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arXiv:1110.5272v1 [math.CO] 24 Oct 2011

THE ALGEBRAIC COMBINATORICS OF SNAKES

MATTHIEU JOSUAT-VERG`ES, JEAN-CHRISTOPHE NOVELLI,

AND JEAN-YVES THIBON

Abstract. Snakes are analogues of alternating permutations defined for any Cox-

eter group. We study these objects from the point of view of combinatorial Hopf

algebras, such as noncommutative symmetric functions and their generalizations.

The main purpose is to show that several properties of the generating functions

of snakes, such as differential equations or closed form as trigonometric functions,

can be lifted at the level of noncommutative symmetric functions or free quasi-

symmetric functions. The results take the form of algebraic identities for type B

noncommutative symmetric functions, noncommutative supersymmetric functions

and colored free quasi-symmetric functions.

1. Introduction

Snakes, a term coined by Arnol’d [2], are generalizations of alternating permuta-

tions. These permutations arose as the solution of what is perhaps the first example

of an inverse problem in the theory of generating functions: given a function whose

Taylor series has nonnegative integer coefficients, find a family of combinatorial ob-

jects counted by those coefficients. For example, in the expansions

(1) tanz =

?

n≥0

E2n+1

z2n+1

(2n + 1)!

andsecz =

?

n≥0

E2n

z2n

(2n)!,

the coefficients Enare nonnegative integers.

It was found in 1881 by D. Andr´ e [1] that En was the number of alternating

permutations in the symmetric group Sn.

Whilst this result is not particularly difficult and can be proved in several ways,

the following explanation is probably not far from being optimal: there exists an

associative (and noncommutative) algebra admitting a basis labelled by all permu-

tations, and such that the map φ sending any σ ∈ Sntozn

this algebra, the formal series

n!is a homomorphism. In

(2)C =

?

n≥0

(−1)nid2n

andS =

?

n≥0

(−1)nid2n+1

(alternating sums of even and odd identity permutations) are respectively mapped

to cosz and sinz by φ. The series C is clearly invertible, and one can see by a direct

calculation that C−1+ C · S−1is the sum of all alternating permutations [8].

Date: October 25, 2011.

M. Josuat-Verg` es was supported by the Austrian Science Foundation (FWF) via the grant Y463.

1

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2M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON

Such a proof is not only illuminating, it says much more than the original statement.

For example, one can now replace φ by more complicated morphisms, and obtain

generating functions for various statistics on alternating permutations.

The symmetric group is a Coxeter group, and snakes are generalizations of al-

ternating permutations to arbitrary Coxeter groups. Such generalizations were first

introduced by Springer [22]. For the infinite series An, Bn, Dn, Arnol’d [2] related

the snakes to the geometry of bifurcation diagrams.

The aim of this article is to study the snakes of the classical Weyl groups (types A,

B and D) by noncommutative methods, and to generalize the results to some series

of wreath products (colored permutations).

The case of symmetric groups (type A) is settled by the algebra of Free quasi-

symmetric functions FQSym (also known the Malvenuto-Reutenauer algebra) which

is based on permutations, and its subalgebra Sym (noncommutative symmetric func-

tions), based on integer compositions. To deal with the other types, we need an

algebra based on signed permutations, and some of its subalgebras defined by means

of the superization map introduced in [17].

After reviewing the necessary background and the above mentioned proof of the

result of Andr´ e, we recover results of Chow [4] on type B snakes, and derive some new

generating functions for this type. This suggests a variant of the definition of snakes,

for which the noncommutative generating series is simpler. These considerations

lead us to some new identities satisfied by the superization map on noncommutative

symmetric functions. Finally, we propose a completely different combinatorial model

for the generating function of type B snakes, based on interesting identities in the

algebra of signed permutations. We also present generalizations of (Arnol’d’s) Euler-

Bernoulli triangle, counting alternating permutations according to their last value,

and extend the results to wreath products and to type D, for which we propose an

alternative definition of snakes.

2. Permutations and noncommutative trigonometry

2.1. Free quasi-symmetric functions. The simplest way to define our algebra

based on permutations is by means of the classical standardization process, familiar

in combinatorics and in computer science. Let A = {a1,a2,...} be an infinite totally

ordered alphabet. The standardized word Std(w) of a word w ∈ A∗is the permu-

tation 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. Alternatively, σ = std(w)−1can be characterized as the unique

permutation of minimal length such that wσ is a nondecreasing word. For example,

std(bbacab) = 341625.

We can now define polynomials

(3)

Gσ(A) :=

?

std(w)=σ

w.

It is not hard to check that these polynomials span a subalgebra of C?A?, denoted

by FQSym(A), an acronym for Free Quasi-Symmetric functions.

