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Journal of Combinatorial Theory, Series A

www.elsevier.com/locate/jcta

Journal o[

Combinatorial

Theory

The algebraic combinatorics of snakes

Matthieu josuat-Vergès3'1, Jean-Christophe Novellib, jean-Yves Thibonb

a Fakültät für Mathematik, Universität Wien, Garnisongasse 3, 1090 Wien, Austria

b Institut Gaspard Monge, Université Paris-Est Marne-la-Vallée, 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France

ARTICLE INFO ABSTRACT

Article history:

Received 8 December 2011

Available online xxxx

Keywords:

Noncommutative symmetric functions

Euler numbers

Snakes

Snakes are analogues of alternating permutations defined for any Coxeter 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.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

Snakes, a term coined by Arnol'd [2], are generalizations of alternating permutations. 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 objects counted by those coefficients. For example, in the expansions

z2n+1 z2n

tanz = £E2n+1 (2n+Ty. and secz = DE2n(2ny., (1)

the coefficients En are nonnegative integers.

It was found in 1881 by D. André [1] that En was the number of alternating permutations in the symmetric group Sn.

E-mail addresses: Matthieu.Josuat-Verges@univie.ac.at (M. Josuat-Verges), novelli@univ-mlv.fr (J.-C. Novelli), jyt@univ-mlv.fr (J.-Y. Thibon).

1 M. Josuat-Verges was supported by the Austrian Science Foundation (FWF) via the grant Y463.

0097-3165/$ - see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/jjcta.2012.05.002

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 labeled by all permutations, and such that the map 0 sending any a e Sn to ni is a homomorphism. In this algebra, the formal series

C = (-1)n id2n and S = (-1)n id2n+i (2)

n>0 n>0

(alternating sums of even and odd identity permutations) are respectively mapped to cos z and sin z by 0. The series C is clearly invertible, and one can see by a direct calculation that C-1 + C-1 • S is the sum of all alternating permutations [7].

Such a proof is not only illuminating, it says much more than the original statement. For example, one can now replace 0 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 alternating permutations to arbitrary Coxeter groups. Such generalizations were first introduced by Springer [18]. 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 as the Malvenuto-Reutenauer algebra) which is based on permutations, and its subalgebra Sym (noncommutative symmetric functions), 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 [15].

After reviewing the necessary background and the above mentioned proof of the result of André, 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 e A* is the permutation obtained by iteratively scanning w from left to right, and labeling 1, 2,... the occurrences of its smallest letter, then numbering the occurrences of the next one, and so on. Alternatively, a = std(w)-1 can be characterized as the unique permutation of minimal length such that w a is a nondecreasing word. For example, std(bbacab) = 341625.

We can now define polynomials

Ga(A) := J2 w. (3)

std(w)=a

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.

The multiplication rule is, for a e Sk and f$ e Si,

Ga Gp = J2 Gy, (4)

Y eaxfi

where a * /) is the set of permutations y e &k+i such that y = u • v with std(u) = a and std(v) = /). This is the convolution of permutations (see [16]). Note that the number of terms in this product depends only on k and l, and is equal to the binomial coefficient (k+l). Hence, the map

0 : o e Sn (5)

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

2.2. Noncommutative symmetric functions

The algebra Sym(A) of noncommutative symmetric functions over A is the subalgebra of FQSym generated by the identity permutations [7,5]

Sn(A) := Gi2...n(A) = aii% ---an. (6)

These polynomials are obviously algebraically independent, so that the products

S1 := Si, Si2... Sir (7)

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 = w, w 2 ...wn e An is 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 S I ( A) is the sum of all words which may have a descents only at places from the set

Des(I) = {i1, i1 + i2,..., i1 + ••• + ir-1}, (8)

called the descent set of I. Another important basis is

Ri (A) = J2 w = J2 Ga, (9)

Des(w)=Des(I) Des(o)=Des(I)

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),

Ri (A)Rj (A) = Rij (A) + RM (A) (10)

with IJ = (i1,...,ir, j1,..., js) and I > J = (i1,...,ir + ju 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. This allows to define noncommutative symmetric functions of A + B, and

