Faà di Bruno Hopf algebras
Héctor Figueroa,1 José M. Gracia-Bondía2,3 and Joseph C. Várilly1∗

1 Escuela de Matemática, Universidad de Costa Rica, San José 11501, Costa Rica
2 CAPA and Departamento de Física Teórica, Universidad de Zaragoza, Zaragoza 50009, Spain

3 Laboratorio de Física Teórica y Computacional, Universidad de Costa Rica, San Pedro 11501, Costa Rica

Rev. Col. Mat. 56 (2022), 1–12

Abstract
This is a short review on the Faà di Bruno formulas, implementing composition of

real-analytic functions, and a Hopf algebra associated to such formulas. This struc-
ture allows, among several other things, a short proof of the Lie–Scheffers theorem,
and relating the Lagrange inversion formulas with antipodes. It is also the maximal
commutative Hopf subalgebra of the one used by Connes and Moscovici to study diffeo-
morphisms in a noncommutative geometry setting. The link of Faà di Bruno formulas
with the theory of set partitions is developed in some detail.

1 The Faà di Bruno formula
Faà di Bruno (Hopf, bi)algebras appear in several branches of mathematics and physics, and
may be introduced in several ways. Here we start from the group 𝐺 of formal exponential
power series like

𝑓 (𝑡) =
∞∑︁
𝑛=1

𝑓𝑛

𝑛! 𝑡
𝑛,

with 𝑓1 > 0. (In view of Borel’s lemma, one may regard them as local representatives of
orientation-preserving diffeomorphisms of ℝ leaving 0 fixed. The question of analyticity is
considered below.)
On this group of power series we consider the coordinate functions

𝑎𝑛 (𝑓 ) := 𝑓𝑛 = 𝑓 (𝑛) (0), 𝑛 ≥ 1.
We wish to compute ℎ𝑛 = 𝑎𝑛 (ℎ), where ℎ is the composition 𝑓 ◦ 𝑔 of two such diffeomor-
phisms, in terms of the 𝑓𝑛 and 𝑔𝑛. Now,

ℎ(𝑡) =
∞∑︁
𝑘=1

𝑓𝑘

𝑘!

( ∞∑︁
𝑙=1

𝑔𝑙

𝑙! 𝑡
𝑙

)𝑘
.

∗Email: joseph.varilly@ucr.ac.cr

1



To compute the 𝑛th coefficient ℎ𝑛 we need only consider the sum up to 𝑘 = 𝑛, since the
remaining terms contain powers of 𝑡 higher than 𝑛. From Cauchy’s product formula,

ℎ𝑛 =

𝑛∑︁
𝑘=1

𝑓𝑘

𝑘!
∑︁

𝑙𝑖≥1, 𝑙1+···+𝑙𝑘=𝑛

𝑛!𝑔𝑙1 · · ·𝑔𝑙𝑘
𝑙1! · · · 𝑙𝑘 !

.

If among the 𝑙𝑖 there are _1 copies of 1, _2 copies of 2, and so on, then the sum 𝑙1+· · ·+𝑙𝑘 = 𝑛
can be rewritten as

_1 + 2_2 + · · · + 𝑛_𝑛 = 𝑛, with _1 + · · · + _𝑛 = 𝑘. (1)
Since there are 𝑘!/_1! · · · _𝑛! contributions from 𝑔 of this type, it follows that

ℎ𝑛 =

𝑛∑︁
𝑘=1

𝑓𝑘

∑︁
_

𝑛!
_1! · · · _𝑛!

𝑔
_1
1 · · ·𝑔_𝑛𝑛

(1!)_1 (2!)_2 · · · (𝑛!)_𝑛 =:
𝑛∑︁
𝑘=1

𝑓𝑘 𝐵𝑛,𝑘 (𝑔1, . . . , 𝑔𝑛+1−𝑘), (2)

where the sum ∑
_ runs over the sequences _ = (_1, . . . , _𝑛) ∈ ℕℕ satisfying (1), and the

𝐵𝑛,𝑘 are called the (partial, exponential) Bell polynomials. Usually these are introduced by
the expansion

exp
(
𝑢
∑︁
𝑚≥1

𝑥𝑚
𝑡𝑚

𝑚!

)
= 1 +

∑︁
𝑛≥1

𝑡𝑛

𝑛!

[ 𝑛∑︁
𝑘=1

𝑢𝑘𝐵𝑛,𝑘 (𝑥1, . . . , 𝑥𝑛+1−𝑘)
]
,

which is a particular case of (2). Each 𝐵𝑛,𝑘 is a homogeneous polynomial of degree 𝑘. (This
is a good moment to declare that the scalar field ℝ may be replaced by any commutative
field of characteristic zero.)
Formula (2) can be recast as

ℎ(𝑛) (𝑡) =
𝑛∑︁
𝑘=1

∑︁
_

𝑛!
_1! · · · _𝑛!