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THE ALGEBRAIC COMBINATORICS OF SNAKES3

The multiplication rule is, for α ∈ Skand β ∈ Sℓ,

(4)

GαGβ=

?

γ∈α∗β

Gγ,

where α ∗ β is the set of permutations γ ∈ Sk+ℓsuch that γ = u · v with std(u) = α

and std(v) = β. This is the convolution of permutations (see [19]). Note that the

number of terms in this product depends only on k and ℓ, and is equal to the binomial

coefficient?k+ℓ

(5)

k

?. Hence, the map

φ : σ ∈ Sn?−→zn

n!

is a homomorphism of algebras FQSym → C[z].

2.2. Noncommutative symmetric functions. The algebra Sym(A) of noncom-

mutative symmetric functions over A is the subalgebra of FQSym generated by the

identity permutations [8, 6]

(6)Sn(A) := G12...n(A) =

?

i1≤i2≤···≤in

ai1ai2...ain.

These polynomials are obviously algebraically independent, so that the products

(7)SI:= Si1Si2...Sir

where I = (i1,i2,...,ir) runs over compositions of n, form a basis of Symn, the

homogeneous component of degree n of Sym.

Recall that a descent of a word w = w1w2...wn ∈ Anis an index i such that

wi > wi+1. The set of such i is denoted by Des(w). Hence, Sn(A) is the sum of

all nondecreasing words of length n (no descent), and SI(A) is the sum of all words

which may have a descents only at places from the set

(8) Des(I) = {i1, i1+ i2,...,i1+ ··· + ir−1},

called the descent set of I. Another important basis is

(9)RI(A) =

?

Des(w)=Des(I)

w =

?

Des(σ)=Des(I)

Gσ,

the ribbon basis, formed by sums of words having descents exactly at prescribed

places. From this definition, it is obvious that if I = (i1,...,ir), J = (j1,...,js)

(10)RI(A)RJ(A) = RIJ(A) + RI⊲J(A)

with IJ = (i1,...,ir,j1,...,js) and I ⊲ J = (i1,...,ir+ j1,j2,...,js).

2.3. Operations on alphabets. If B is another totally ordered alphabet, we denote

by A + B the ordinal sum of A and B.

symmetric functions of A + B, and

This allows to define noncommutative

(11)Sn(A + B) =

n

?

i=0

Si(A)Sn−i(B)(S0= 1).

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4M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON

If we assume that A and B commute, this operation defines a coproduct, for which

Sym is a graded bialgebra, hence a Hopf algebra. The same is true of FQSym.

Symmetric functions of the virtual alphabet (−A) are defined by the condition

(12)

?

n≥0

Sn(−A) =

??

n≥0

Sn(A)

?−1

and more generally, for a difference A − B,

(13)

?

n≥0

Sn(A − B) =

??

k≥0

Sk(B)

?−1?

l≥0

Sl(A)

(note the reversed order, see [13] for detailed explanations).

2.4. Noncommutative trigonometry.

2.4.1. Andr´ e’s theorem. One can now define “noncommutative trigonometric func-

tions” by

(14)

cos(A) =

?

n≥0

(−1)nS2n(A) and

sin(A) =

?

n≥0

(−1)nS2n+1(A).

The image by φ of these series are the usual trigonometric functions. With the help

of the product formula for the ribbon basis, it is easy to see that

(15)

sec := cos−1=

?

n≥0

R(2n)

and

tan := cos−1sin =

?

n≥0

R(2n1)

which implies Andr´ e’s theorem: the coefficient ofzn

of alternating permutations of Sn(if we choose to define alternating permutations

as those of shape (2n) and (2n1)). In FQSym,

n!in sec(z)+tan(z) is the number

(16)

sec + tan =

?

σ alternating

Gσ.

2.4.2. Differential equations. If ∂ is the derivation of Sym such that ∂Sn= Sn−1,

then

?

satisfy the differential equations

∂X = 1 + X2,

(17)X = tan =

m≥0

R(2m1) and Y = sec =

?

m≥0

R(2m)

(18) ∂Y = XY .

These equations can be lifted to FQSym, actually to its subalgebra PBT, the Loday-

Ronco algebra of planar binary trees (see [9] for details). Solving them in this algebra

provides yet another combinatorial proof of Andr´ e’s result.

Let us sketch it for the tangent. The original proof of Andr´ e relied upon the

differential equation

dx

dt= 1 + x2

(19)

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THE ALGEBRAIC COMBINATORICS OF SNAKES5

whose x(t) = tan(t) is the solution such that x(0) = 0. Equivalently, x(t) is the

unique solution of the functional equation

?t

which can be solved by iterated substitution.

In general, given an associative algebra R, we can consider the functional equation

for the power series x ∈ R[[t]]

(20)x(t) = t +

0

x(s)2ds

(21)x = a + B(x,x)

where a ∈ R and B(x,y) is a bilinear map with values in R[[t]], such that the

valuation of B(x,y) is strictly greater than the sum of the valuations of x and y.

