Page 1
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
Page 2
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.
Page 3
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).
Page 4
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)
Page 5
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 ).
Page 6
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
Page 7
THE ALGEBRAIC COMBINATORICS OF SNAKES7
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
Page 8
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:
Page 9
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.
Page 10
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′′.
Page 11
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.
Page 16
16M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON
The lift of the differential equation for y(1) is given by a map δ similar to d, with
δunv = ¯ uv and δu¯ nv = u¯ v. Then
(90)δY = YX
and we have a fixed point equation
(91)
Y = 1 +ˆB(Y,X)
for an appropriate bilinear mapˆB.
6. Another combinatorial model
6.1. An analogue of cosz − sinz in FQSym(2).
Definition 6.1. A signed permutation π ∈ Bnis a valley-signed permutation if, for
any i ∈ [n], π(i) < 0 implies that
• either i > 2, and πi−1> 0, and |πi−2| > πi−1< |πi|,
• or i = 2, and 0 < π1< |π2|.
We denote by Vnthe set of valley-signed permutations of size n.
Here are these signed permutations, up to n = 4:
(92)1,
12,1¯2,21,
(93) 123,1¯23,132,1¯32,213,21¯3,231,2¯31,312,31¯2,321,
1234,1¯234,1243,1¯243,1324,132¯4,1¯324,1¯32¯4,1342,1¯342,1423,142¯3,
1¯423,1¯42¯3,1432,1¯432,2134,21¯34,2143,21¯43,2314,231¯4,2¯314,2¯31¯4,
2341,2¯341,2413,241¯3,2¯413,2¯41¯3,2431,2¯431,3124,31¯24,3142,31¯42,
3214,321¯4,3241,32¯41,3412,341¯2,3¯412,3¯41¯2,3421,3¯421,4123,41¯23,
4132,41¯32,4213,421¯3,4231,42¯31,4312,431¯2,4321
(94)
The terminology is explained by the following remark. Let σ ∈ Sn, and let us
examine how to build a valley-signed permutation π so that πi= ±σi. It turns out
that for each valley σ(i−1) > σ(i) < σ(i+1), we can choose independently the sign
of π(i + 1) (here 1 ≤ i < n and 1 is a valley if σ(1) < σ(2)).
The goal of this section is to obtain the noncommutative generating functions for
the sets Vn.
Theorem 6.2. The following series
(95)U = 1 − [G1+ G1¯2− G12¯3− G1¯23¯4+ G12¯34¯5+ G1¯23¯45¯6+ ...]
is again a lift of cosz − sinz in FQSym(2). It satisfies
(96)U−1=
?
n≥0
?
π∈Vn
Gπ.
Hence the π occuring in this expansion are in bijection with snakes of type B.
This result is a consequence of the next two propositions.
Page 17
THE ALGEBRAIC COMBINATORICS OF SNAKES17
Definition 6.3. Let R2n⊂ B2nbe the set of signed permutations π of size 2n such
that |π| is of shape 2n, and for any 1 ≤ i ≤ n, we have π(i) > 0 iff i is odd. Let
(97)V =
?
n≥0
?
π∈R2n
Gπ= Gǫ+ G1¯2+ G1¯32¯4+ G1¯42¯3+ ...
Let R2n+1⊂ B2n+1be the set of signed permutations π of size 2n+1 such that |π| is
of shape 12n, π1> 0, and for any 2 ≤ i ≤ n, we have π(i) > 0 iff i is even. Let
?
Note that Rn⊂ Vn. Clearly, #Rnis the number of alternating permutations of
Snsince, given |π|, there is only one possible choice for the signs of each πi. So V
and W respectively lift sec and tan in FQSym(2). Now, given that the product rule
of the Gσdoes not affect the signs, a simple adaptation of the proof of the Ancase
shows:
(98)W =
n≥0
?
π∈R2n+1
Gπ= G1+ G21¯3+ G31¯2+ G21¯43¯5+ ...
Proposition 6.4. We have
V−1= 1 − G1¯2+ G1¯23¯4− G1¯23¯45¯6+ ...,
WV−1= G1− G12¯3+ G12¯34¯5− G12¯34¯56¯7+ ...
(99)
(100)
Note that U = (1 − W)V−1. So, to complete the proof of the theorem, it remains
to show:
Proposition 6.5. We have
(101)V (1 − W)−1=
?
n≥0
?
π∈Vn
Gπ.