𝑓 (𝑘) (𝑔(𝑡))
(
𝑔(1) (𝑡)
1!

)_1 (𝑔(2) (𝑡)
2!

)_2
· · ·

(
𝑔(𝑛) (𝑡)
𝑛!

)_𝑛
. (3)

Expression (3) is the famous formula attributed to Faà di Bruno (1855, 1857), who in
fact followed previous authors; his original contribution was a determinant form of it.
Apparently (3) goes back to Arbogast (1800); we refer the reader to [1] —and references
therein— for these historical matters. Note that if 𝑔, 𝑓 are differentiable up to order 𝑛, then
ℎ is also differentiable up to order 𝑛, and the expressions for its derivatives hold.
The formula shows that the composition of two real-analytic functions is real-analytic.

Indeed, by use of (2) or (3) with 𝑓 (𝑡) = ∑∞
𝑘=1 𝑡

𝑘 = 𝑡/(1−𝑡) and 𝑔(𝑡) = ∑∞
𝑙=1 𝑥𝑡

𝑙 = 𝑥𝑡/(1−𝑡)
one sees that

𝑛∑︁
𝑘=1

∑︁
_

𝑘!
_1!_2! · · · _𝑛!

𝑥𝑘 = 𝑥 (1 + 𝑥)𝑛−1. (4)

Now, a smooth function 𝑔 on an open interval 𝐼 ⊆ ℝ is analytic [2, Chap. 1] if and only if
for each 𝑦 ∈ 𝐼 there is an open interval 𝐽𝑦 with 𝑦 ∈ 𝐽𝑦 ⊆ 𝐼 and constants 𝐴, 𝐵 such that

|𝑔( 𝑗) (𝑡) | ≤ 𝐴 𝑗!
𝐵 𝑗

for all 𝑡 ∈ 𝐽𝑦 (5)

2



which guarantees local uniform convergence of the Taylor series of 𝑔. Assume further that
𝑔 takes values in an open interval on which the smooth function 𝑓 is defined. If 𝑓 is also
analytic with |𝑓 (𝑚) (𝑠) | ≤ 𝐶𝑚!/𝐷𝑚 for all𝑚 at 𝑠 = 𝑔(𝑡), it follows from (3) and (4) that

|ℎ(𝑛) (𝑡) | ≤
𝑛∑︁
𝑘=1

∑︁
_

𝑛!
_1! · · · _𝑛!

𝐶 𝑘!
𝐷𝑘

(
𝐴

𝐵

)_1
· · ·

(
𝐴

𝐵𝑛

)_𝑛
= 𝑛! 𝐶

𝐵𝑛
𝐴

𝐷

(
1 + 𝐴

𝐷

)𝑛−1
=
𝐸 𝑛!
𝐹𝑛

,

with 𝐸 = 𝐴𝐶/(𝐴 + 𝐷) and 𝐹 = 𝐵𝐷/(𝐴 + 𝐷). Hence 𝑓 ◦ 𝑔 is analytic on the domain of 𝑔.

2 Hopf algebras
Introduce the notation, with (1) understood:(

𝑛

_;𝑘

)
:= 𝑛!
_1! · · · _𝑛! (1!)_1 (2!)_2 . . . (𝑛!)_𝑛

.

A Hopf algebra dual to 𝐺 is obtained when we define a coproduct Δ on the polynomial
algebra ℝ[𝑎1, 𝑎2, . . . ] of coordinate functions by requiring that Δ𝑎𝑛 (𝑔, 𝑓 ) = 𝑎𝑛 (𝑓 ◦ 𝑔), or
equivalently, 𝑎𝑛 (𝑓 ◦ 𝑔) =𝑚(Δ𝑎𝑛 (𝑔 ⊗ 𝑓 )) where𝑚 means multiplication. This entails that

Δ𝑎𝑛 =
𝑛∑︁
𝑘=1

∑︁
_

(
𝑛

_;𝑘

)
𝑎
_1
1 𝑎

_2
2 . . . 𝑎

_𝑛
𝑛 ⊗ 𝑎𝑘 .

