Table Of ContentTREES, FUNCTIONAL EQUATIONS, AND COMBINATORIAL
HOPF ALGEBRAS
7
FLORENT HIVERT, JEAN-CHRISTOPHE NOVELLI, AND JEAN-YVES THIBON
0
0
2 Abstract. One of the main virtues of trees is to represent formal solutions of
n variousfunctional equationswhichcanbe castinthe formoffixedpoint problems.
a Basic examples include differential equations and functional (Lagrange) inversion
J
in power series rings. When analyzed in terms of combinatorialHopf algebras,the
9
simplest examples yield interesting algebraic identities or enumerative results.
1
]
O
C 1. Introduction
.
h Let R be an associative algebra, and consider the functional equation for the power
t
a series x ∈ R[[t]]
m
(1) x = a+B(x,x)
[
1 wherea ∈ RandB(x,y)isabilinearmapwithvaluesinR[[t]],suchthatthevaluation
v of B(x,y) is strictly greater than the sum of the valuations of x and y. Then, (1) has
9
a unique solution
3
5
(2) x = a+B(a,a)+B(B(a,a),a)+B(a,B(a,a))+··· = B (a)
1 T
0 T∈XCBT
7
0 where CBT is the set of (complete) binary trees, and for a tree T, BT(a) is the result
/ of evaluating the expression formed by labeling by a the leaves of T and by B its
h
t internal nodes.
a
m Of course, the same can be done with m-ary trees, or more generally with plane
: trees. We are in particular interested in those counted by the little Schr¨oder numbers,
v
that is, plane trees without vertex of arity 2 [18, A001003], which solve equations of
i
X
the form
r
a (3) x = a+ F (x,x,...,x)
n
Xn≥2
each F being an n-linear operation.
n
All this is well-known and rather trivial. However, the simplest example has still
something to tell us. Consider the differential equation (for x ∈ K[[t]])
dx
(4) = x2, x(0) = 1.
dt
Its solution is obviously x = (1 − t)−1, but let us ignore this for the moment, and
recast it as a fixed point problem
t
(5) x = 1+ x2(s)ds = 1+B(x,x),
Z
0
1
2 F. HIVERT,J.-C. NOVELLI,ANDJ.-Y. THIBON
t
where B(x,y) = x(s)y(s)ds. Then, for a binary tree T with n+ 1 leaves, B (1)
0 T
is the monomial oRbtained by putting 1 on each leaf and integrating at each internal
node the product of the evaluations of its subtrees:
t4/8
trrrrr OOOOOt2/2
(6) (cid:7)(cid:7)(cid:7) 777 ||| 333
1 1 t 3
xxxx FFFF 33
1 1 1
One can observe that
tn
(7) B (1) = c ,
T T′
n!
where T′ is the incomplete binary tree with n nodes obtained by removing the leaves
of T, and c is the number of permutations σ ∈ S whose decreasing tree has shape
T′ n
T′. Indeed, c is explicitly given by a hook length formula [8], which can becompared
T′
with the easily obtainedclosed formforB (1). The hook lengths ofT′ arethe number
T
of nodes of all the subtrees
' &!4%"$#
(8) ' &!1%"$#(cid:2)(cid:2)(cid:2) <<<' &!2%"$#
' &!1%"(cid:2)$#(cid:2)(cid:2)
and c is n! over the product of the hook lengths, here 4!/(4 · 2 · 1 · 1) = 3, the
T′
corresponding decreasing trees being
4 4 4
(cid:2)(cid:2) << (cid:2)(cid:2) << (cid:2)(cid:2) <<
(9) 1 3 2 3 3 2
(cid:2)(cid:2) (cid:2)(cid:2) (cid:2)(cid:2)
2 1 1
Our starting point will be the following question: can one use this observation
to derive the hook length formula for binary trees, and if yes, can we use the same
method to obtain more interesting results ?