Then, Equation (21) has a unique solution

(22)x = a + B(a,a) + B(B(a,a),a) + B(a,B(a,a)) + ··· =

?

T∈CBT

BT(a)

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

internal nodes. Pictorially,

x = a + B(a,a) + B(B(a,a),a) + B(a,B(a,a)) + ...

= a +

B? ?

??

aa

+

B

? ? ?

??

B

?

?

??

a

aa

+

B? ?

???

a

B? ?

??

aa

+ ...

It is proved in [9] that if one defines

(23)

where σ′is obtained from σ by erasing its maximal letter n, then ∂ is a derivation

of FQSym. Its restriction to Sym coincides obviously with the previous definition.

For α ∈ Sk, β ∈ Sℓ, and n = k + ℓ, set

∂Gσ:= Gσ′

(24)

B(Gα,Gβ) =

?

γ=u(n+1)v

std(u)=α,std(v)=β

Gγ.

Clearly,

(25)∂B(Gα,Gβ) = GαGβ,

and our differential equation for the noncommutative tangent is now replaced by the

fixed point problem

(26)X = G1+ B(X,X).

of which it is the unique solution. Again, solving it by iterations gives back the sum

of alternating permutations. As an element of the Loday-Ronco algebra, tan appears

as the sum of all permutations whose decreasing tree is complete.

The same kind of equation holds for Y :

(27)Y = 1 + B(X,Y ).

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6 M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON

Hence, the noncommutative secant sec is therefore an element of PBT, so a sum

of binary trees. The trees are well-known: they correspond to complete binary trees

(of odd size) where one has removed the last leaf.

2.5. Derivative polynomials. For the ordinary tangent and secant, the differential

equations imply the existence [11] of two sequences of polynomials Pn, Qnsuch that

dn

dzn(tanz) = Pn(tanz)

Since ∂ is a derivation of Sym, we have as well for the noncommutative lifts

(28)and

dn

dzn(secz) = Qn(tanz)secz.

(29)∂n(X) = Pn(X) and∂n(Y ) = Qn(X)Y .

Hoffman [11] gives the exponential generating functions

(30)

?

The noncommutative version of these identities can be readily derived as follows. We

want to compute

Pn(X)tn

P(u,t) =

n≥0

Pn(u)tn

n!=sint + ucost

cost − usintand Q(u,t) =

?

n≥0

Qn(u)tn

n!=

1

cost − usint.

(31)P(X,t) =

?

n≥0

n!= et∂X .

Since ∂ is a derivation, et∂is an automorphism of Sym. It acts on the generators Sn

by

(32)et∂Sn(A) =

n

?

k=0

Sn−k(A)tk

k!= Sn(A + tE)

where tE is the “virtual alphabet” such that Sn(tE) =tn

n!. Hence,

P(X,t) = tan(A + tE) = cos(A + tE)−1sin(A + tE)

= (cost − X sint)−1(sint + X cost)

(33)

as expected. Similarly,

Qn(X,t) =

?

n≥0

Qn(X)tn

n!= (et∂Y )Y−1

(34)

= cos(A + tE)−1cos(A) = (cost − X sint)−1.(35)

3. The uniform definition of snakes for Coxeter groups

Before introducing the relevant generalizations of Sym and FQSym, we shall com-

ment on the definitions of snakes and alternating permutations for general Coxeter

groups.

It is apparent in Springer’s article [22] that alternating permutations can be defined

in a uniform way for any Coxeter group. Still, little attention has been given to this

fact. For example, 20 years later, Arnol’d [2] gives separately the definitions of snakes

of type A, B and D, even though there is no doubt that he was aware of the uniform

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definition. The goal of this section is to give some precisions and to simplify the proof

of Springer’s result in [22].

Let (W,S) be an irreducible Coxeter system. Recall that s ∈ S is a descent of

w ∈ W if ℓ(ws) < ℓ(w) where ℓ is the length function. When J ⊂ S, we denote

by DJ the descent class defined as {w ∈ W : ℓ(ws) < ℓ(w) ⇔ s ∈ J}. Following

Arnol’d, let us consider the following definition.

Definition 3.1. The Springer number of the Coxeter system (W,S) is

(36)K(W) := max

J⊆S(#DJ).

The aim of [22] is to give a precise description of sets J realizing this maximum.

The result is as follows:

Theorem 3.2 (Springer [22]). Let J ⊆ S. Then, K(W) = #DJif and only if J and

S\J are independent subsets of the Coxeter graph S (i.e., they contain no two adja-

cent vertices). In particular, there are two such subsets J which are complementary

to each other.