Proof – We can write V (1 − W)−1= V + V W + V W2+ ..., and expand everything
in terms of the Gπ, using their product rule (see Equation (53)). We obtain a sum
of Gu1...ukwhere Std(u1) ∈ R2∗and Std(ui) ∈ R2∗+1for any i ≥ 2 (where ∗ is any
integer). The sum is a priori over lists (u1,...,uk) such that u = u1...uk ∈ Bn.
Actually, if the words u1,...,uk satisfy the previous conditions, then u is in Vn.
Indeed, the first two letters of each ui are not signed, each ui is a valley-signed
permutation, which implies that u itself is a valley-signed permutation.
Conversely, it remains to show that this factorization exists and is unique for any
u ∈ Vn. First, observe that in u = u1...uk, ukis the only suffix of odd length having
the same signs as an element of R2∗+1(since the first two letters are positive and the
other alternate in signs, it cannot be itself a strict suffix of an element of R2∗+1). So
the factorization can be obtained by scanning u from right to left.
Let
(102)Z := G¯1− G¯12− G¯12¯3+ G¯123¯4+ G¯12¯34¯5− G¯123¯45¯6+ ...
Precisely, the n-th term in this expansion is the permutation σ of Bn such that,
|σ| = id, σ(1) =¯1, σ(2) = 2 and, for 3 ≤ i ≤ n, σ(i) < 0 iff n − i is even. The sign
Page 18
18 M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON
corresponds to the sign of znin the expansion of cosz + sinz − 1, so that it is a lift
of cosz + sinz − 1 in FQSym(2).
Theorem 6.6. The series Z U−1is a sum of Gπwithout multiplicities in the Hopf
algebra FQSym(2). Hence the π occuring in this expansion are in bijection with
snakes of type D (see Section 9).
Moreover, the series (1 + Z)U−1is also a sum of Gπwithout multiplicities in the
Hopf algebra FQSym(2). Hence the π occuring in this expansion are in bijection with
alternating permutations of type B.
Here are the elements of Z U−1, up to n = 4:
(103)
¯1,
¯21,
(104)
¯213,¯21¯3,¯312,¯31¯2,¯321,
¯21¯34,¯2134,¯21¯43,¯2143,¯31¯24,¯3124,¯31¯42,¯3142,¯321¯4,¯3214,¯32¯41,¯3241,
¯41¯23,¯4123,¯41¯32,¯4132,¯421¯3,¯4213,¯42¯31,¯4231,¯431¯2,¯4312,¯4321.
(105)
Proof – Let εibe the linear operator sending a word w to the word w′where w′is
obtained from w by sending its ith letter to its opposite and not changing the other
letters.
Then if one writes 1 + Z = E + O as a sum of an even and an odd series, one has
(106)E = ε1ε2V−1
O = ǫ1(WV−1).
Since εi(ST) = εi(S)T if S contains only terms of size at least i, an easy rewriting
shows that
(107) (1 + Z)V = V + ε1(W) + ε1ε2(1 − V ).
Then, one gets
(108) (1 + Z)V (1 − W)−1= U−1+ ε1(W + ε2(1 − V ))(1 − W)−1.
So it only remains to prove that Q = ε1(W + ε2(1 − V ))(1 − W)−1has only positive
terms and has no term in common with U−1. This last fact follows from the fact
that all terms in Q have a negative number as first value. Let us now prove that
Q has only positive terms. First note that W + ε2(1 − V ) is an alternating sum
of permutations of shapes 2nand 12n. Hence, any permutation of shape 2ncan
be associated with a permutation of shape 12n−1by removing its first entry and
standardizing the corresponding word. Now, all negative terms −GvGwcome from
−ε2(V )(1 − W)−1and are annihilated by the term Gv′G1Gwwhere v′is obtained
from v by the removal-and-standardization process described before.
Page 19
THE ALGEBRAIC COMBINATORICS OF SNAKES19
6.2. Another proof of Theorem 6.2. Let Sgnbe the group algebra of {±1}n. We
identify a tuple of signs with a word in the two symbols 1,¯1, and the direct sum of
the Sgnwith the free associative algebra on these symbols.
We can now define the algebra of signed noncommutative symmetric functions as
(109)
sSym :=
?
n≥0
Symn⊗ Sgn
endowed with the product
(110)(f ⊗ u) · (g ⊗ v) = fg ⊗ uv.
It is naturally embedded in FQSym(2)by
(111)RI⊗ u =
?
C(σ)=I
Gσ,u.