Sn(A + B) = J2 Si(A)Sn-i (B) (So = 1). (11)

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

£ Sn (— A) = ( £ Sn (A)] (12)

and more generally, for a difference A — B,

£ Sn (A — B) = { £ Sk (B)\ £ S|(A) (13)

(note the reversed order, see [12] for detailed explanations). 2.4. Noncommutative trigonometry

2.4.1. Andre's theorem

One can now define "noncommutative trigonometric functions" by

cos(A) = J2 (—1)nS 2n (A) and sin(A) = £ (—1)nS 2n+1(A). (14)

n>0 n>0

The image by 0 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

sec := cos-1 = ^ R(2n) and tan := cos-1sin = ^ R(2n1) (15)

n>0 n>0

which implies André's theorem: the coefficient of nn in sec(z) + tan(z) is the number of alternating permutations of &n (if we choose to define alternating permutations as those of shape (2n) and (2n1)). In FQSym,

sec + tan = Ga. (16)

a alternating

2.4.2. Differential equations

If d is the derivation of Sym such that d Sn = Sn—1, then

X = tan = ^ R(2mj) and Y = sec = ^ R(2m) (17)

m>0 m>0

satisfy the differential equations

dX = 1 + X2, d Y = XY. (18)

These equations can be lifted to FQSym, actually to its subalgebra PBT, the Loday-Ronco algebra of planar binary trees (see [8] for details). Solving them in this algebra provides yet another combinatorial proof of André's result.

Let us sketch it for the tangent. The original proof of André relied upon the differential equation

dX = 1 + x2 (19)

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

x(t) = t + j x(s)2 ds (20)

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 e R[[t]],

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

where a e 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, Eq. (21) has a unique solution

x = a + B(a, a) + B (B(a, a), a) + B (a, B(a, a)) + •••= ^ BT(a) (22)

T eCBT

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)) +----

BB B / \ / \

= a + / \ + B a + a B +•••. a a / \ / \

a a a a

It is proved in [9] that if one defines

9Ga := Gff, (23)

where a' is obtained from a by erasing its maximal letter n, then 9 is a derivation of FQSym. Its restriction to Sym coincides obviously with the previous definition. For a e &k, P e &e, and n = k +1, set

B(Ga, Gp) = J2 Gy. (24)

Y=u(n+1)v std(u)=a,std(v)=p

Clearly,

9 B(Ga, Gp) = Ga Gp, (25)

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

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

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:

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

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 [10] of two sequences of polynomials Pn, Qn such that

-—(tanz) = Pn(tanz) and -— (secz) = Qn(tanz) secz. (28)

dzn dzn

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

dn (X) = Pn (X) and dn(Y) = Qn(X)Y. (29)

Hoffman [10] gives the exponential generating functions tn sin t + u cos t

P(u, t) = £ Pniu)^ = Sintt + UC0S t and Q (u, t) = £ Qn(u) ^ = t 1 , t.

n\ cos t — u sin t n! cos t — u sin t

n! cos t — u sin t

n>0 n>0

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

P(X, t) = v Pn(X)- = etdX. (31)

Since d is a derivation, etd is an automorphism of Sym. It acts on the generators Sn by n tk

etd Sn (A) = Sn-k (A) - = Sn (A + te) (32)

where tE is the "virtual alphabet" such that Sn (tE) = nn. Hence,

P(X, t) = tan(A + te) = cos(A + te)-1sin(A + te)

= (cos t - X sin t)—1 (sin t + X cos t) (33)

as expected. Similarly,

Qn(X, t) = J2 Qn(X)- = (etdY)Y-1 (34)

= cos(A + te)-1cos(A) = (cos t - X sin t)-1. (35)

3. The uniform definition of snakes for Coxeter groups

Before introducing the relevant generalizations of Sym and FQSym, we shall comment on the definitions of snakes and alternating permutations for general Coxeter groups.

It is apparent in Springer's article [18] 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 definition. The goal of this section is to give some precisions and to simplify the proof of Springer's result in [18].