Therefore we can choose a subset J such that K(W) = #DJand call snakes of W

the elements of DJ. The other choice S\J would essentially lead to the same objects,

since there is a simple involution on W exchanging the subsets DJand DS\J.

We are thus led to the following definition:

Definition 3.3. Let (W,S) be a Coxeter group, and J be a maximal independent

subset of S. The snakes of (W,S) are the elements of the descent class DJ.

This definition depends on the choice of J, so that we can consider two families

of snakes for each W. In the case of alternating permutations, these are usually

called the up-down and down-up permutations, and are respectively defined by the

conditions

(37)σ1< σ2> σ3< ... orσ1> σ2< σ3> ...

It is natural to endow a descent class with the restriction of the weak order, and

this defines what we can call the snake poset of (W,S). Known results show that this

poset is a lattice [3]. Let us now say a few words about the proof of Theorem 3.2,

which relies upon the following lemma.

Lemma 3.4 (Springer [22]). Let J ⊆ S, and assume that there is an edge e of the

Coxeter graph whose endpoints are both in J or both not in J. Let S = S1∪S2be the

connected components obtained after removing e. Let J′= (S1∩ J) ∪ (S2∩ (S\J)).

Then, #DJ< #DJ′.

Using the above lemma, we see that if J or complementary is not independent, we

can find another subset J′having a strictly bigger descent class, and Theorem 3.2

then follows. Whereas Lemma 3.4 is the last one out of a series of 5 lemmas in

Springer’s article, we give here a simple geometric argument.

Let R be a root system for (W,S), and let Π = (αs)s∈Sbe a set of simple roots.

There is a bijection w ?→ i(w) between W and the set of Weyl chambers, so that s ∈ S

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8 M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON

is a descent of w ∈ W if and only if i(w) lies in the half-space {v ∈ Rn: ?v,αs? ≥ 0}.

For any J ⊂ S, let

(38)CJ= {v ∈ Rn: ?v,αs? ≥ 0 if s ∈ J, and ?v,αs? ≤ 0 if s / ∈ J}.

It is the closure of the union of Weyl chambers i(w) where w ∈ DJ. Now, let e, S1,

S2 and J′be as in the lemma, and let x ∈ S1and y ∈ S2be the endpoints of e.

Note that either x or y is in J′but not both. Let σ be the orthogonal symmetry

through the linear span of {αj : j ∈ S1}. We claim that σ(CJ) ? CJ′, and this

implies #DJ< #DJ′ since CJcontains strictly less Weyl chambers than CJ′.

So it remains to show that σ(CJ) ? CJ′. It is convenient to use the notion of dual

cone, which is defined for any closed convex cone C ⊂ Rnas

(39)C∗:= {v ∈ Rn: ?v,w? ≥ 0 for any w ∈ C}.

The map C ?→ C∗is an inclusion-reversing involution on closed convex cones, and it

commutes with any linear isometry, so that we have to prove that σ(C∗

(C1∩ C2)∗= C∗

the half-line R+w, the dual of CJ′ is

J′) ? C∗

J. Since

1+ C∗

2and since the dual of the half-space {v ∈ Rn: ?v,w? ≥ 0} is

(40)C∗

J′ =

??

s∈S

usαs : us≥ 0 if s ∈ J′, and us≤ 0 if s / ∈ J′

?

,

and the same holds for C∗

ity since σ(αs) = αs if s ∈ S1, σ(αs) = −αs if s ∈ S2\y, and σ(αy) = −αy+

2?αx,αy??αx,αx?−1αx. Indeed, let v =?

(41)σ(v) =

?

Since uxand uyhave different signs and ?αx,αy? < 0, we obtain σ(v) ∈ C∗

thus proved σ(C∗

in C∗

σ(v), if uy?= 0 there is a nonzero term in αxas well. This completes the proof.

J. A description of σ(C∗

J′) is obtained easily by linear-

s∈Susαs∈ C∗

J′, we have:

s∈S1

usαs−

?

s∈S2\y

usαs− uyαy+ 2uy?αx,αy??αx,αx?−1αx.

J. We have

J′) ⊂ C∗

J. To show the strict inclusion, note that either αyor −αyis

J. But none of these two elements is in σ(C∗

J′), because in the above formula for

4. Signed permutations and combinatorial Hopf algebras

Whereas the constructions of the Hopf algebras Sym, PBT, and FQSym ap-

pearing when computing the usual tangent are almost straightforward, the situation

is quite different in type B. First, there are at least three different generalizations

of Sym to a pair of alphabets, each with its own qualities either combinatorial or

algebraic. Moreover, there are also two different ways to generalize FQSym. The

generalizations of PBT are not (yet) defined in the literature but the computations

done in the present paper give a glimpse of what they should be.