With this at hand, writing U = P − Q with
(112)P =
?
m≥0
(−1)mR2m⊗ (1¯1)m
andQ =
?
m≥0
(−1)mR2m+1⊗ 1(1¯1)m,
it is clear that
(113)P−1=
?
m≥0
R(2m)⊗ (1¯1)m= V and W =
?
m≥0
R(12m)⊗ 1(1¯1)m= QV,
so that U = (1−W)V−1, and U−1= V (1−W)−1can now be computed by observing
that
(114)(1 − W)−1=
?
I
RI⊗ pI(1,¯1)
where pIis the sum of words in 1,¯1 obtained by associating with each tiling of the
ribbon shape of I by tiles of shapes (1,2m1,...,1,2mk) the word 1(1¯1)m1...1(1¯1)mk.
For example, p12= 111 + 11¯1, for there are two tilings, one by three shapes (1) and
one by (12). The proof of Theorem 6.6 can be reformulated in the same way.
6.3. Some new identities in BSym. Let us have a closer look at the map¯A ?→ tA.
By definition, RI(A|tA) = RI((1 − q)A)|q=−t, so that the generating series for
one-part compositions is
(115)
?
n≥0
Rn((1 − q)A)xn= σx((1 − q)A) = λ−qx(A)σx(A).
One can now expand this formula in different bases. Tables are given in Section 10.
6.3.1. Image of the˜R in the S basis of Sym. The first identity gives the image of a
ribbon in the S basis:
(116)RI(A|tA) =
?
J
(−1)l(I)+l(J)(1 − (−t)jr) (−t)
?
A(I,J)jkSJ
Page 20
20 M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON
where A(I,J) = {p|j1+ ··· + jp?∈ Des(I)}. Indeed, expanding λ−qx(A) on the basis
SJ, one finds
(117)
?
n≥0
Rn((1 − q)A)xn=
?
n≥0
xn
?
?
k+m=n
(−q)k?
K?k
(−1)k−ℓ(K)SKSm
?
Each composition of a given n occurs twice in the sum, so that
(118)Rn((1 − q)A) =
?
J?n
(−1)1−ℓ(J)(qn−jr− qn)SJ.
Hence, Equation (116) is true for one-part compositions. The general case follows by
induction from the product formula
(119)RIRj= RIj+ RI⊲j.
Now, as for the˜R, we have
Proposition 6.7. Let I be a type-B composition and set I =: (i0,I′). Then the
expansion of the˜RI(A,tA) on the SJ(A) is
(120)
where C2is the set of compositions of n different from (n) whose first part is a sum of
any prefix of I, and C1the set complementary to C2in all compositions of n different
from (n).
(−1)ℓ(I′)+ntnSn+
?
(−1)ℓ(I)+ℓ(J)(1 − (−t)jr) (−t)
J∈C1
(−1)ℓ(I)+ℓ(J)(1 − (−t)jr) (−t)
?
A(I,J)jkSJ
if i0= 0,
(−1)ℓ(I)+1Sn+
?
J∈C2
?
A(I,J)jkSJ
otherwise,
For example, with I = (1321), C2(I) is the set of compositions of 7 whose first part
is either 1, 4, or 6.
Proof – By induction on the length of I. First, Rn(A,tA) = Rn(A) = Sn(A) and
(121)
˜Rn(A,tA) = Rn(A) and
˜R0n(A,tA) = Rn(A|tA) − Rn(A).
Now, the formula˜RIRj=˜RIj+˜RI⊲jtogether with Equation (116) implies the general
case.
Note that this also means that one can compute the matrices recursively. Indeed,
if one denotes by K0(n) (resp. K1(n)) the matrix expanding the˜RI(A,tA) where
i0= 0 (resp. i0?= 0) on the SJ, one has the following structure:
(122)
?
(K0(n+1)K1(n+1)) =
−tK0(n)
t(K0(n)+K1(n))
tK0(n) K1(n)
0
−K1(n)
K0(n)+K1(n)0
?
Page 21
THE ALGEBRAIC COMBINATORICS OF SNAKES 21
6.3.2. Image of the˜R in the Λ basis of Sym. Expanding Equation (115) on the basis
ΛJ, the same reasoning gives as well
(123)RI(A|tA) =
?
J
(−1)n+1+l(I)+l(J)(1 − (−t)j1) (−t)
?
k∈A′(I,J)jkΛJ
where A′(I,J) = {ℓ|j1+ ··· + jℓ−1∈ Des(I)}.