For this, we have to lift our problem to the combinatorial Hopf algebra of Free
quasi-symmetric functions FQSym. We can then derive in the same way the q-hook
length formulas of Bj¨orner and Wachs [1,2]. The case of plane trees can bedealt with
in the same way, the relevant Hopf algebra being there WQSym, the Word Quasi-
Symmetric invariants (or quasi-symmetric functions in noncommutative variables),
and here the resulting formula is believed to be new. Finally, we give new proofs
of some identities of Postnikov [16] and Du-Liu [6] by relating these to appropriate
functional equations.
Notations. The symmetric group is denoted by S . The standardized Std(w) of a word
n
w of length n is the permutation obtained by iteratively scanning w from left to right, and
labelling 1,2,... the occurrences of its smallest letter, then numbering the occurrences of
the next one, and so on. All algebras are over a field K of characteristic 0.
TREES AND COMBINATORIAL HOPF ALGEBRAS 3
2. Free quasi-symmetric functions and hook length formulas
2.1. A derivation of FQSym. Recall from [5] that for an (infinite) totally ordered
alphabet A, FQSym(A) is the subalgebra of KhAi spanned by the polynomials
(10) G (A) = w
σ
X
Std(w)=σ
the sum of all words in An whose standardization is the permutation σ ∈ S . The
n
multiplication rule is, for α ∈ S and β ∈ S ,
k l
(11) G G = G .
α β γ
γ∈SkX+l;γ=u·v
Std(u)=α,Std(v)=β
This sum has k+l terms. Hence, the linear map
k
(cid:0) (cid:1) tn
(12) φ : G 7−→ (σ ∈ S )
σ n
n!
is a homomorphism of algebras FQSym → K[[t]]. It is convenient to introduce the
notation F = G and a scalar product satisfying hF ,G i = δ . As a graded
σ σ−1 σ τ σ,τ
bialgebra, FQSymisself-dual,anditscoproduct∆satisfieshFG,Hi = hF⊗G,∆Hi.
Let ∂ be the linear map defined by
(13) ∂G = G
σ σ′
where σ′ is the permutation whose word is obtained by erasing the letter n in σ ∈ S .
n
Obviously,
d
(14) φ(∂F) = φ(F)
dt
for all F ∈ FQSym, and moreover:
Proposition 2.1. The map ∂ is a derivation of FQSym. It is the adjoint of the
linear map F 7→ F ·F .
1
Proof – By definition, h∂G ,F i = δ is equal to 1 if σ occurs in τ n and to 0
σ τ σ′,τ
otherwise. Hence,
(15) h∂G ,F i = hG ,F F i
σ τ σ τ 1
whence the second part of the proposition. Now, F is a primitive element, so that
1
∂ is a derivation.
The Leibniz relation
(16) ∂(G G ) = ∂G ·G +Gα·∂G
α β α β β
can be interpreted in terms of the dendriform structure of FQSym. Recall [10] that
the product G G can be split into two parts (the dendriform operations)
α β
(17) G G = G ≺ G +G ≻ G ,
α β α β α β
(18) G ≺ G = G ,
α β γ
X
γ=u·v, max(u)>max(v)
Std(u)=α,Std(v)=β
4 F. HIVERT,J.-C. NOVELLI,ANDJ.-Y. THIBON
(19) G ≻ G = G .
α β γ
X
γ=u·v, max(u)≤max(v)
Std(u)=α,Std(v)=β
Then,
(20) ∂(G ≺ G ) = ∂G ≺ G , ∂(G ≻ G ) = G ≻ ∂G .
α β α β α β α β
It will be convenient to consider the half products as also defined on permutations,
so that their sum is then the convolution α∗β.
2.2. A differential equation in FQSym. It follows from Proposition 2.1 that if
we set X = (1−G )−1, we have
1
(21) ∂X = X2
with X = 1 (constant term), and φ(X) = (1−t)−1.