Here follows how they embed each in the other. All embeddings are embeddings

of Hopf algebras except the two embeddings concerning BSym which is not itself an

algebra. However, the embedding of Sym(A|¯A) into Sym(2)obtained by composing

the two previous embeddings is a Hopf embedding:

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THE ALGEBRAIC COMBINATORICS OF SNAKES9

(42)

Sym(A|¯A) ֒→ BSym ֒→ Sym(2)֒→ FQSym(A|¯A) ֒→ FQSym(2).

4.1. The Mantaci-Reutenauer algebra of type B. The most straightforward

definition of Sym in type B is to generalize the combinatorial objects involved in the

definition: change compositions into signed compositions.

We denote by Sym(2)= MR [15] the free product Sym⋆Sym of two copies of the

Hopf algebra of noncommutative symmetric functions. In other words, MR is the

free associative algebra on two sequences (Sn) and (S¯ n) (n ≥ 1). We regard the two

copies of Sym as noncommutative symmetric functions on two auxiliary alphabets:

Sn= Sn(A) and S¯ n= Sn(¯A). We denote by F ?→¯F the involutive automorphism1

which exchanges Snand S¯ n. And we denote the generators of Sym(2)by S(k,ǫ)where

ǫ = {±1}, so that S(k,1)= Skand S(k,−1)= S¯k.

4.2. Noncommutative supersymmetric functions. The second generalization of

Sym comes from the transformation of alphabets sending A to a combination of A

and¯A. It is the algebra containing the type B alternating permutations.

We define Sym(A|¯A) as the subalgebra of Sym(2)generated by the S#

any F ∈ Sym(A),

nwhere, for

(43)F#= F(A|¯A) = F(A − q¯A)|q=−1,

called the supersymmetric version, or superization, of F [17].

The expansion of an element of Sym(A|¯A) as a linear combination in Sym(2)is

done thanks to generating series. Indeed,

(44)σ#

1=¯λ1σ1=

??

k≥0

Λk

???

m≥0

Sm

?

where Λk=?

(45)

I|=k(−1)ℓ(I)−kSI, as follows from¯λ1= (¯ σ−1)−1(see [13]). For example,

S#

1= S1+ S1,S#

2= S2+ S11− S2+ S11,

(46)S#

3= S3+ S12+ S111− S21+ S111− S21− S12+ S3.

4.3. Noncommutative symmetric functions of type B. The third generaliza-

tion of Sym is not an algebra but only a cogebra but is the generalization one gets

with respect to the group Bn: its graded dimension is 2nand, as we shall see later

in this paragraph, a basis of BSym is given by sums of permutations having given

descents in the type B sense. This algebra contains the snakes of type B.

Noncommutative symmetric functions of type B were introduced in [4] as the right

Sym-module BSym freely generated by another sequence (˜Sn) (n ≥ 0,˜S0= 1) of

homogeneous elements, with ˜ σ1grouplike. This is a coalgebra, but not an algebra.

1This differs from the convention used in some references.

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10 M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON

We embed BSym as a sub-coalgebra and right sub-Sym-module of MR as follows.

The basis element˜SIof BSym, where I = (i0,i1,...,ir) is a B-composition (that

is, i0may be 0), can be embedded as

˜SI= Si0(A)Si1i2...ir(A|¯A).

In the sequel, we identify BSym with its image under this embedding.

As in Sym, one can define by triangularity the analog of the ribbon basis ([4]):

(47)

(48)

˜SI=

?

J≤I

˜RJ,

where J ≤ I if the B-descent set of J is a subset of the B-descent set of I. Note that

we have in particular˜S0n=˜R0n+˜Rn.

Note also that, thanks to that definition, SI#=˜SIand, thanks to the transitions

between all bases,

R#

(49)

I=˜R0I+˜RI.

4.4. Type B permutations and descents in Bn. The hyperoctahedral group Bnis

the group of signed permutations. A signed permutation can be denoted by w = (σ,ǫ)

where σ is an ordinary permutation and ǫ ∈ {±1}n, such that w(i) = ǫiσ(i). If we

set w(0) = 0, then, i ∈ [0,n − 1] is a B-descent of w if w(i) > w(i + 1). Hence, the

B-descent set of w is a subset D = {i0,i0+ i1,...,i0+ ··· + ir−1} of [0,n − 1]. We

then associate with D the type-B composition (i0− 0,i1,...,ir−1,n − ir−1).

4.5. Free quasi-symmetric functions of level 2. Let us now move to generaliza-

tions of FQSym. As in the case of Sym, the most natural way is to change the usual

alphabet into two alphabets, one of positive letters and one of negative letters and

to define a basis indexed by signed permutations as a realization on words on both

alphabets. This algebra is FQSym(2), the algebra of free quasi-symmetric functions

of level 2, as defined in [16].