Now, as for the˜R, we have
Proposition 6.8. Let I be a type-B composition and set I =: (i0,I′). Then the
expansion of the˜RI(A,tA) on the ΛJ(A) is
Note that this is coherent with the fact that 0 ∈ Des(I) if i0= 0.
(124)
?
?
J
(−1)n+l(I′)+l(J)(−t)j1+?
k∈A′(I′,J)jkΛJ
if i0= 0,
J
(−1)n+1+l(I′)+l(J)(−t)
?
k∈A′(I′,J)jkΛJ
otherwise.
Proof – The proof is again by induction of the length of I following the same steps
as in the expansion on the S.
Again, one can compute the matrices recursively. If one denotes by L0(n) (resp.
L1(n)) the matrix expanding the˜RI(A,tA) where i0= 0 (resp. i0?= 0) on the ΛJ,
one has the following structure:
?
6.3.3. Image of the˜R in the R basis of Sym. The expansion of RI(A|tA) has been
discussed in [13] and [17]. Recall that a peak of a composition is a cell of its ribbon
diagram having no cell to its right nor on its top (compositions with one part have
by convention no peaks) and that a valley is a cell having no cell to its left nor at its
bottom.
The formula is the following:
(125)(L0(n+1)L1(n+1)) =
tL0(n) −tL0(n) −L1(n) L1(n)
tL1(n) tL0(n)L1(n)L0(n)
?
.
(126)RI(A|tA) =
?
J
(1 + t)v(J)tb(I,J)RJ(A),
where the sum is over all compositions J such that I has either a peak or a valley
at each peak of J. Here v(J) is the number of valleys of J and b(I,J) is the number
of values d such that, either d is a descent of J and not a descent of I, or d − 1 is a
descent of I and not a descent of J.
In the case of the˜R, the matrices satisfy a simple induction.
by M0(n) (resp. M1(n)) the matrix expanding the˜RI(A,tA) where i0 = 0 (resp.
i0 ?= 0) on the R, one has the following structure which follows directly from the
interpretation in terms of signed permutations:
(127)
?
If one denotes
(M0(n+1)M1(n+1)) =
tM1(n)tM0(n) M1(n)
0
M0(n)
t(M0(n)+M1(n))0M0(n)+M1(n)
?
.
Page 22
22 M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON
For example, one can check this result on Figure 3. One then recovers the matrix
of RI(A|tA) on the R as M0+ M1.
7. Euler-Bernoulli triangles
7.1. Alternating permutations of type B. Counting ordinary (type A) alternat-
ing permutations according to their last value yields the Euler-Bernoulli triangle,
sequence A010094 or A008281 of [20] depending on whether one requires a rise or a
descent at the first position.
The same can be done in type B for alternating permutations and snakes. Since
usual snakes begin with a descent, we shall count type B permutations of ribbon
shape 2mor 2m1 according to their last value. We then get the table
(128)
n\p −6 −5
1
2
3
4
5
6
0
−4−3−2−1
1
1
3
11
57
0
0
0
0
0
0
1
1
1
3
23456
0
4
8
68
2
2
14
46
4
4
0
16
32
0
80
160 236 304 361 0 361 418 464 496 512 512
11
57
16
1680
80
760
Proposition 7.1. The table counting type B alternating permutations by their last
value is obtained by the following algorithm: first separate the picture by the column
p = 0 and then compute two triangles. Put 1 at the top of each triangle and compute
the rest as follows: fill the second row of the left (resp. right) triangle as the sum of
the elements of the first row (resp. strictly) to their left. Then fill the third row of
the right (resp. left) triangle as the sum of the elements of the previous row (resp.
strictly) to their right. Compute all rows successively by reading from left to right
and right to left alternatively.
This is the analogue for alternating permutations of Arnol’d’s construction for
snakes of type B [2].
Proof – Let S(n,p) be the set of alternating permutations of Bnending with p.
The proof is almost exactly the same as for type A, with one exception: it is obvious
that S(n,1) = S(n,−1). Since the reading order changes from odd rows to even
rows, let us assume that n is even and consider both sets S(n,p) and S(n,p−1). The
natural injective map of S(n,p−1) into S(n,p) is simple: exchange p−1 with p while
leaving the possible sign in place. The elements of S(n,p) that were not obtained
previously are the permutations ending by p−1 followed by p. Now, removing p−1
and relabeling the remaining elements in order to get a type B permutation, one gets
elements that are in bijection with elements of either S(n − 1,p) or S(n − 1,p − 1),
depending on the sign of p.