Let (W, S) be an irreducible Coxeter system. Recall that s e S is a descent of w e W if ¿(ws) < ¿(w) where I is the length function. When J c S, we denote by Dj the descent class defined as {w e W: ¿(ws) < ¿(w) -o- s e J}. Following Arnol'd, let us consider the following definition.

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

K(W) := max(#D,). (36)

J CS J

The aim of [18] is to give a precise description of sets J realizing this maximum. The result is as follows:

Theorem 3.2. (See Springer [18].) Let J c S. Then, K(W) = #D j if and only if J and S\J are independent subsets of the Coxeter graph S (i.e., they contain no two adjacent vertices). In particular, there are two such subsets J which are complementary to each other.

We are thus led to the following definition:

Definition 3.3. Let (W, S) be a Coxeter group, and J be as in Theorem 3.2. 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

a1 < a2 > a3 < ••• or a1 > a2 < a3 > •••. (37)

It is natural to endow a descent class with the restriction of the weak order, and this defines what we can call the snakeposet 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. (See Springer [18].) Let J c S, and assume that there is an edge e of the Coxeter graph whose endpoints are both in J or both not in J.LetS = S1 U S2 be the connected components obtained after removing e. Let J' = (S1 n J) U (S2 n (S\ J)). Then, #Dj < #Dji.

Using the above lemma, we see that if J or its complement 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 n = (as)seS be a set of simple roots. There is a bijection w ^ i(w) between W and the set of Weyl chambers, so that s e S is a descent of w e W if and only if i(w) lies in the half-space {v e Rn: (v, as} > 0}. For any J c S, let

CJ = {v e rn: (v,as} > 0 if s e J, and (v,as} < 0 ifs e J}. (38)

It is the closure of the union of Weyl chambers i(w) where w e Dj. Now, let e, S1, S2 and J' be as in the lemma, and let x e S1 and y e S2 be the endpoints of e. Note that either x or y is in J' but not both. Let a be the orthogonal symmetry through the linear span of {aj: j e S1}. We claim that a(Cj) C Cji, and this implies #Dj < #Dji since Cj contains strictly fewer Weyl chambers than Cji.

So it remains to show that a(Cj) C Cji. It is convenient to use the notion of dual cone, which is defined for any closed convex cone C c Rn as

C* := {v e rn: (v, w} > 0 for any w e C}. (39)

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 a(C*l) C C*. Since (C1 n C2)* = C* + C* and since the dual of the half-space {v e Rn: (v, w} > 0} is the half-line R+ w, the dual of CJi is

C* = usas: us > 0 if s e Jl, and us < 0 if s / J4, (40)

I seS '

and the same holds for C**. A description of a(C*l) is obtained easily by linearity since a(as) = as if s e S1, a(as) = -as if s e S2\y, and a(ay) = —ay + 2(ax,ay}(ax,ax}~1ax. Indeed, let v = Y>seSusas e C*i, we have:

a(v) =Y1 usas — usas — uy ay + 2uy (ax, ay }(ax,ax} 1ax. seS1 seS2\y

Since ux and uy have different signs and (ax,ay) < 0, we obtain a(v) e Cj. We have thus proved a(C j,) c C j .To show the strict inclusion, note that either ay or -a y is in C j. But none of these two elements is in a(C), because in the above formula for a(v), if uy = 0 there is a nonzero term in ax as well. This completes the proof.

4. Signed permutations and combinatorial Hopf algebras

Whereas the constructions of the Hopf algebras Sym, PBT, and FQSym appearing 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 into each 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:

Sym(A |A) BSym Sym(2) FQSym(A |A) FQSym(2). (42)

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 [13] 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 (Sn) (n > 1). We regard the two copies of Sym as noncommutative symmetric functions on two auxiliary alphabets: Sn = Sn(A) and Sn = Sn(A). We denote by F ^ F the involutive automorphism2 which exchanges Sn and Sn. And we denote the generators of Sym(2) by S(k,e) where e = {±1}, so that S(k,1) = Sk and S(k,-1) = Sk.