Let us set

A(0)= A = {a1< a2< ··· < an< ...}, (50)

A(1)=¯A = {··· < ¯ an< ··· < ¯ a2< ¯ a1}, (51)

and order A =¯A ∪ A by ¯ ai< ajfor all i,j. Let us also denote by std the standard-

ization of signed words with respect to this order.

We shall also need the signed standardization Std, defined as follows. Represent a

signed word w ∈ Anby a pair (w,ǫ), where w ∈ Anis the underlying unsigned word,

and ǫ ∈ {±1}nis the vector of signs. Then Std(w,ǫ) = (std(w),ǫ).

Then, FQSym(2)is spanned by the polynomials in A ∪¯A

(52)

Gσ,u:=

?

w∈An;Std(w)=(σ,u)

w∈ Z?A?.

Let (σ′,u′) and (σ′′,u′′) be signed permutations. Then (see [16, 17])

(53)

Gσ′,u′ Gσ′′,u′′ =

?

σ∈σ′∗σ′′

Gσ,u′·u′′.

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THE ALGEBRAIC COMBINATORICS OF SNAKES11

We denote by m(ǫ) the number of entries −1 in ǫ.

4.6. Free super-quasi-symmetric functions. The second algebra generalizing the

algebra FQSym is FQSym(A|¯A). It comes from the transformation of alphabets

applied to FQSym as Sym(A|¯A) comes from Sym. To do this, we first need to

recall that FQSym(2)is equipped with an internal product.

Indeed, viewing signed permutations as elements of the group {±1} ≀ Sn, we have

the internal product

(54)

Gα,ǫ∗ Gβ,η= G(β,η)◦(α,ǫ)= Gβ◦α,(ηα)·ǫ,

with ηα = (ηα(1),...,ηα(n)) and ǫ · η = (ǫ1η1,...,ǫnηn).

We can now embed FQSym into FQSym(2)by

(55)

Gσ?→ G(σ,1n),

which allows us to define

(56)

so that FQSym(A|¯A) is the algebra spanned by the Gσ(A|¯A).

G#

σ:= Gσ(A|¯A) = Gσ∗ σ#

1,

Theorem 4.1 ([17], Thm. 3.1). The expansion of Gσ(A|¯A) on the basis Gτ,ǫis

(57)

Gσ(A|¯A) =

?

std(τ,ǫ)=σ

Gτ,ǫ.

4.6.1. Embedding Sym#and BSym into FQSym(2). One can embed BSym into

FQSym(2)as one embeds Sym into FQSym (see [4]) by

(58)

˜RI=

?

Bdes(π)=I

Gπ,

where I is any B-composition.

Given Equation (49) relating R#

Iand the˜RI, one has

?

(59)R♯

I=

Des(π)=I

Gπ,

where I is any (usual) composition.

5. Algebraic theory in type B

5.1. Alternating permutations of type B.

5.1.1. Alternating shapes. Let us say that a signed permutation π ∈ Bnis alternating

if π1<π2>π3<... (shape 2mor 2m1).

Here are the alternating permutations of type B for n ≤ 4:

(60)

¯1,1,

12,¯12,¯21,¯2¯1

(61)12¯3,¯12¯3,132,13¯2,¯132,¯13¯2,¯21¯3,¯2¯1¯3,231,23¯1,¯231,¯23¯1,¯31¯2,¯3¯1¯2,¯321,¯32¯1,

Page 12

12 M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON

12¯34,¯12¯34,12¯43,12¯4¯3,¯12¯43,¯12¯4¯3,1324,13¯24,¯1324,¯13¯24,13¯42,13¯4¯2,

¯13¯42,¯13¯4¯2,1423,14¯23,¯1423,¯14¯23,14¯32,14¯3¯2,¯14¯32,¯14¯3¯2,¯21¯34,¯2¯1¯34,

¯21¯43,¯21¯4¯3,¯2¯1¯43,¯2¯1¯4¯3,2314,23¯14,¯2314,¯23¯14,23¯41,23¯4¯1,¯23¯41,¯23¯4¯1,

2413,24¯13,¯2413,¯24¯13,24¯31,24¯3¯1,¯24¯31,¯24¯3¯1,¯31¯24,¯3¯1¯24,¯31¯42,¯31¯4¯2,

¯3¯1¯42,¯3¯1¯4¯2,¯3214,¯32¯14,¯32¯41,¯32¯4¯1,¯3¯2¯41,¯3¯2¯4¯1,3412,34¯12,¯3412,¯34¯12,

34¯21,34¯2¯1,¯34¯21,¯34¯2¯1,¯41¯23,¯4¯1¯23,¯41¯32,¯41¯3¯2,¯4¯1¯32,¯4¯1¯3¯2,¯4213,¯42¯13,

¯42¯31,¯42¯3¯1,¯4¯2¯31,¯4¯2¯3¯1,¯4312,¯43¯12,¯43¯21,¯43¯2¯1

(62)

Hence, π is alternating iff ¯ π is a β-snake in the sense of [2]. Hence, the sum in

FQSym(2)of all Gπlabeled by alternating signed permutations is, as already proved

in [4]

(63)

X = (X + Y )#= sec#+ tan#= sec#(1 + sin#) =

?

m≥0

(R#

(2m)+ R#

(2m1)).

