(T )
j.
By translating the general results for nestohedra into the above language for binary trees,
we have the following:
Corollary 5.9. The q-h-polynomial for the associahedron is
hPath(n)(t, q) =
∑
T
tr(T )qrindex(T )
where the sum ranges over all rooted unlabelled binary tree T on n vertices.
Remark 5.10. This theorem gives rise to a q-analogue of the Narayana numbers. We say
the (alternative) q-Narayana number is
N(n, k, q) =
∑
T
r(T )=k−1
qrindex(T ).
E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 555
It is clear that the substitution q = 1 yields N(n, k) as desired. We call these the alter-
native q-Narayana numbers because, while this is the natural q-analogue in the context of
generalized permutohedra as it arises from the major index, this does not agree with the
usual q-Narayana number in the literature (see, e.g., [8, 22]).
5.4 The stellahedron
The star graph on n + 1 vertices is the complete bipartite graph K1,n. The stellohedron
is the graph associahedron associated to K1,n. Let K1,n be labeled such that the center
vertex is labeled n + 1. Recall that a partial permutation of [n] is a linear ordering of a
k-subset L ⊆ [n] for some k = 1, 2, . . . n. The B-trees for K1,n are in bijection with partial
permutations of [n]. In particular, the structure of a B-tree is given by the ordinal sum of
an antichain with a totally ordered chain An−k−1 ⊕Ck+1 for some k = 0, . . . , n such that
the minimal element of Ck+1 has label n+ 1.
2
35
6
1
4
7
2 3 4 5
6
1
7
Figure 4: A tubing of K1,6 and its corresponding B-tree.
To see this, note that we can identify the B-trees with graph tubings. Any tubing of
K1,n is either
(i) the tubing where each vertex i = 1, 2, . . . , n is in a singleton tube and n + 1 is the
root, or
(ii) some vertex i is the root and we have a tube containing all other vertices.
In the case of (ii), once i is chosen, then the tubing directly arises from a tubing of K1,n−1
on the labels [n+ 1] \ {i}. Thus, by induction, we will have B-trees of the proposed form.
For example, consider the tubing and B-tree given in Figure 4, which corresponds to the
partial permutation π = 61 on [6].
Subsequently, the elements of the Ck+1 above the n+1 are the partial permutation (see
[21, Section 10.4]) With this in mind, we can state the q-analogue of the h-polynomial for
the stellohedron.
Proposition 5.11. The q-h-polynomial for the stellohedron is
hK1,n(t, q) = 1 +
∑
w
tdes(w)+1qmaj(w)+2des(w)+2
where the sum is over all nonempty partial permutations of [n].
556 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559
Proof. The labels on Ck+1 correspond to a partial permutation of w̃ of [n + 1] where
w̃1 = n + 1. Thus, we consider w to be the partial permutation of [n] with this first
element omitted. If w = ∅, the corresponding B-tree has no descents. If w ̸= ∅, then the
corresponding B-tree T has precisely des(w) + 1 descents, due to the guaranteed descent
between n + 1 and w1. When computing the major index, note that if i ∈ Des(w), this
means that we have an element of rank i + 2 where a descent occurs in T . Hence, the
contribution to the major index is
∑
i∈Des(w)(i + 2) = maj(w) + 2des(w). Additionally,
the descent between n+ 1 and w1 contributes 2, as ρ(w1) = 2. Thus, we have the desired
formula.
5.5 The Stanley-Pitman polytope
Introduced by Stanley and Pitman in [23], the Stanley-Pittman polytope is a integral poly-
tope defined by the equations
PS(n) :=
{
x ∈ Rn : xi ≥ 0 and
j∑
i=1
xi ≤ j for each 1 ≤ j ≤ n
}
.