Page 23
THE ALGEBRAIC COMBINATORICS OF SNAKES23
7.2. Snakes of type B. The classical algorithm computing the number of type B
snakes (also known as Springer numbers, see Sequence A001586 of [20]) makes use of
the double Euler-Bernoulli triangle.
Proposition 7.2. The table counting snakes of type B by their last value is obtained
by the following algorithm: first separate the picture by the column p = 0 and then
compute two triangles. Put 1 at the top of the left triangle and 0 at the top of the
right one and compute the rest as follows: fill the second row of the left (resp. right)
triangle as the sum of the elements of the first row (resp. strictly) to their left. Then
fill the third row of the right (resp. left) triangle as the sum of the elements of the
previous row (resp. strictly) to their right. Compute all rows successively by reading
from left to right and right to left alternatively.
Here are the first rows of both triangles:
(129)
n\p −6 −5
1
2
3
4
5
6
0
−4−3−2−1
1
1
2
8
40
1
0
1
2
8
40
256 296 328 350 361 361
23456
0
3
6
1
13
3
54
0
11
22
0 10
32
11
11 57
57
57
114 168 216 256
480
Proof – The proof is essentially the same as in the case of alternating permutations
of type B: it amounts to a bijection between a set of snakes on the one side and two
sets of snakes on the other side.
One also sees that each row of the triangles of the alternating type B permutations
presented in Equation (128) can be obtained, up to reversal, by adding or subtracting
the mirror image of the left triangle to the right triangle. For example, on the fifth
row, the sums are 40 + 40 = 80, then 48 + 32 = 80, then 54 + 22 = 76, then
57 + 11 = 68, and 57 + 0 = 57; the differences are 57 − 0 = 57, then 57 − 11 = 46,
then 54−22 = 32, then 48−32 = 16, and 40−40 = 0. These properties follow from
the induction patterns.
These numerical properties indicate that one can split alternating permutations
ending with (−1)n−1i into two sets and obtain alternating permutations beginning
with (−1)ni by somehow taking the ”difference” of these two sets. On the alternating
permutations, the construction can be as follows: assume that n is even. If p > 1,
the set S(n,p) has a natural involution I without fixed points: change the sign of ±1
in permutations.
Then define two subsets of S(n,p) by
S′(n,p) = {σ ∈ S(n,p)|σn−1< −p or − 1 ∈ σ},
S′′(n,p) = {σ ∈ S(n,p)|σn−1> −p and − 1 ?∈ σ}.
(130)
Page 24
24M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON
Then S′(n,p)∪S′′(n,p) is S(n,p) and S′(n,p)/I(S′′(n,p)) is S(n,−p) up to the sign
of the first letter of each element. In the special case p = −1, both properties still
hold, even without the involution since S′′is empty.
Let us illustrate this with the example n = 4 and p = 2. We then have:
S(4,2) = {13¯42,¯13¯42,14¯32,¯14¯32,¯31¯42,¯3¯1¯42,3412,
34¯12,¯3412,¯34¯12,¯41¯32,¯4¯1¯32,¯4312,¯43¯12},
(131)
S′(4,2) = {13¯42,¯13¯42,14¯32,¯14¯32,¯31¯42,¯3¯1¯42,
34¯12,¯34¯12,¯41¯32,¯4¯1¯32,¯43¯12},
(132)
(133)S′′(4,2) = {3412,¯3412,¯4312},
so that S′(4,2) ∪ S′′(4,2) = S(4,2) and
S′(4,2)/I(S′′(4,2)) = {13¯42,¯13¯42,14¯32,¯14¯32,¯31¯42,¯3¯1¯42,¯41¯32,¯4¯1¯32},
(134)
which is S(4,−2) up to the sign of the first letter of each permutation.
8. Extension to more than two colors
8.1. Augmented alternating permutations. Let
(135)
A = A(0)⊔ A(1)⊔ ··· ⊔ A(r−1)= A × C,
with C = {0,...,r − 1} be an r-colored alphabet. We assume that A(i)= A× {i} is
linearly ordered and that
(136)A(0)> A(1)> ··· > A(r−1).
Colored words can be represented by pairs w = (w,u) where w ∈ Anand u ∈ Cn.
We define r-colored alternating permutations σ = (σ,u) by the condition
(137)
σ1< σ2> σ3< ...
hence, as permutations of shape 2nor 2n1 as words over A. Let A(r)
such permutations. Their noncommutative generating series in FQSym(r)is then
n be the set of
(138)
?
n≥0
?