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# where, for any F e Sym(A),

F# = F(A|A) = F(A - qA)|,=_1, (43)

is the so-called the supersymmetric version, or superization, of F [15].

The expansion of an element of Sym(A|A) on the S-basis of Sym(2) is done by means of a generating series. Indeed,

a# = Mat = (£ ^V £ S^ (44)

yk>0 ' ^m>0

where Ak = Y1 ¡<Fk(-1)l(>) kS1, as follows from X1 = (a-1) 1 (see [12]). For example,

S# = S1 + S1, S# = S2 + S11 - S2 + S11, (45)

S# = S3 + S12 + S111 - S21 + S111 - S21 - S12 + S3. (46)

2 This differs from the convention used in some references.

4.3. Noncommutative symmetric functions of type B

The third generalization of Sym is not an algebra but only a coalgebra. However, it is the generalization encoding the descent algebra of the Coxeter group Bn: its graded dimension is 2n and, as we shall see later in this section, 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 <a1 grouplike. This is a coalgebra, but not an algebra.

We embed BSym as a sub-coalgebra and right sub-Sym-module of MR as follows. The basis element S1 of BSym, where I = (i0, i1,..., ir) is a B-composition (that is, i0 may be 0), can be embedded as

S1 = Si0 (A) Sil i2...ir (A\A). (47)

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]):

S1 = £ R J, (48)

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 = R 0n + R n.

I # ~ i

Note also that, thanks to that definition, S1 = S1 and, thanks to the transitions between all bases, R# = R 0I + RI. (49)

4.4. Type B permutations and descents in Bn

The hyperoctahedral group Bn is the group of signed permutations. A signed permutation can be denoted by w = (a ,e) where a is an ordinary permutation and e e{±1}n, such that w (i) = ei a(i). If we set w(0) = 0, then, i e[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 generalizations 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 [14]. Let us set

A(0) = A = {ai < a2 < ••• < an < ■■■}, (50)

A(1) = a = {- <an < ••• <a2 <a 1}, (51)

and order A = AU A by a < a¡ for all i, j. Let us also denote by std the standardization of signed words with respect to this order.

We shall also need the signed standardization Std, defined as follows. Represent a signed word w e An by a pair (w, e), where w e An is the underlying unsigned word, and e e {±1}n is the vector of signs. Then Std(w, e) = (std(w), e).

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

Ga,u := Y^ w e Z(A). (52)

weAn;Std(w)=(a,u)

Let (au') and (a", u") be signed permutations. Then (see [14,15])

Ga',u' Ga",u" = ^^ Ga,u'u"■ (53)

a ea ;*a

We denote by m(e) the number of entries —1 in e.

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 in the same way 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

Ga,e * GM = G(fi,n)o(a,e) = Gfioa,(na)-e, (54)

with na = (tfa(1), ■■■, tfa(n)) and e • n = (e1 Vn).

We can now embed FQSym into FQSym(2) by

Ga ^ G(a1n), (55)

which allows us to define

G# := Ga(A|A) = Ga * a#, (56)

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

Theorem 4.1. (See [15], Theorem 3.1.) The expansion of Ga (A|A) on the basis GT e is

Ga(A| A) = J2 Gr,e■ (57)

std(T,e)=a

4.6.1. EmbeddingSym(A|A) and BSym into FQSym(2)

One can embed BSym into FQSym(2) as one embeds Sym into FQSym (see [4]) by

R, = J2 Gx, (58)

Bdes(n)=l

where I is any B-composition.

Given Eq. (49) relating R# and the RI, one has

R1} = £ Gn, (59)

Des(n)=I

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 n e Bn is alternating if n1 < n2 > n3 < ••• (shape 2m or 2m1).