0
Note that thanks to the multiplication formula (11),
(22) X = G = w
σ
Xσ wX∈A∗
is the sum of all permutations (interpreted as G’s), that is, the sum of all words. If
t
we can lift to FQSym the scalar bilinar map B(x,y) = x(s)y(s)ds, it will also be
0
interpretable as the sum of all complete binary trees. R
2.3. The bilinear map. The required map is given by a simple operation, already
introduced in [5], precisely with the aim of providing a better understanding of the
Loday-Ronco algebra [10] of planar binary trees.
For α ∈ S , β ∈ S , and n = k +l, set
k l
(23) B(G ,G ) = G .
α β γ
X
γ=u(n+1)v
Std(u)=α,Std(v)=β
Clearly,
(24) ∂B(G ,G ) = G G ,
α β α β
and our differential equation is now equivalent to the fixed point problem
(25) X = 1+B(X,X).
Theorem 2.2. In the binary tree solution (2) of (25),
(26) B (1) = G ,
T σ
X
T(σ)=T
where T (σ) denotes the shape of the decreasing tree of the permutation σ. In partic-
ular, B (1) coincides with P , the natural basis of the Loday-Ronco algebra (in the
T T
notation of [7]).
TREES AND COMBINATORIAL HOPF ALGEBRAS 5
Proof – By induction on the number n of internal nodes of T. For n = 1 the result
is obvious, and if n > 1,
B (1) = B(B (1),B (1)),
T T′ T′′
where T′ and T′′ are the left an right subtrees of T. Hence, B (1) is the sum of the
T
G for σ = αnβ such that G occurs in B (1) and G occurs in B (1).
σ Std(α) T′ Std(β) T′′
Since we have assummed that (26) holds for T′ and T′′, this implies that it holds for
T as well.
Corollary 2.3 (The hook length formula). The number of permutations whose de-
creasing tree has shape T is
n!
(27) ,
h
v∈T v
where for a vertex v of T, h is the nuQmber of nodes of the subtree with root v.
v
2.4. The q-hook length formula. Recall that under the q-specialization
1
(28) A = := {... < qn < qn−1 < ... < q < 1}
1−q
we have [9, (125)]
1 qimaj(σ)
(29) G =
σ
(cid:18)1−q(cid:19) (q)
n
where imaj(σ) = maj(σ−1), maj(σ)istheclassical majorindex(sumofthedescents)
of σ ∈ S and (q) = (1−q)(1−q2)···(1−qn).
n n
Hence, the map
qimaj(σ)tn 1
(30) φ (G ) = = (t(1−q))nG
q σ σ
[n] ! (cid:18)1−q(cid:19)
q
is a homomorphism of algebras. The image of (25) under φ reads
q
(31) x = 1+B (x,x)
q
where the bilinear map is now a q-integral
t
(32) B (f,g) = d sf(s)g(qs),
q q
Z
0
where the q-integral is defined by
t tn+1
(33) snd s = .
q
Z [n+1]
0 q
To show this, we have to compute φ (B(G ,G )).
q α β
Lemma 2.4. Let α ∈ S , β ∈ S . The inverse major index is distributed over the
k l
half-products according to
k +l−1
(34) qimaj(γ) = qimaj(α)+imaj(β) ,
(cid:20) l−1 (cid:21)
γ∈Xα≻β q
6 F. HIVERT,J.-C. NOVELLI,ANDJ.-Y. THIBON
and
k +l−1
(35) qimaj(γ) = qimaj(α)+imaj(β)+l .
(cid:20) l (cid:21)
γ∈Xα≺β q
Proof – Straightformward by induction on n = k +l.