σ∈A(r)
n
Gσ(A) = X(A) =
?
m≥0
R2m(A) + R2m1(A).
Thus, if we send A(i)to qiE, the exponential generating function of the polynomials
(139)αn(q0,...,qr−1) =
?
σ∈A(r)
n
n?
i=1
qui
is
(140)α(z;q0,...,qr−1) =sin(q0+ ··· + qr−1)z + 1
cos(q0+ ··· + qr−1)z
Iterating the previous constructions, we can define generalized snakes as colored
alternating permutations such that σ1∈ A(0), or, more generally σ1∈ A(0)⊔···⊔A(i).
.
Page 25
THE ALGEBRAIC COMBINATORICS OF SNAKES25
Setting Bi = A(0)⊔ ··· ⊔ A(i)and Bi = A(i+1)⊔ ··· ⊔ A(r−1), we have for these
permutations the noncommutative generating series
(141)
Y(Bi,Bi) = (cos + sin)(Bi) sec(Bi|Bi)
which under the previous specialization yields the exponential generating series
(142)yi(t;q0,...,qr−1) =cos(q0+ ··· + qi)t + sin(qi+1+ ··· + qr−1)t
cos(q0+ ··· + qr−1)t
.
Setting all the qiequal to 1, we recover sequences A007286 and A007289 for r = 3
and sequence A006873 for r = 4 of [20], counting what the authors of [7] called
augmented alternating permutations.
8.2. Triangles of alternating permutations with r colors. One can now count
alternating permutations of shapes 2kand 2k1 by their last value. With r = 3, the
following tables present the result:
(143)
n
1
2
3
4
5
1
1
0
9
0
23451
1
1
8
26
352
23451
1
2
5
2345
1
9
9
2
7
3
3
51
159
8
18
396
5
41
292
0
54
108
26
378
34
326
46
251
46
205
54
54 405 405 3522050
Proposition 8.1. The table counting alternating permutations with r colors by their
last value is obtained by the following algorithm: first separate the picture by the
column p = 0 and then compute r triangles. Put 1 at the top of each triangle and
compute the rest as follows: fill the second row of all triangles as the sum of the
elements of the first row strictly to their left. Then fill the third row of all triangles
as the sum of the elements of the previous row to their right. Compute all rows
successively by reading from left to right and right to left alternatively.
Proof – Same argument as for Propositions 7.1 and 7.2.
Applying the same rules to the construction of three triangles but with only one 1
at the top of one triangle gives the following three tables. Note that this amounts to
split the alternating permutations, first by the number of bars of their first element,
then, inside the triangle, by their last value.
(144)
n
1
2
3
4
5
1
1
0
5
0
23451
0
1
4
14
176
23451
0
1
2
23
100
2345
1
5
5
1
3
18
162
1
1
25
77
4
10
200
2
21
144
0
26
52
14
190
23
123
26
26205205176 1000
Page 26
26 M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON
(145)
n
1
2
3
4
5
1
0
0
3
0
23451
1
0
3
9
23451
0
1
2
17
76
2345
0
3
3
1
3
12
120
1
1
19
59
3
6
2
15
108
0
20
40
9 17
93
20
20 147147 144138 129129760
(146)
n
1
2
3
4
5
1
0
0
1
0
23451
0
0
1
3
23451
1
0
1
6
29
2345
0
1
1
53
0
1
4
44
1
1
7
1
2
52
1
5
40
0
8
16
3
50
6
35
8
8 5347 4729230
Arnol’d [2] has found remarkable arithmetical properties of the Euler-Bernoulli
triangles. The study of the properties of these new triangles remains to be done.
9. Snakes of type D
9.1. The triangle of type D snakes. The Springer numbers of type D (Sequence
A007836 of [20]) are given by exactly the same process as for type B Springer num-
bers, but starting with 0 at the top of the left triangle and 1 at the top of the right
triangle. Since all operations computing the rows of the triangles are linear in the first
entries, we have in particular that the sum of the number of snakes of type Dnand
the number of snakes on type Bnis equal to the number of alternating permutations
of type B.
We have even more information related to the triangles: both B and D triangles
can be computed by taking the difference between the triangle of Equation (128) and
of Equation (129). We obtain
(147)
n/p −6 −5 −4 −3 −2
1
2
3
4
5 23
6
023
−1
0
0
1
3
17
105
1
1
0
1
3
23456
0
1
2
20
88
1
1
4
14
1
1
0
5
10
0
23
46
5
5 22
68
17
105 122 136 146 151 151
0
9.1.1. Snakes of type D. From our other sets having the same cardinality as type
B snakes, we can deduce combinatorial objects having same cardinality as type D
snakes by taking the complement in the alternating permutations of type B.