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

1,1, 12,12, 21, 21, (60)

123,123, 132,132,132,132, 213, 213, 231, 231, 231, 231, 312, 312, 321, 321, (61)

1234,1234,1243,1243,1243,1243, 1324,1324,1324,1324,1342,1342,

1342,1342, 1423,1423,1423,1423,1432,1432,1432,1432, 2134, 2134,

2143,2143, 2143,2143, 2314,2314,2314,2314,2341,2341, 2341,2341,

2413, 2413, 2413, 2413, 2431, 2431, 2431, 2431, 3124, 3124, 3142,3142,

3142, 3142, 3214, 3214, 3241, 3241, 3241, 3241, 3412, 3412, 3412, 3412,

3421,3421, 3421,3421, 4123,4123,4132,4132, 4132,413 2, 4213,4213,

4231,4231, 4231,4231, 4312,4312,4321, 4321. (62)

Hence, n is alternating iff n is a f$-snake in the sense of [2]. Hence, the sum in FQSym(2) of all Gn labeled by alternating signed permutations is, as already proved in [4]

X = (X + Y )# = sec# + tan# = sec#(1 + sin#) = ^ (R #m) + R #m1)). (63)

5.1.2. Quasi-differential equations

Let d be the linear map acting on Gn as follows:

dGn = | Guv if ^ = UnV, (64)

n |GUV if n = un v. K '

This map lifts to FQSym(2) the derivation 9 of (23), although it is not itself a derivation. We then have

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

dX = 1 + X2. (65)

Proof. Indeed, let us compute what happens when applying d to R#2m), the property being the same with dR#2m1). Let us fix a permutation a of shape (2m). If n appears in a, let us write a = unv. Then

dGa appears in the product Gstd(u)Gstd(V) and u and v are of respective shapes (2n 1) and (2m-n-1). If n appears in a, let us write again a = un v. Then dGa appears 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 (2n 1) 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.

This is not enough to characterize X but we have the analog of fixed point Eq. (26)

X = 1 + G1 + B(X, X), (66)

Hy=u(n+1)v, Std(u)=a, Std(v)=ß GY if \a I is odd'

B(Ga, Gß) H -Y=u(n_+1)v' Std(u)=a Std(v)=ß GY if ¡a ' (67)

[ Z^y=u(n+1)V, Std(u)=a, Std(v)=ß GY 11 Ia 1 ls even-

Indeed, applying d to the fixed point equation brings back Eq. (65) and It ls clear from the definition of B that all terms in B(Ga, Gß) are alternating signed permutations.

Solving this equation by iterations gives back the results of [11, Section 4]. Indeed, the iteration of Eq. (66) yields the solution

X = J^ Bt(Go = 1, G1), (68)

T eCBT

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 Pn defined in [11, 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 specialization A = tA, X goes to the series

X(t; A) = £( J2 tm(K)JRI(A) (69)

I n alternating, C(std(n))=I where m(n) is the number of negative letters of n. If we further set A = zE, we obtain 1 + sin((1 + t)z)

x(t, z) =--(70)

cos((1 + t)z)

which reduces to

1 + sin2z cos z + sin z

x(1, z) =-— =-— (71)

cos2z cos z — sin z

for t = 1, thus giving a t-analogue different from the one of [11].

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

J2zn £ tm(n) = sec((1 +1 )z) + tan((1 + t)z)■ (72)

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

J2zn £ tm(n) = sec((1 + t)z) + tan((1 + t)z)■ (73)

n n s.t. |n| is alternating in An

It is thus in order to give a bijection proving the equality of the generating functions. Let n be an alternating signed permutation. We can associate with n the pair (std(n),e) where e is the sign vector such that ei = 1 if n—1(i) > 0 and ei = —1 otherwise. The image of {1,..., n} by n is {eii: 1 < i < n}. Since n can be recovered from std(n) and the image of {1,...,n}, this map is a bijection between signed alternating permutations and pairs (a, e) where a is alternating and e is a sign vector. Then, with such a pair (a,e), one can associate a signed permutation t such that |t| is alternating simply by taking Ti = aiei. The composition n ^ (a,e) ^ t gives the desired bijection.