The unnecessary flip of 𝑓 and 𝑔 is traditional. This Faà di Bruno bialgebra, so called by Joni
and Rota [3], is commutative but not cocommutative. Since 𝑎1 is a grouplike element, it
must be invertible for this to be a Hopf algebra. For that, one must either adjoin an inverse
𝑎−11 , or else put 𝑎1 = 1, as we do from now on. That is, we consider only the subgroup 𝐺1
of diffeomorphisms tangent to the identity at 0. The first instances of the coproduct are,
accordingly,

Δ𝑎2 = 𝑎2 ⊗ 1 + 1 ⊗ 𝑎2,
Δ𝑎3 = 𝑎3 ⊗ 1 + 1 ⊗ 𝑎3 + 3𝑎2 ⊗ 𝑎2,
Δ𝑎4 = 𝑎4 ⊗ 1 + 1 ⊗ 𝑎4 + 6𝑎2 ⊗ 𝑎3 + (3𝑎22 + 4𝑎3) ⊗ 𝑎2,
Δ𝑎5 = 𝑎5 ⊗ 1 + 1 ⊗ 𝑎5 + 10𝑎2 ⊗ 𝑎4 + (10𝑎3 + 15𝑎22) ⊗ 𝑎3 + (5𝑎4 + 10𝑎2𝑎3) ⊗ 𝑎2. (6)

The resulting graded connected Hopf algebra F is called the Faà di Bruno Hopf algebra; the
degree # being given by #𝑎𝑛 = 𝑛 − 1.
Consider the graded dual Hopf algebra F′. Its space of primitive elements has a basis

{𝑎′𝑛 : 𝑛 ≥ 2} defined by ⟨𝑎′𝑛, 𝑎𝑚⟩ = 𝛿𝑛𝑚 and ⟨𝑎′𝑛, 𝑎𝑚1𝑎𝑚2 . . . 𝑎𝑚𝑟
⟩ = 0 for 𝑟 > 1. Their product

is given by the duality recipe ⟨𝑏′𝑐′, 𝑎⟩ := ⟨𝑏′ ⊗ 𝑐′,Δ𝑎⟩, leading to:

𝑎′𝑛𝑎
′
𝑚 =

(
𝑚 − 1 + 𝑛

𝑛

)
𝑎′𝑛+𝑚−1 + (1 + 𝛿𝑛𝑚) (𝑎𝑛𝑎𝑚)′.

3



In particular, taking 𝑏′𝑛 := (𝑛 +1)!𝑎′
𝑛+1 for 𝑛 ≥ 1, we are left with the commutator relations

[𝑏′𝑛, 𝑏′𝑚] = (𝑚 − 𝑛)𝑏′𝑛+𝑚 . (7)

The Milnor–Moore theorem implies that F′ is isomorphic to the enveloping algebra of the
Lie algebra A spanned by the 𝑏′𝑛 with these commutators.
A curious consequence of (7) is that the space 𝑃 (F) of primitive elements of F just has

dimension 2. Indeed, 𝑃 (F) = (ℝ1 ⊕ F′
+
2)⊥, where F′

+ is the augmentation ideal of F′. But
(7) entails that there is a basis of F′ made of products, except for its first two elements:
therefore, dim 𝑃 (F) = 2. A basis of 𝑃 (F) is given by {𝑎2, 𝑎3 − 3

2𝑎
2
2}. The second of these

corresponds to the Schwarzian derivative, which is known [4] to be invariant under the
projective group 𝑃𝑆𝐿(2,ℝ). Nonexistence of more primitive elements of F is related to
the affine linear and Riccati equations being the only Lie–Scheffers systems [5,6] over the
real line.
The Faà di Bruno Hopf algebra F reappears as the maximal commutative Hopf subalge-

bra of the (noncommutative geometry) Hopf algebra 𝐻 of Connes and Moscovici [7]. Their
description of F uses a different set of coordinates 𝛿𝑛 (𝑓 ) := [log 𝑓 ′(𝑡)] (𝑛) (0), 𝑛 ≥ 1. Since

ℎ(𝑡) :=
∑︁
𝑛≥1

𝛿𝑛 (𝑓 )
𝑡𝑛

𝑛! = log 𝑓
′(𝑡) = log

(
1 +

∑︁
𝑛≥1

𝑎𝑛+1(𝑓 )
𝑡𝑛

𝑛!

)
,

it follows from formula (2), for logarithm and exponential functions respectively, that

𝛿𝑛 =

𝑛∑︁
𝑘=1

(−1)𝑘−1(𝑘 − 1)!𝐵𝑛,𝑘 (𝑎2, . . . , 𝑎𝑛+2−𝑘) =: 𝐿𝑛 (𝑎2, . . . , 𝑎𝑛+1),

inverted by

𝑎𝑛+1 =
𝑛∑︁
𝑘=1

𝐵𝑛,𝑘 (𝛿1, . . . , 𝛿𝑛+1−𝑘) =: 𝑌𝑛 (𝛿1, . . . , 𝛿𝑛),

where the 𝐿𝑛 and the 𝑌𝑛 are respectively called the logarithmic polynomials and the
(complete, exponential) Bell polynomials. In this way we get 𝛿1 = 𝑎2; 𝛿2 = 𝑎3 − 𝑎22;
𝛿3 = 𝑎4−3𝑎2𝑎3 +2𝑎32; 𝛿4 = 𝑎5−3𝑎23−4𝑎2𝑎4 +12𝑎22𝑎3−6𝑎42; and so on. Since the coproduct
is an algebra morphism, by use of (6) wemay obtain the coproduct in the Connes–Moscovici
coordinates. For instance,