Since the generating series of type B alternating permutations is
(148)
X =
?
m≥0
(R#
(2m)+ R#
(2m1)).
Page 27
THE ALGEBRAIC COMBINATORICS OF SNAKES 27
and the generating series of type B snakes defined in Section 5.2.2 is
(149)
Y = 1 +
?
n≥0
(˜R02n1+˜R02n+1),
we easily get one definition of the generating series type D snakes:
(150)
D = X − Y =
?
n≥0
(˜R2n1+˜R2n+1).
In other words, our first sort of type D snakes corresponds to permutations of
ribbon shape 2n1 or 2nwhose first letter is positive.
Here are these elements for n ≤ 4:
(151) 12,
12¯3,132,231,13¯2,23¯1
¯12¯34,12¯43,12¯4¯3,1324,13¯24,13¯42,13¯4¯2,1423,14¯23,14¯32,14¯3¯2,2314,
23¯14,23¯41,23¯4¯1,2413,24¯13,24¯31,24¯3¯1,3412,34¯12,34¯21,34¯2¯1.
(152)
Since both alternating permutations and snakes of type B can be interpreted as
solutions of a differential equation and a fixed point solution involving the same
bilinear form, one then concludes that these snakes of type D satisfy
(153)dD = 1 + DX,
and
(154)
D = G1+ B(D,X).
The iteration of (154) brings up a solution close to (68) and (87):
(155)
D =
?
T∈CBT
BT(G0,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. Note that in this
case, the first leaf needs to have label b.
9.1.2. The usual snakes of type D. The previous type D snakes are not satisfactory
since, even if they fit into the desired triangle, they do not belong to Dn. The
classical snakes of type D of Arnol’d belong to Dn and are easily defined: select
among permutations of ribbon shape 12nand 12n1 the elements with an even number
of negative signs and such that σ1+ σ2< 0.
One then gets the following elements for n ≤ 4:
(156)1
¯1¯2,
¯1¯23,1¯3¯2,¯1¯32,2¯3¯1,¯2¯31,
1¯23¯4,1¯24¯3,¯1¯243,1¯32¯4,¯1¯3¯2¯4,1¯34¯2,¯1¯342,1¯42¯3,¯1¯4¯2¯3,1¯43¯2,¯1¯432,2¯31¯4,
¯2¯3¯1¯4,2¯34¯1,¯2¯341,2¯41¯3,¯2¯4¯1¯3,2¯43¯1,¯2¯431,3¯41¯2,¯3¯4¯1¯2,3¯42¯1,¯3¯421.
(157)
It is easy to go from these last elements to the other type D snakes: change all
values into their opposite and then change the first element s to −|s|. Conversely,
change (resp. do not change) the sign of the first element depending whether it is
not (resp. it is) in Dnand then change all values into their opposite.
Page 28
28M. JOSUAT-VERG`ES, J.-C. NOVELLI, AND J.-Y. THIBON
10. Tables
Here follow the tables of the maps A ?→ tA from BSym to Sym.
All tables represent in columns the image of ribbons indexed by type-B compo-
sitions, where the first half begins with a 0 and the other half does not. So, with
N = 3, compositions are in the following order:
(158)[0,3], [0,2,1], [0,1,2], [0,1,1,1], [3], [2,1], [1,2], [1,1,1].
Note that the zero entries have been represented by dots to enhance readability.
(159)
?
−t2
t2+ t
t2
.
1
.
−1
t + 1
?
t3
−t3
.
t3−t
t2+t
−t3
t3+t2
.
.
t3
.
.
.
1
.
.
.
−1
t+1
.
.
−1
.
1−t2
t2+t
1
−t3−t2
t−t3
t3+t2
−t−1
t2−1
t+1
Figure 1. Matrices of˜RI(A,tA) on the S basis for n = 2, 3.
(160)
?
t2
t
−t2
t2
−1 1
1t
?
t3
t2
−t
t
−t3
t3
t
t2
−t3
−t2
t3
t2
t3
1−1 −1
−t
1
t
1
t−t3
−t3
t3
−1
−1
1
1
t2
t
−t2
t2
Figure 2. Matrices of˜RI(A,tA) on the Λ basis for n = 2, 3.
(161)
?
tt2
.
1
.
t
t2+ tt + 1
?
t
.
t2
t2
t3
.
.
.
1
.