For example, here follow the 16 permutations obtained by applying the bijection to the 16 alternating permutations of size 3 (see Eq. (61)):

231, 231, 132, 231,132, 231, 231, 231, 231, 231,132,132,132, 132, 132, 132■

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 cos z + sin z

y(1, z) =-:— =----(75)

cos z - sin z cos2z

after the same sequence of specializations. A natural choice, simple enough and given by a series in BSym, is to set

Y = (cos + sin) • sec# = ( £(- 1)k(S2k + S2k+i)) • £ R#. (76) Now, Y lives in BSym and expands in the ribbon basis R of BSym as

Y = ( £(-1)k (R 2k + R 2k+l)) £ R#n

= ^2 (R#n + R 12" + R32") + £ (-1)k (R2k2n + R2k+22"-1 + R2k+12n + R2k+32"-1 ) n>0 k>1;n>0

= £(R#n + R i2" + RR 32n) - £(R 2"+i + R 32n) (77)

which simplifies into

Y = 1 +£ (R 12" + RR 02"+1 ). (78)

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

{0 > n1 < n2 > ••• if n is even, (7„)

0 < n1 > n2 < ••• if n is odd.

Thus, for n odd, n is exactly a Bn-snake in the sense of [2], and for n even, n 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. Here are these modified snakes for n < 4:

1, 12, 21, 21, (80)

123,132,132, 213, 213, 231, 231, 312, 312, 321, 321, (81)

1234,1243,1243,1324,1324,1342,1342,1423,1423,1432,1432, 2134, 2134,2143,2143, 2143,2143, 2314,2314,2341,2341, 2413,2413,2431, 2431, 3124,3124,3142,3142, 3142,3142, 3214,3214,3241,3241, 3 241, 3241, 3412,3412,3421,3421, 4123,4123,4132,4132, 413 2,413 2, 4213, 4213,4231,4231, 4231,4231, 4312,4312,4321,4321. (82)

5.2.2. Snakes as particular alternating permutations

Note that in the previous definition, snakes are not alternating permutations for odd n. So, instead, let us consider the generating series

Y = cos • sec# + sin • sec#, (83)

where f ^ f is the involution of FQSym(2) inverting the signs of permutations. Expanding Y in the R basis, one gets

Y = 1 + £ (R 02n 1 + R 02"+1). (84)

As for type B alternating permutations (see Eq. (65)), the series Y satisfies a differential equation with the same linear map d as before (see Eq. (64)):

dY = YX. (85)

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

Y = 1 + B (Y, X). (86) The iteration of (86) brings up a solution close to (68):

Y = J^ Bt(G0 = 1, G1), (87)

T eCBT

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 [11, Section 4] since one can again interpret the G0 leaves 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

cos • sec# + sin • sec#, (88)

and can be written as

Y = 1 (R 12" + R 12n 1) (89)

on the ribbon basis.

The lift of the differential equation for y(1) is given by a map 8 similar to d, with Sunv = uuv and Sufiv = uv. Then

8Y = YX (90)

and we have a fixed point equation

Y = 1 + B (Y, X) (91) for an appropriate bilinear map B.

6. Another combinatorial model

6.1. An analogue of cos z — sin z in FQSym(2)

Definition 6.1. A signed permutation n e Bn is a valley-signed permutation if, for any i e [n], n(i) < 0 implies that

• either i > 2, and ni—1 > 0, and |ni—2| > ni—1 < |ni|,

• or i = 2, and 0 < n1 < |n2|.

We denote by Vn the set of valley-signed permutations of size n. Here are these signed permutations, up to n = 4:

1, 12,12,21, (92)

123,123,132,132, 213, 213, 231, 231, 312, 312, 321, (93)

1234,1234,1243,1243,1324,1324, 13 24,13 24, 1342,1342,1423,1423,

1423,1423, 1432,1432, 2134, 2134, 2143, 2143, 2314, 2314, 2314, 2314,

2341, 2341, 2413, 2413, 2413, 2413, 2431, 2431, 3124, 3124, 3142,3142,

3214, 3214, 3241,3241,3412,3412, 3412,3412, 3421, 3421, 4123,4123,

4132, 4132,4213,4213, 4231, 4231, 4312,4312, 4321. (94)

The terminology is explained by the following remark. Let a e Sn, and let us examine how to build a valley-signed permutation n so that ni = ±ai. It turns out that for each valley a(i — 1) > a(i) < a(i + 1), we can choose independently the sign of n(i + 1) (here 1 < i < n and 1 is a valley if

a(1)<a(2)).