From this, on deduces immediately
k +l
(36) qimaj(γ) = qimaj(α)+imaj(β)+l ,
(cid:20) k (cid:21)
γ=uX·(n+1)·v q
Std(u)=α,Std(v)=β
which in turn implies the following:
Lemma 2.5. If f(t) = φ (F) and g(t) = φ (G), then
q q
t
(37) φ (B(F,G)) = d sf(s)g(qs).
q q
Z
0
Corollary 2.6 (The q-hook length formula of [1]). The inverse major index polyno-
mial of the set of permutations whose decreasing tree has shape T is
qδv
(38) qimaj(σ) = [n] ! ,
q
[h ]
T(Xσ)=T vY∈T v q
where δ is the number of nodes in the right subtree of v.
v
2.5. Another approach. It has been observed in [5] that FQSym had a natural
q-deformation, obtained by replacing the ordinary shuffle by the q-shuffle in
q
the product formula for the basis F . That is, FQSym is the algebra with basis
σ q
F = G and product rule
σ σ−1
(39) F F = (γ|α β[k])F = (γ|α β[k])ql(γ)−l(β)−l(α)F
α β q γ γ
Xγ Xγ
where (γ|f) means the coefficient of γ in f, k is the length of α and β[k] = (β +
1
k)···(β +k), (the shifted word), l(σ) being the number of inversions of σ.
l
Then, the map φ : FQSym → K[[t]] defined by
q q
tn
(40) φ (G ) =
q σ
[n] !
q
is a homomorphism of algebras.
One has now
(41) φ (∂F) = D φ (F)
q q q
where D is the q-derivative
q
f(qt)−f(t)
(42) D f(t) = .
q
qt−t
In FQSym , ∂ is not anymore a derivation, but satisfies
q
(43) ∂(FG) = ∂F(A)·G(qA)+F(A)·∂G(A)
TREES AND COMBINATORIAL HOPF ALGEBRAS 7
so that the noncommutative functional equation is now
(44) ∂X(A) = X(A)X(qA), X = 1
0
and its one-variable projection under φ is
q
(45) D x(t) = x(t)x(qt), x(0) = 1.
q
This is equivalent to
(46) x = 1+B (x,x)
q
where we have again
t
(47) B (x,y) = d sx(s)y(qs).
q q
Z
0
Theorem 2.7 (q-hook length formula for inversions [2]). The inversion polynomial
of the set of permutations having a decreasing tree of shape T is given by the same
hook length formula as for the inverse major index,
qδv
(48) ql(σ) = [n] ! ,
q
[h ]
T(Xσ)=T vY∈T v q
In particular imaj and l are equidistributed on these sets.
This is a refinement of a classical result of Foata and Schu¨tzenberger.
3. Word quasi-symmetric functions and plane trees
Tointerpret(3),weneedtoworkinWQSym,thealgebraofWordQuasi-Symmetric
functions, which contains an algebra of plane trees (the free dendriform trialgebra on
one generator [11]) in the same way as FQSym contains an algebra of binary trees
[14].
The basis elements M of WQSym are labeled by packed words u, or if one
u
prefers, surjections [n] → [k], set compositions, or facets of the permutohedron [3].
These objects are counted by the ordered Bell numbers [18, A000262]. There is a
canonical way to associate a plane tree to such an object [14], and the sums over the
fibers of this map span a Hopf subalgebra of WQSym. Hence, we need to define on
WQSym an analogue of our derivation ∂ of FQSym.
Recall that a word w over the aphabet of positive integers is said to be packed if
the set of letters occuring in w is an initial interval [a ,a ] of the alphabet A. The
1 k
packed word u = pack(w) associated to a word w ∈ A∗ is obtained by the following
process. If b < b < ... < b are the letters occuring in w, u is the image of w by
1 2 k
the semigroup homomorphism b 7→ a . For example, pack(34364) = 12132. A word
i i
u is said to be packed if pack(u) = u. To such a word is associated a polynomial M ,
u
defined as the sum of all words w such that pack(w) = u.
The product on WQSym is given by
(49) M M = M ,
u′ u′′ u
u∈Xu′⋆u′′
8 F. HIVERT,J.-C. NOVELLI,ANDJ.-Y. THIBON
where the convolution u′ ⋆u′′ of two packed words is defined as
(50) u′ ⋆u′′ = u.
v,w;u=v·w∈PW,paXck(v)=u′,pack(w)=u′′
For example,
(51) M M = M +M +M +M +M .