Δ𝛿4 = 𝛿4 ⊗ 1 + 1 ⊗ 𝛿4 + 6𝛿1 ⊗ 𝛿3 + (7𝛿21 + 4𝛿2) ⊗ 𝛿2 + (3𝛿1𝛿2 + 𝛿31 + 𝛿3) ⊗ 𝛿1.

It is not easy to find a closed formula for Δ(𝛿𝑛) directly from (6). Fortunately, through F′

another method is available. Using 𝐵𝑛,1(𝑎2, . . . , 𝑎𝑛+1) = 𝑎𝑛+1, one finds that ⟨𝑏′𝑛, 𝛿𝑚⟩ =

(𝑛 + 1)!𝛿𝑛,𝑚. Let 𝐴 be the graded free Lie algebra generated by primitive elements 𝑋𝑛,
𝑛 ≥ 1. Its enveloping algebra U(𝐴) is the concatenation Hopf algebra. A linear basis for

4



U(𝐴), indexed by all vectors with positive integer components 𝑛 = (𝑛1, . . . , 𝑛𝑟 ), is made of
products 𝑋𝑛 := 𝑋𝑛1𝑋𝑛2 . . . 𝑋𝑛𝑟 , together with the unit element 𝑋∅ = 1. Its coproduct is

Δ(𝑋𝑛) :=
∑︁
𝑛1,𝑛2
sh𝑛1,𝑛2
𝑛

𝑋𝑛1 ⊗ 𝑋𝑛2,

with sh𝑛1,𝑛2
𝑛

denoting the number of shuffles of the vectors 𝑛1, 𝑛2 that produce 𝑛. Let 𝑢𝑛
denote a dual basis to𝑋𝑛; the graded dual ofU(𝐴) is the shuffle Hopf algebra𝐻 with product
and coproduct respectively given by

𝑢𝑛
1
𝑢𝑛
2 :=

∑̄︁
𝑛

sh𝑛1,𝑛2
𝑛

𝑢𝑛, Δ(𝑢𝑛) :=
∑︁

𝑛1𝑛2=𝑛

𝑢𝑛
1 ⊗ 𝑢𝑛2,

where 𝑛1𝑛2 is the concatenation of the vectors 𝑛1, 𝑛2. The surjective morphism 𝜌 : 𝐴 → A

defined by 𝜌 (𝑋𝑛) := 𝑏′𝑛 extends, by the universal property of enveloping algebras, to a
surjective morphism 𝜌 : U(𝐴) → F′, whose transpose is the injective Hopf map 𝜌𝑡 : F → 𝐻

given by 𝛿𝑛 ↦→ Γ𝑛 := 𝛿𝑛 ◦ 𝜌. We may thus regard F as a Hopf subalgebra of 𝐻 , and thereby
compute the coproduct of F from that of 𝐻 . The argument may look circular, since we
seem to need an expression for the Γ𝑛, which in turn requires computing Δ(𝛿𝑛). But we
can write

⟨Γ𝑚, 𝑋𝑛⟩ = ⟨𝛿𝑚, 𝜌 (𝑋𝑛)⟩ = ⟨𝛿𝑚, 𝑏′𝑛1 · · ·𝑏
′
𝑛𝑟
⟩ = ⟨Δ(𝛿𝑚), 𝑏′𝑛1 ⊗ 𝑏

′
𝑛2 · · ·𝑏

′
𝑛𝑟
⟩. (8)

Thus, to compute Γ𝑛, the only terms we need in the expansion of Δ(𝛿𝑛) are 𝛿𝑛 ⊗ 1 + 1 ⊗ 𝛿𝑛
and the bilinear terms, namely multiples of 𝛿𝑖 ⊗ 𝛿 𝑗 ; the remaining terms are of the form
(constant) 𝛿𝑟1

𝑖1
· · · 𝛿𝑟𝑘

𝑖𝑘
⊗ 𝛿 𝑗 , where 𝑟1𝑖1 + · · · + 𝑟𝑘𝑖𝑘 + 𝑗 = 𝑛. The bilinear part 𝐵(𝛿𝑛) may be

computed by induction [7] to be

𝐵(𝛿𝑛) =
𝑛−1∑︁
𝑖=1

(
𝑛

𝑖 − 1

)
𝛿𝑛−𝑖 ⊗ 𝛿𝑖 . (9)

Substituting (9) repeatedly in (8), one obtains

Γ𝑛 = 𝑛!
∑︁

𝑛:𝑛1+···+𝑛𝑟=𝑛
𝐶𝑛 𝑢𝑛 with coefficients 𝐶𝑛 := (𝑛𝑟 + 1)

𝑟∏
𝑖=2

(𝑛𝑖 + · · · + 𝑛𝑟 ).