The goal of this section is to obtain the noncommutative generating functions for the sets Vn.

Theorem 6.2. The following series

U = 1 —[G1 + gh2 — G123 — G1234 + G12345 + G1234 56 + ] (95)

is again a lift of cos z — sin z in FQSym(2). It satisfies

U—1 = ^^ Gn. (96)

n '^0 n eVn

Hence the n occurring in this expansion are in bijection with snakes of type B.

This result is a consequence of the next two propositions.

Definition 6.3. Let R2n c B2n be the set of signed permutations n of size 2n such that |n | is of shape 2n, and for any 1 < i < n, we have n(i) > 0 iff i is odd. Let

V = E H Gn = + G12+ G1324 + G1423 + -. (97)

Let R2n+1 c B2n+1 be the set of signed permutations n of size 2n + 1 such that |n| is of shape 12n, n1 > 0, and for any 2 < i < n, we have n(i) > 0 iff i is even. Let

W = £ J2 Gn = G1 + G213 + G312+ G21435 + • • . (98)

n '^0 n er2n+1

Note that Rn c Vn. Clearly, #Rn is the number of alternating permutations of &n since, given |n|, there is only one possible choice for the signs of each ni .So V and W respectively lift sec and tan in FQSym(2). Now, given that the product rule of the Ga does not affect the signs, a simple adaptation of the proof of the An case shows:

Proposition 6.4. We have

V-1 = 1 - G12 + G1234 - G123456 + •" , (99)

W V-1 = G1 - Gj23 + G!2345 - G1234567 + ••• . (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

V(1 - W)-1 = £ J^Gn. (101)

n'^0 n eVn

Proof. We can write V(1 - W)-1 = V + VW + VW2 +----, and expand everything in terms of the Gn,

using their product rule (see Eq. (53)). We obtain a sum of Gu1...uk where Std(u1) e R2* and Std(ui) e R2*+1 for any i > 2 (where * is any integer). The sum is a priori over lists (u1,...,uk) such that u = u1 ...uk e 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 e Vn. First, observe that in u = u1 ...uk, uk is 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. □

Z := G 1 — G 12 - G 123 + G ï234 + Gï23345 - G123456 + • ( 1 02)

Precisely, the n-th term in this expansion is the permutation a of Bn such that, |a| = id, a( 1 ) = 1, a(2) = 2 and, for 3 < i < n, a(i) < 0 iff n — i is even. The sign corresponds to the sign of zn in the expansion of cos z + sin z — 1 , so that it is a lift of cos z + sin z — 1 in FQSym(2).

Theorem 6.6. The series Z U—1 is a sum of Gn without multiplicities in the Hopf algebra FQSym(2). Hence the n occurring in this expansion are in bijection with snakes of type D (see Section 9).

Moreover, the series (1 + Z ) U—1 is also a sum of Gn without multiplicities in the Hopf algebra FQSym(2). Hence the n occurring in this expansion are in bijection with alternating permutations of type B.

Here are the elements of ZU—1, up to n = 4:

ï, 21, (103)

213, 213, 312, 312, 321, (104)

2134, 2134, 2143, 2143,3124,3124,3142,3142, 3214, 3214, 3241,3241, 4123,4123,413 2,4132,4213, 4213,4231,4231,4312, 4312,4 321

Proof. Let si be 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

E = s1s2 V-1, O = e1(WV-1). (106)

Since Si (ST) = Si (S)T if S contains only terms of size at least i, an easy rewriting shows that

(1 + Z)V = V + s1(w) + s1s2(1 - V). (107)

Then, one gets

(1 + Z)V(1 - W)-1 = U-1 + si(w + s2(1 - V))(1 - W)-1. (108)

So it only remains to prove that Q = s1(W + s2(1 - V))(1 - W)-1 has 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 + s2(1 - V) is an alternating sum of permutations of shapes 2n and 12n. Hence, any permutation of shape 2n can be associated with a permutation of shape 12n-1 by removing its first entry and standardizing the corresponding word. Now, all negative terms —GvGw come from -S2(V)(1 - W)-1 and are annihilated by the term GvG1Gw where v' is obtained from v by the removal and standardization process described before. □