This polytope is combinatorially equivalent to an n-cube, as illustrated in Figure 5. How-
ever, this polytope is of particular interest as it appears naturally when studying empirical
distributions in statistics and has connections to many combinatorial objects, such as park-
ing functions and plane trees. Postnikov [20, Section 8.5] observed that this polytope can
be realized as the nestohedron from the building set
BPS = {[i, n], {i} : i ∈ [n]},
where [i, n] = {i, i + 1, . . . , n}. Notably, this is not a graph associahedron. Given that
this polytope is combinatorially equivalent to an n-cube, we have hBPS(t) = (1 + t)
n−1
[23, Theorem 20]. We now give the q-analogue.
x1
x2
x1
x2
x3
Figure 5: PS(2) and PS(3).
Proposition 5.12. The q-h-polynomial for the Stanley-Pitman polytope is
hBPS(t, q) =
n−2∑
ℓ=0
(
n− 2
ℓ
)
tℓq
ℓ2+3ℓ+2
2
(
t+ qℓ
)
.
E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 557
3
4
6
75
21
3
4
7
65
21
Figure 6: Two BPS-trees for n = 7 from the increases sequences I1 = {3 < 4 < 6 < 7}
and I2 = {3 < 4 < 7}. Alternatively, these are the two trees from the set {3, 4} ⊂ [5].
Proof. First note that hBPS(t, 1) = (t + 1)
n−1, so this agrees with the known results.
To compute this, we will need BPS-trees, which as determined by Postnikov, Reiner, and
Williams [21, Section 10.5], are formed in the following way. Given any increasing se-
quence of positive integers I = {i1 < i2 < · · · < ik = n} where we let i1 be the root and
form the chain of edges (i1, i2), (i2, i3), . . . , (ik−1, ik) and for all j ∈ [n] \ I we have the
edge (is, j) where is is the minimal element of I such that is > j. An example can be seen
in Figure 6.
It is clear that all descents will be occur along the chain of edges. So, we must consider
two cases:
(i) ik−1 = n− 1 and
(ii) ik−1 ≤ n− 2.
In case (i), for convenience let ℓ = k − 2. We form a tree T by choosing a subset J ∈(
[n−2]
ℓ
)
and arranging it increasing order to form a chain of edges which ends in (iℓ, n−1),
(n− 1, n). By definition, depth(T ) = ℓ+ 1, des(T ) = ℓ+ 1, and maj(T ) = (ℓ+ 1)2 −∑ℓ
i=0 i =
ℓ2+3ℓ+2
2 . So, the contribution of trees of this form to the q-h-polynomials is
n−2∑
ℓ=0
(
n− 2
ℓ
)
tℓ+1q
ℓ2+3ℓ+2
2 . (5.1)
In case (ii) where ik−1 ̸= n − 1, for ease of notation, let ℓ = k − 1. Similarly, we
form such a tree T by choosing J ∈
(
[n−2]
ℓ
)
and arranging it increasing order to form a
chain of edges which ends in (iℓ, n). Note that, when including the elements not in the
chain, we gain edges from the vertex n going away from the root, in particular, the edge
(n, n− 1). So, we again have depth(T ) = ℓ+1. However, we now have des(T ) = ℓ, and
maj(T ) = (ℓ + 1)2 −
∑ℓ−1
i=0 i =
ℓ2+5ℓ+2
2 . So the contribution of trees of this type to the
q-h-polynomial is
n−2∑
ℓ=0
(
n− 2
ℓ
)
tℓq
ℓ2+5ℓ+2
2 . (5.2)
Summing (5.1) and (5.2) and simplifying gives the desired expression.
558 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559
Remark 5.13. We conclude our discussion by noting that our computation produces an
alternative q-analogue of
(
n−1
ℓ
)
, namely(
n− 2
ℓ− 1
)
q
ℓ2+ℓ
2 +
(
n− 2
ℓ
)
q
ℓ2+5ℓ+2
2 .
This, of course, reduces to
(
n−1
ℓ
)
when q = 1 and arises quite naturally from generalizing
the major index statistic. However, this is not the usual q-analogue of a binomial coefficient
which arises in many natural ways, such as bit string inversions and lattice path areas.
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