For instance, Γ1 = 2𝑢1 and Γ3 = 12(2𝑢3 + 𝑢 (2,1) + 3𝑢 (1,2) + 2𝑢 (1,1,1)).
Another calculation of the Γ𝑛 was sketched in [8]; it eventually allows to improve (9) to

Δ(𝛿𝑛) = 𝛿𝑛 ⊗ 1 + 1 ⊗ 𝛿𝑛 +
∑̄︁
𝑛∈𝑁𝑛

𝑛!
𝑛1! . . . 𝑛𝑟 !

𝐾𝑛1,...,𝑛𝑟−1𝑛𝑟
𝛿𝑛1 . . . 𝛿𝑛𝑟−1 ⊗ 𝛿𝑛𝑟 ,

where 𝑁𝑛 := {𝑛 : 𝑛1 + · · · + 𝑛𝑟 = 𝑛, 𝑟 > 1 } and, mindful that
(𝑛𝑟
𝑘

)
= 0 when 𝑛𝑟 < 𝑘,

𝐾𝑛1,...,𝑛𝑟−1𝑛𝑟
=

𝑟−1∑︁
𝑘=1

(
𝑛𝑟

𝑘

) ∑︁
𝑛1···𝑛𝑘=(𝑛1,...,𝑛𝑟−1)

1
𝑟1! . . . 𝑟𝑘 !

𝑘∏
𝑖=1

1
1 + 𝑛𝑖1 + · · · + 𝑛𝑖

𝑟 𝑖

.

5



For 𝑟 = 2, this becomes 𝐾𝑛−𝑖𝑖 = 𝑖
1+𝑛−𝑖 , thus the coefficient of 𝛿𝑛−𝑖 ⊗ 𝛿𝑖 is

(𝑛
𝑖

)
𝑖

1+𝑛−𝑖 =
( 𝑛
𝑖−1

) , as
in (9).
From the combinatorial viewpoint, the Faà di Bruno Hopf algebra is the incidence

Hopf algebra corresponding to intervals formed by partitions of finite sets. This is no
surprise, since the coefficients of a Bell polynomial 𝐵𝑛,𝑘 just count the number of partitions
of {1, . . . , 𝑛} into 𝑘 blocks. A partition 𝜋 ∈ Π(𝑆), of a finite set 𝑆 with 𝑛 elements, is a
collection {𝐵1, 𝐵2, . . . , 𝐵𝑘} of nonempty disjoint subsets, called blocks, such that

⋃𝑘
𝑖=1 𝐵𝑖 = 𝑆 .

We simply write 𝜋 ⊢ 𝑛 for such, with |𝜋 | being the number of blocks in 𝜋 .
We say that 𝜋 is of type (𝛼1, . . . , 𝛼𝑛) if exactly 𝛼𝑖 of these 𝐵 𝑗 have 𝑖 elements; thus

𝛼1 + 2𝛼2 + · · · + 𝑛𝛼𝑛 = 𝑛 and 𝛼1 + · · · + 𝛼𝑛 = 𝑘 [9]. We say that 𝜋 refines 𝜏 , and write
{𝐴1, . . . , 𝐴𝑛} = 𝜋 ≤ 𝜏 = {𝐵1, . . . , 𝐵𝑚}, if each 𝐴𝑖 is contained in some 𝐵 𝑗 . A subinterval
[𝜋, 𝜏] = {𝜎 : 𝜋 ≤ 𝜎 ≤ 𝜏} of the lattice P of partitions of finite sets is isomorphic to the
poset Π_11 × · · · × Π_𝑛𝑛 , where Π 𝑗 := Π({1, . . . , 𝑗}) and _𝑖 blocks of 𝜏 are unions of exactly
𝑖 blocks of 𝜋 . One assigns to each interval the sequence _ = (_1, . . ., _𝑛) and declares
two intervals in P to be equivalent when their vectors _ are equal. From the matching�[𝜋, 𝜏] ↔ _ ↔ Π̃_11 Π̃

_2
2 · · · Π̃_𝑛𝑛 of equivalence classes, one may regard the family P̃ of

equivalence classes as the algebra of polynomials of infinitely many variablesℝ[Π̃1, Π̃2, . . . ].
Bymeans of the general theory of coproducts for incidence bialgebras [10] one then recovers
the Faà di Bruno algebra under the identifications 𝑎𝑛 ↔ Π̃𝑛. The cardinality in the sense
of category theory [11] of the groupoid of finite sets equipped with a partition is given by

∞∑︁
𝑛=0

𝑛∑︁
𝑘=1

1
𝑛! 𝐵𝑛,𝑘 (1, . . . , 1) = 𝑒

𝑒−1.