11 21 1121 1132 2221 2231 3321
The coproduct can be defined by the usual trick of noncommutative symmetric
functions, considering the alphabet A as an ordered sum of two mutually commuting
alphabets A′+ˆA′′. First, by direct inspection, one finds that
(52) M (A′+ˆA′′) = M (A′)M (A′′),
u (u|[1,k]) pack(u|[k+1,max(u))
X
0≤k≤max(u)
where u| denote the subword obtained by restricting u to the subset B of the
B
alphabet.
For a packed word u, let u′ be the word obtained from u by erasing all the oc-
curences of the maximal letter m = max(u), e.g., (5211354)′ = 21134. Now, define a
linear map δ by
(53) δM = M .
u u′
This is not anymore a derivation, but rather a finite difference operator: indeed, it
follows from (52) that
(54) δM (A) = M (A+ˆ1)−M (A),
u u u
where A+ˆ1 is the ordered sum of A and {1} (the scalar 1, so that M (1) = 1 if u is
u
of the form 11···1, and is 0 otherwise). Alternatively, δ is the adjoint of the right
multiplication by M∗ , where M∗ is the dual basis of M .
n≥1 1n u u
This implies thaPt δ satisfies
(55) δ(FG) = (δF)G+(δF)(δG)+F(δG),
but this formula can be refined in terms of the tridendriform structure of WQSym
[14]. Indeed, it is known that WQSym+ is a sub-dendriform trialgebra of KhAi+,
the partial products being given by
(56) M ≺ M = M ,
w′ w′′ w
w=u·v∈w′⋆w′′,|u|X=|w′|;max(v)<max(u)
(57) M ◦M = M ,
w′ w′′ w
w=u·v∈w′⋆w′′,|u|X=|w′|;max(v)=max(u)
(58) M ≻ M = M r,.
w′ w′′ w
w=u·v∈w′⋆w′′,|u|X=|w′|;max(v)>max(u)
and it follows from the multiplication rule (49) that
(59) δ(F ≺ G) = (δF)G, δ(F ◦G) = (δF)(δG), δ(F ≻ G) = F(δG).
TREES AND COMBINATORIAL HOPF ALGEBRAS 9
Now, let
(60) X = (1−qM )−1 = q|u|M = q|w|w.
1 u
Xu Xw
It follows from (55) that
(61) δX = qX2(1−qX)−1 = qn−1Xn.
Xn≥2
For packed words u ,...,u , define
1 k
(62) F (M ,...,M ) = M
k u1 uk w
X
where the sums runs over packed words w such that
(63) w = w mw m···mw , pack(w ) = u , m = max(w ,...,w )+1.
1 2 k i i 1 k
For example,
(64) F (M ,M ) = M +M +M +M +M .
2 11 21 11321 11432 22321 22431 33421
Then, obviously,
(65) X = 1+ qn−1F (X...,X).
n
Xn≥2
which does indeed give back (61), since
(66) δF (M ,...,M ) = M ···M .
k u1 uk u1 uk
It follows from (49) that the linear map ψ : WQSym → K[[t]] defined by
t
(67) ψ(M ) =
u
(cid:18)max(u)(cid:19)
isahomomorphismofalgebras. Moreover, itmapsδ overthefinitedifferenceoperator
(68) ψ(δF) = ∆ψ(F)
where ∆f(t) = f(t+1)−f(t). Hence, the images of (61) and (65) by ψ are
(69) ∆x = qn−1xn
Xn≥2
(70) x = 1+ qn−1F (x,x,...,x),
n
Xn≥2
where
(71) F (x ,...,x ) = Σtx (s)x (s)···x (s)δs,
n 1 n 0 1 2 n
the discrete integral being defined by
t−1
(72) Σtf(s)δs = f(i).