5.1.2. Quasi-differential equations. Let d be the linear map acting on Gπas follows:

(64)dGπ=

?

Guv

Gu¯ v

if π = unv,

if π = u¯ nv.

This map lifts to FQSym(2)the derivation ∂ of (23), although it is not itself a

derivation. We then have

Theorem 5.1. The series X satisfies the quasi-differential equation

(65)dX = 1 + X2.

Proof – Indeed, let us compute what happens when applying d to R#

being the same with dR#

in σ, let us write σ = unv. Then dGσappears in the product GStd(u)GStd(v)and u

and v are of respective shapes (2n1) and (2m−n−1). If n appears in σ, let us write

again σ = unv. Then dGσappears in the product GStd(u)GStd(v)and u and v are of

respective shapes (2n) and (2m−n−11). Conversely, any permutation belonging to u∗v

with u and v of shapes (2n1) and (2m−n−1) has a shape (2m) if one adds n in position

2n + 2. The same holds for the other product, hence proving the statement.

(2m), the property

(2m1). Let us fix a permutation σ of shape (2m). If n appears

This is not enough to characterize X but we have the analog of fixed point equa-

tion (26)

(66)

X = 1 + G1+ B(X,X),

where

(67)B(Gα,Gβ) =

??

γ=u(n+1)v, Std(u)=α, Std(v)=βGγ

?

if |α| is odd,

if |α| is even.

γ=u(n+1)¯ v, Std(u)=α, Std(v)=βGγ

Indeed, applying d to the fixed point equation brings back Equation (65) and it is

clear from the definition of B that all terms in B(Gα,Gβ) are alternating signed

permutations.

Page 13

THE ALGEBRAIC COMBINATORICS OF SNAKES13

Solving this equation by iterations gives back the results of [12, Section 4]. Indeed,

the iteration of Equation (66) yields the solution

(68)

X =

?

T∈CBT

BT(G0= 1,G1),

where, for a tree T, BT(a,b) is the result of the evaluation of all expressions formed

by labeling by a or b the leaves of T and by B its internal nodes. This is indeed the

same as the polynomials Pndefined in [12, Section 4] since one can interpret the G0

leaves as empty leaves in this setting, the remaining nodes then corresponding to all

increasing trees of the same shape, as can be seen on the definition of the operator B.

5.1.3. Alternating signed permutations counted by number of signs. Under the spe-

cialization¯A = tA, X goes to the series

(69)X(t;A) =

?

I

?

π alternating, C(std(π))=I

tm(π)

RI(A)

where m(π) is the number of negative letters of π. If we further set A = zE, we

obtain

x(t,z) =1 + sin((1 + t)z)

cos((1 + t)z)

(70)

which reduces to

(71)x(1,z) =1 + sin2z

cos2z

=cosz + sinz

cosz − sinz

for t = 1, thus giving a t-analogue different from the one of [12].

5.1.4. A simple bijection. From (70), we have

(72)

?

n

zn

?

π alternating in Bn

tm(π)= sec((1 + t)z) + tan((1 + t)z).

But another immediate interpretation of the series in the right-hand side is

(73)

?

n

zn

?

π s.t. |π| is alternating in An

tm(π)= sec((1 + t)z) + tan((1 + t)z).

It is thus in order to give a bijection proving the equality of the generating func-

tions. Let π be an alternating signed permutation. We can associate with π the pair

(std(π),ǫ) where ǫ is the sign vector such that ǫi = 1 if π−1(i) > 0 and ǫi = −1

otherwise. The image of {1,...,n} by π is {ǫii : 1 ≤ i ≤ n}. Since π can be recov-

ered from std(π) and the image of {1,...,n}, this map is a bijection between signed

alternating permutations and pairs (σ,ǫ) where σ is alternating and ǫ is a sign vector.

Then, with such a pair (σ,ǫ), one can associate a signed permutation τ such that |τ|

is alternating simply by taking τi= σiǫi. The composition π ?→ (σ,ǫ) ?→ τ gives the

desired bijection.