The characters of F form a group Homalg(F,ℝ) under the convolution operation of Hopf
algebra theory. The action of a character 𝑓 is determined by its values on the 𝑎𝑛. The map
𝑓 ↦→ 𝑓 (𝑡) = ∑∞

𝑛=1 𝑓𝑛𝑡
𝑛/𝑛!, where 𝑓𝑛 := ⟨𝑓 , 𝑎𝑛⟩, matches characters with exponential power

series over ℝ such that 𝑓1 = 1. This correspondence is an anti-isomorphism of groups:
indeed, the convolution 𝑓 ∗ 𝑔 of 𝑓 , 𝑔 ∈ Homalg(F,ℝ) is given by

⟨𝑓 ∗ 𝑔, 𝑎𝑛⟩ :=𝑚(𝑓 ⊗ 𝑔)Δ𝑎𝑛 = ⟨𝑔 ◦ 𝑓 , 𝑎𝑛⟩.

This is just the𝑛th coefficient ofℎ(𝑡) = 𝑔(𝑓 (𝑡)). Also, the algebra endomorphisms Endalg(F)
form a group under the convolution of the unital algebra End(F) of linear endomorphisms.
The inverse under functional composition of an exponential series is given by the rever-

sion formula of Lagrange [12], one of whose forms [13] states that if 𝑓 and 𝑔 are two such
series and if 𝑓1 = 1, 𝑓 ◦ 𝑔(𝑡) = 𝑔 ◦ 𝑓 (𝑡) = 𝑡 , then

𝑔𝑛 =

𝑛−1∑︁
𝑘=1

(−1)𝑘𝐵𝑛−1+𝑘,𝑘 (0, 𝑓2, 𝑓3, . . . ). (10)

Now, the inverse under convolution of 𝑓 ∈ Endalg(F) is 𝑔 = 𝑓 ◦ 𝑆 , with 𝑆 the antipode map

6



of F. The multiplicativity of 𝑓 forces

𝑆 (𝑎𝑛) =
𝑛−1∑︁
𝑘=1

(−1)𝑘𝐵𝑛−1+𝑘,𝑘 (0, 𝑎2, 𝑎3, . . . ).

One may reverse the roles and prove the combinatorial identity (10) from Hopf algebra
theory [14]. The real-analytic inverse function theorem is stated in a completely similar
way to the standard inverse function theorem for differentiable functions, and it can be
proved by use of the formula of Faà di Bruno, with the help of the estimates (5).
Use of partitions with special properties may lead to other incidence algebras: for

instance, if we restrict to noncrossing partitions, we obtain a cocommutative Hopf algebra,
with the commutative group operation on characters essentially corresponding to Lagrange
reversion of the Cauchy product of reverted series [15].

3 Faà di Bruno formulas in several variables
To go to higher Faà di Bruno formulas means to consider exponential 𝑁 ′-series in 𝑁 ′

variables (“colours”) of the form

𝑓 (𝑡1, . . . , 𝑡𝑁 ′) =
(
𝑡1 +

∑︁
|�̄� |>1

𝑓 1�̄�
𝑡�̄�

�̄�! , 𝑡2 +
∑︁
|𝑛 |>1

𝑓 2𝑛
𝑡𝑛

𝑛! , . . . , 𝑡𝑁 ′ +
∑︁
|𝑝 |>1

𝑓 𝑁
′

𝑝

𝑡𝑝

𝑝!