0
Xi=0
10 F. HIVERT,J.-C. NOVELLI,ANDJ.-Y. THIBON
The realization of the free dendriform trialgebra given in [14] involves the following
construction. With any word w of length n, associate a plane tree T(w) with n+1
leaves, as follows: if m = max(w) and if w has exactly k −1 occurences of m, write
(73) w = v mv ···v mv ,
1 2 k−1 k
where the v may be empty. Then, T(w) is the tree obtained by grafting the subtrees
i
T (v ),T(v ),...,T(v ) (in this order) on a common root, with the initial condition
1 2 k
T (ǫ) = ∅ for the empty word. For example, the tree associated with 243411 is
(74) *{{{4{{*JJJ4JJJ4
(cid:20)(cid:20)(cid:20)*** (cid:19)(cid:19)(cid:19)*** (cid:8)(cid:8)(cid:8) 444
(cid:20)2 (cid:19)3 (cid:8)1 1
From the previous considerations, one can now deduce a closed formula for the
number of packed words yielding a given plane tree, which can beregardedas another
generalization of the hook length formula for binary trees:
Theorem 3.1. If a term F (1) in the plane tree solution has the decomposition
T
t
(75) F (1) = c
T k
(cid:18)k(cid:19)
Xk
then, c is the number of packed words u with maximal letter k such that T (u) = T.
k
Proof – A straightforward induction, from (63) and (73).
For example, the following tree
Σt3
(76) F (F (1,1),F (1,1),F (1,1,1)) = tppppppppp t PPPPPPPPPPt
3 2 2 3 1 : 9
(cid:13)(cid:13)(cid:13) 111 (cid:4)(cid:4)(cid:4) ::: (cid:5)(cid:5)(cid:5) 999
(cid:13) (cid:4) (cid:5)
1 1 1 1 1 1 1
gives
t t t
(77) Σts3δs = 6 +6 +
0 (cid:18)4(cid:19) (cid:18)3(cid:19) (cid:18)2(cid:19)
so that there are 6+6+1 = 13 packed words whose plane trees have this shape:
*{{{2{{*JJJ2JJJ4
(cid:20)(cid:20)(cid:20)*** (cid:19)(cid:19)(cid:19)*** (cid:8)(cid:8)(cid:8) 444
(cid:20)1 (cid:19)1 (cid:8)1 1
*{{{3{{*JJJ3JJJ4 *{{{3{{*JJJ3JJJ4 *{{{3{{*JJJ3JJJ4 *{{{3{{*JJJ3JJJ4 *{{{3{{*JJJ3JJJ4 *{{{3{{*JJJ3JJJ4
(cid:20)(cid:20)(cid:20)*** (cid:19)(cid:19)(cid:19)*** (cid:8)(cid:8)(cid:8) 444 (cid:20)(cid:20)(cid:20)*** (cid:19)(cid:19)(cid:19)*** (cid:8)(cid:8)(cid:8) 444 (cid:20)(cid:20)(cid:20)*** (cid:19)(cid:19)(cid:19)*** (cid:8)(cid:8)(cid:8) 444 (cid:20)(cid:20)(cid:20)*** (cid:19)(cid:19)(cid:19)*** (cid:8)(cid:8)(cid:8) 444 (cid:20)(cid:20)(cid:20)*** (cid:19)(cid:19)(cid:19)*** (cid:8)(cid:8)(cid:8) 444 (cid:20)(cid:20)(cid:20)*** (cid:19)(cid:19)(cid:19)*** (cid:8)(cid:8)(cid:8) 444
(cid:20)1 (cid:19)1 (cid:8)2 2 (cid:20)1 (cid:19)2 (cid:8)1 1 (cid:20)1 (cid:19)2 (cid:8)2 2 (cid:20)2 (cid:19)1 (cid:8)1 1 (cid:20)2 (cid:19)1 (cid:8)2 2 (cid:20)2 (cid:19)2 (cid:8)1 1