.
.
ttt2
.t2+ t
t3+ t2
t2+ t
t3+ t2
.
.
t + 1 t2+ t
.
.
t2+ t
t3+ t2
t + 1
t2+ t
t2+ t
t + 1
Figure 3. Matrices of˜RI(A,tA) on the R basis for n = 2, 3.
References
[1] D. Andr´ e, Sur les permutations altern´ ees, J. Math. Pures Appl. 7, (1881), 167–184.
[2] V. I. Arnol’d, The calculus of snakes and the combinatorics of Bernoulli, Euler, and Springer
numbers for Coxeter groups, Russian Math. Surveys 47 (1992), 1–51.
[3] A. Bj¨ orner, Orderings of Coxeter groups, Combinatorics and algebra (1983), Amer. Math.
Soc., Providence, RI, 1984, 175–195.
[4] C.-O. Chow, Noncommutative symmetric functions of type B, Thesis, MIT, 2001.
[5] G. Duchamp, F. Hivert, J.-C. Novelli, and J.-Y. Thibon, Noncommutative Symmet-
ric Functions VII: Free Quasi-Symmetric Functions Revisited, Ann. Comb., to appear. Also
preprint arXiv:0809.4479.
Page 29
THE ALGEBRAIC COMBINATORICS OF SNAKES29
[6] G. Duchamp, F. Hivert, and J.-Y. Thibon, Noncommutative symmetric functions VI: free
quasi-symmetric functions and related algebras, Internat. J. Alg. Comput. 12 (2002), 671–717.
[7] R. Ehrenborg and M. Readdy, Sheffer posets and r-signed permutations, Ann. Sci. Math.
Qu´ ebec 19 (1995), 173–196.
[8] I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, and J.-Y. Thibon,
Noncommutative symmetric functions, Adv. in Math. 112 (1995), 218–348.
[9] F. Hivert, J.-C. Novelli, and J.-Y. Thibon, The algebra of binary search trees, Theoret.
Comput. Sci. 339 (2005), 129–165.
[10] F. Hivert, J.-C. Novelli and J.-Y. Thibon, Trees, functional equations and combinatorial
Hopf algebras, Europ. J. Comb. 29 (1) (2008), 1682–1695.
[11] M. Hoffman, Derivative polynomials, Euler polynomials, and associated integer sequences,
Elec. J. Comb. 6 (1999), #R21.
[12] M. Josuat-Verg` es, Enumeration of snakes and cycle-alternating permutations, preprint
arXiv:1011.0929.
[13] D. Krob, B. Leclerc, and J.-Y. Thibon, Noncommutative symmetric functions II: Trans-
formations of alphabets, Internal J. Alg. Comput. 7 (1997), 181–264.
[14] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University
Press, 1995.
[15] R. Mantaci and C. Reutenauer, A generalization of Solomon’s descent algebra for hyper-
octahedral groups and wreath products, Comm. Algebra 23 (1995), 27–56.
[16] J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions and descent algebras for
wreath products, and noncommutative multi-symmetric functions, Disc. Math. 30 (2010), 3584–
3606.
[17] J.-C. Novelli and J.-Y. Thibon, Superization and (q,t)-specialization in combinatorial Hopf
algebras, Elec. J. Comb., R21, 16 (2) (2009).
[18] S. Poirier, Cycle type and descent set in wreath products, Disc. Math., 180 (1998), 315–343.
[19] C. Reutenauer, Free Lie algebras, Oxford University Press, 1993.
[20] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (electronic),
http://www.research.att.com/∼njas/sequences/
[21] L. Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra, 41 (1976),
255–268.
[22] T. A. Springer, Remarks on a combinatorial problem, Nieuw Arch. Wisk. 19 (1971), 30–36.
[23] R. P. Stanley, A survey of alternating permutations, to appear in Contemporary Mathematics
(2010).
Fak¨ ult¨ at f¨ ur Mathematik, Universit¨ at Wien, Garnisongasse 3, 1090 Wien, Austria
Institut Gaspard Monge, Universit´ e Paris-Est Marne-la-Vall´ ee, 5 Boulevard Des-
cartes, Champs-sur-Marne, 77454 Marne-la-Vall´ ee cedex 2, France
E-mail address, Matthieu Josuat-Verg` es: Matthieu.Josuat-Verges@univie.ac.at
E-mail address, Jean-Christophe Novelli: novelli@univ-mlv.fr
E-mail address, Jean-Yves Thibon: jyt@univ-mlv.fr
Download full-text