)
, (11)

where �̄�, 𝑛, . . . , 𝑝 ∈ ℕ𝑁
′ . The simplest way to go about this is to rewrite Eq. (3) in terms of

partitions:
(𝑓 ◦ 𝑔) (𝑛) (𝑡) =

∑︁
𝜋⊢𝑛

𝑓 ( |𝜋 |) (𝑔(𝑡))
∏
𝑙∈𝜋

𝑔( |𝑙 |) (𝑡), (12)

where the coefficients ( 𝑛
_;𝑘

) of (3) count partitions yielding equal summands. (Note that∑
|_ |=𝑘

( 𝑛
_;𝑘

)
=
{
𝑛
𝑘

}
is the Stirling number of the second kind [16, Chap. 6].) For instance, for

𝑛 = 4 one immediately finds:
(𝑓 ◦ 𝑔) (iv) (𝑡) = 𝑓 ′(𝑔(𝑡)) 𝑔(iv) (𝑡) + 4𝑓 ′′(𝑔(𝑡)) 𝑔′(𝑡)𝑔′′′(𝑡) + 3𝑓 ′′′(𝑔(𝑡)) 𝑔′′(𝑡)2

+ 6𝑓 ′′′(𝑡) 𝑔′(𝑡)2𝑔′′(𝑡) + 𝑓 (iv) (𝑡) 𝑔′(𝑡)4.
The above can be generalized to formal series in several indeterminates as follows. Consider
the set of maps from {1, . . . , 𝑛} to a set of colours {1′, . . . , 𝑁 ′}. This allows consideration
of coloured partitions of {1, . . . , 𝑛}, with monocoloured partitions being of the same type
as before. There are 𝑚 families of those series, one for each colour, with tangency at the
identity being enforced by:

𝜕𝑡𝑖 𝑓
𝑖 (0, . . . , 0) = 1, 𝜕𝑡 𝑗 𝑓

𝑖 (0, . . . , 0) = 0 for 𝑗 ≠ 𝑖; with 𝑖, 𝑗 ∈ {1′, . . . , 𝑁 ′}.

Then the very formula (12) is valid, provided we understand now |𝑙 | as a vector of colours.
A fully multivariate treatment in this vein is provided by [17]. See the simplest case 𝑁 ′ = 2
in the foreword to the book [18] – this book contains much interesting information besides.

7



The series (11) can be regarded as characters of “coloured” Faà di Bruno Hopf algebras
F(𝑁 ′) [14]. For any finite set 𝑋 gifted with a colouring map \ : 𝑋 → {1′, . . . , 𝑁 ′}, one
considers partitions 𝜋 whose sets of blocks are also coloured, provided \ ({𝑥}) = \ (𝑥) for
singletons. Such coloured partitions form a poset, with 𝜋 ≤ 𝜌 if 𝜋 refines 𝜌 as partitions,
and if \𝜋 (𝐵) = \𝜌 (𝐵) for each block 𝐵 of 𝜋 which is also a block of 𝜌; this condition entails
that 𝜌 induces a coloured partition 𝜌 | 𝜋 of the set of blocks of 𝜋 . Coloured partitions 𝜋 of 𝑋
with \ (𝑋 ) = 𝑟 (i.e., the one-block partition of 𝑋 is assigned the colour 𝑟) form a poset Π𝑟

𝑛

where 𝑛 ∈ ℕ𝑁
′ counts the colours of its elements; their types Π̃𝑟

𝑛
generate the Hopf algebra

F(𝑁 ′), with coproduct given by:

ΔΠ̃𝑟𝑛 :=
∑︁
𝜋∈Π𝑟

𝑛

(∏
𝐵∈𝜋

Π̃\ (𝐵)|𝐵 |

)
⊗ Π̃𝑟|𝜋 | .

A character 𝑓 of F(𝑁 ′) is specified by its values on algebra generators 𝑓 𝑟
𝑛

= 𝑓 (Π̃𝑟
𝑛
),

which yield coefficients of the 𝑁 ′-series (11). The convolution of two such characters 𝑔, 𝑓
has coefficients

𝑔 ∗ 𝑓
(
Π̃𝑟𝑛

)
=

∑︁
𝜋∈Π𝑟

𝑛

𝑓 𝑟|𝜋 |

∏
𝐵∈𝜋

𝑔
\ (𝐵)
|𝐵 | =

∑︁
|𝑘 |≤|𝑛 |

𝑓 𝑟
𝑘
/𝑘!

∏
(𝐵1,...,𝐵 |𝑘 | )

𝑔1|𝐵1 | . . . 𝑔
𝑁 ′

|𝐵 |𝑘 | |

where the second product ranges over ordered coloured partitions of a set with |𝑛 | elements;
since there are 𝑛!/∏𝑖 (�̄�𝑖!)_𝑖,�̄�𝑖 of these with prescribed colours, rearrangement of the right
hand side yields a formula for (11). Thus, the character group of F(𝑁 ′) is anti-isomorphic
to the group of 𝑁 ′-series like (11) under composition. Also, the antipode on F(𝑁 ′) provides
Lagrange reversion in several variables [14].
For the applications of Faà di Bruno algebras to real analytic function theory we rec-

ommend [2]. The Faà di Bruno algebras (perhaps involving functional derivatives) have
applications in quantum field theory. Some elementary ones are described in [10]. Deeper
ones related to renormalization theory were broached in [19, 20], and further explored
in [21]. A nice, relatively recent work in this respect is [22].
Much remains to be delved into. Noncommutative Bell polynomials were studied

in [23]. Faà di Bruno’s formulas for operads have been developed in [24]. An application
to control theory is found in [25]. A recent comprehensive treatise on Bell polynomials and
generalized Lagrange inversion [26] is also recommended.