Page 14

14M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON

For example, here follow the 16 permutations obtained by applying the bijection

to the 16 alternating permutations of size 3 (see Equation (61)):

23¯1,¯23¯1,132,2¯31,¯132,¯2¯31,2¯3¯1,¯2¯3¯1,231,¯231,1¯32,¯1¯32,1¯3¯2,¯1¯3¯2,13¯2,¯13¯2. (74)

5.2. Type B snakes.

5.2.1. An alternative version. The above considerations suggest a new definition of

type B snakes, which is a slight variation of the definition of [2]. We want to end up

with the generating series

1

cosz − sinz=cosz + sinz

after the same sequence of specializations. A natural choice, simple enough and given

by a series in BSym, is to set

(75)y(1,z) =

cos2z

(76)

Y = (cos + sin) · sec#=

??

k≥0

(−1)k(S2k+ S2k+1)

?

·

?

n≥0

R#

2n.

Now, Y lives in BSym and expands in the ribbon basis˜R of BSym as

Y =

??

?

?

?

k≥0

(R#

(−1)k(˜R2k+˜R2k+1)

??

n≥0

R#

2n

=

n≥0

2n +˜R12n +˜R32n)

+

k≥1;n≥0

(−1)k(˜R2k2n +˜R2k+22n−1 +˜R2k+12n +˜R2k+32n−1)

=

n≥0

(R#

2n +˜R12n +˜R32n) −

?

n≥0

(˜R2n+1 +˜R32n)

(77)

which simplifies into

(78)

Y = 1 +

?

n≥0

(˜R12n +˜R02n+1).

In FQSym(2), this is the sum of all Gπsuch that

?

Thus, for n odd, π is exactly a Bn-snake in the sense of [2], and for n even, ¯ π is a

Bn-snake. Clearly, the number of sign changes or of minus signs in snakes and in

these modified snakes are related in a trivial way so we have generating series for

both statistics in all cases.

(79)

0 > π1< π2> ... if n is even,

0 < π1> π2< ... if n is odd.

Here are these modified snakes for n ≤ 4:

(80)1,

¯12,¯21,¯2¯1,

(81)1¯23,1¯32,1¯3¯2,213,2¯13,2¯31,2¯3¯1,312,3¯12,3¯21,3¯2¯1,

Page 15

THE ALGEBRAIC COMBINATORICS OF SNAKES15

¯12¯34,¯12¯43,¯12¯4¯3,¯1324,¯13¯24,¯13¯42,¯13¯4¯2,¯1423,¯14¯23,¯14¯32,¯14¯3¯2,¯21¯34,

¯2¯1¯34,¯21¯43,¯21¯4¯3,¯2¯1¯43,¯2¯1¯4¯3,¯2314,¯23¯14,¯23¯41,¯23¯4¯1,¯2413,¯24¯13,¯24¯31,

¯24¯3¯1,¯31¯24,¯3¯1¯24,¯31¯42,¯31¯4¯2,¯3¯1¯42,¯3¯1¯4¯2,¯3214,¯32¯14,¯32¯41,¯32¯4¯1,¯3¯2¯41,

¯3¯2¯4¯1,¯3412,¯34¯12,¯34¯21,¯34¯2¯1,¯41¯23,¯4¯1¯23,¯41¯32,¯41¯3¯2,¯4¯1¯32,¯4¯1¯3¯2,¯4213,

¯42¯13,¯42¯31,¯42¯3¯1,¯4¯2¯31,¯4¯2¯3¯1,¯4312,¯43¯12,¯43¯21,¯43¯2¯1.

(82)

5.2.2. Snakes as particular alternating permutations. Note that in the previous defi-

nition, snakes are not alternating permutations for odd n. So, instead, let us consider

the generating series

(83)

Y = cos · sec#+ sin · sec#,

where f ?→¯f is the involution of FQSym(2)inverting the signs of permutations.

Expanding Y in the˜R basis, one gets

(84)

Y = 1 +

?

n≥0

(˜R02n1+˜R02n+1).

As for type B alternating permutations (see Equation (65)), the series Y satisfies

a differential equation with the same linear map d as before (see Equation (64)):

(85)dY = YX.

It is then easy to see that Y also satisfies a fixed point equation similar to (26):

(86)

Y = 1 + B(Y,X).

The iteration of (86) brings up a solution close to (68):

(87)

Y =

?

T∈CBT

BT(G0= 1,G1),

where, for a tree T, BT(a,b) is now the result of the evaluation of all expressions

formed by labeling by a or b the leaves of T and by B its internal nodes. Note that

in this case, the first leaf needs to have label a. This is the same as the trees defined

in [12, Section 4] since one can again interpret the G0leaves as empty leaves in this

setting, the remaining nodes then corresponding to all increasing trees of the same

shape.

5.2.3. Snakes from [2]. The generating series of the snakes of [2], also in BSym is

(88)

cos · sec#+ sin · sec#,

and can be written as

(89)

Y = 1 +

?

n≥0

(˜R12n +˜R12n1)

on the ribbon basis.

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