References
[1] A. D. D. Craik, “Prehistory of Faà di Bruno’s formula”, Amer. Math. Monthly 112 (2005), 119–130.
[2] S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, 2nd edition, Birkhäuser, Boston, 2002.
[3] S. A. Joni and G.-C. Rota, “Coalgebras and bialgebras in combinatorics”, Contemp. Math. 6 (1982),

1–47.
[4] E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley, New York, 1976.

8



[5] S. Lie and G. Scheffers, Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwen-
dungen, Teubner, Leipzig, 1893.

[6] J. F. Cariñena, K. Ebrahimi-Fard, H. Figueroa and J. M. Gracia-Bondía, “Hopf algebras in dynamical
systems theory”, Int. J. Geom. Methods Mod. Phys. 4 (2007), 577–646.

[7] A. Connes and H. Moscovici, “Hopf algebras, cyclic cohomology and the transverse index theorem”,
Commun. Math. Phys. 198 (1998), 198–246.

[8] F. Menous, “Formulas for the Connes–Moscovici Hopf algebra”, C. R. Acad. Sci. (Paris) 341 (2005),
75–78.

[9] R. P. Stanley, “Generating functions”, in Studies in Combinatorics, Gian-Carlo Rota, ed., MAA, Washing-
ton, DC, 1978; pp. 100–141.

[10] H. Figueroa and J. M. Gracia-Bondía, “Combinatorial Hopf algebras in quantum field theory”, Rev.
Math. Phys. 17 (2005), 881–976.

[11] J. C. Baez and J. Dolan, “From finite sets to Feynman diagrams”, in Mathematics Unlimited – 2001 and
Beyond, B. Engquist and W. Schmid, eds., Springer, Berlin, 2001; pp. 29–50.

[12] J.-L. de Lagrange, “Nouvelle méthode pour résoudre les équations littérales par la moyen des séries”,
Mém. Acad. Royale des Sciences et Belles-Lettres de Berlin 24 (1770), 251–326.

[13] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel, Dordrecht, 1974.
[14] M. Haiman and W. R. Schmitt, “Incidence algebra antipodes and Lagrange inversion in one and several

variables”, J. Combin. Theory A 50 (1989), 172–185.
[15] R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999.
[16] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley,

Reading, MA, 1994.
[17] G. M. Constantine and T. H. Savits, “A multivariate Faà di Bruno formula with applications”, Trans.

Amer. Math. Soc. 348 (1996), 503–520.
[18] Faà di Bruno Hopf algebras, Dyson–Schwinger equations, and Lie–Butcher Series, K. Ebrahimi-Fard and F.

Fauvet, eds., IRMA Lectures in Mathematics and Theoretical Physics 21, EMS Press, Zürich, 2015.
[19] W. E. Caswell and A. D. Kennedy, “Simple approach to renormalization theory”, Phys. Rev. D 25 (1982),

392–408.
[20] A. Connes and D. Kreimer, “Renormalization in quantum field theory and the Riemann–Hilbert prob-

lem II: the 𝛽-function, diffeomorphisms and the renormalization group”, Commun. Math. Phys. 216
(2001), 215–241.

[21] L. Foissy, “Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson–
Schwinger equations”, Adv. Math. 218 (2008), 136–162.

[22] D. Kreimer and K. Yeats, “Diffeomorphisms of quantum fields”, Math. Phys. Anal. Geom. 20:16 (2017).
[23] K. Ebrahimi-Fard, A. Lundervold and D. Manchon, “Noncommutative Bell polynomials, quasidetermi-

nants and incidence Hopf algebras”, Int. J. Alg. Comp. 24 (2014), 671–705.
[24] J. Kock and M. Weber, “Faà di Bruno for operads and internal algebras”, J. London Math. Soc. 99 (2019),

919–944.
[25] K. Ebrahimi-Fard and W. S. Gray. “Center problem, Abel equation and the Faà di Bruno Hopf algebra

for output feedback”, Int. Math. Res. Notices 2017 (2017), 5415–5450.
[26] A. Schreiber, “Inverse relations and reciprocity laws involving partial Bell polynomials and related

extensions”, Enumer. Combin. Appl. 1:1 (2021).

9