6.2. Another proof of Theorem 6.2

Let Sgn be 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 Sgn with the free associative algebra on these symbols. We can now define the algebra of signed noncommutative symmetric functions as

sSym := 0Symn ® Sgn (109)

endowed with the product

(f ® u) ■ (g ® v) = fg® uv. (110)

It is naturally embedded in

FQSym(2) by

Ri ® u = Y, Ga,u. (111)

C (a)=I

With this at hand, writing U = P - Q with

P =J2 (-1)mR2m ® (11 )m and Q = J2 (-1)mR2m+1 ® 1(11 )m, (112)

m>0 m>0

it is clear that

P-1 =J2 R(2m) ® (11 )m = V and W =J2 R(12m) ® 1(11 )m = QV, (113)

m>0 m>0

so that U = (1 - W)V-1, and U-1 = V(1 - W)-1 can now be computed by observing that

(1 - W )-1 =Y Ri ® Pi (1,1) (114)

where p I is the sum of words in 1 , 11 obtained by associating with each tiling of the ribbon shape of I by tiles of shapes (1, 2m1,..., 1, 2mk) the word 1(11 )m1 ...1(11 )mk. For example, p12 = 111 + 111, 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

J^Rn((1 - q)A)xn = Ox((1 - q)A) = x-qX(A)ax(A). (115)

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:

Ri(A|tA) = J2(-1)l(i)+l(J}(1 - (-t)jr)(-t)EA(1,J)jkSJ (116)

where A(i, J) = {p| j1 +----+ jp e Des(i)}. Indeed, expanding X-qx(A) on the basis SJ, one finds

£ Rn((1 - q)A)xn = £xn( £ (-q)k £(-1)k-*(K)SKSm). (117)

n>0 n>0 k+m=n K Nk '

Each composition of a given n occurs twice in the sum, so that

Rn((1 - q)A) = £(-1)1-£(J)(qn-jr - qn)SJ. (118)

Hence, Eq. (116) is true for one-part compositions. The general case follows by induction from the product formula

RiRj = Rij + Ri >j. (119)

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 R i (A, tA) on the SJ (A) is

(-1)£(I')+ntnSn + £ (-1/CD+m)(1 - (-t)jr) (-t)ZA(i,J) jkSJ ifi0 = 0,

JeC1 (120) (-1/C0+1 Sn + £ (-1)^(I)+^(J)(1 - (-t)jr) (-t)ZA(i,j) jkSJ otherwise,

where C2 is the set of compositions ofn different from (n) whose first part is a sum of any prefix of i, and C1 the set complementary to C2 in all compositions ofn different from (n).

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

R n( A, tA) = Rn(A) and R 0n(A, tA) = Rn( A|tA) - Rn (A). (121)

Now, the formula RiRj = Rij + Ri>j together with Eq. (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:

(K (nj- 1)K (nj-1)) ( -tK0(n) tK0(n) K\ (n) -K\(n) \ (122)

{K0(n + 1)K1(n + ^ = \ t(K0(n) + K1(n)) 0 0 K0(n) + Kx(n)). (122)

6.3.2. Image of the R in the A basis of Sym

Expanding Eq. (115) on the basis AJ, the same reasoning gives as well

RI(A\tA) = J2(-1)n+1+l(I)+l(J) (1 - (-t)j1 )(-t)ZksA^(I,j)jkaJ (123)

where A'(I, J) = {¿j + ••• + j- e 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 A J ( A) is

^(-1)n+l(I')+l(J)(-t)h+T,keAi(Ii, J) jk a J ifi0 = 0,

J (124)

^(-1)n+i+l(I0+l(J) (-t)ZkeA>(j>,j)haJ otherwise. J

Note that this is coherent with the fact that 0 e Des(I) if i0 = 0.

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 AJ, one has the following structure:

(f (n.L. 1)i („4.1)) (tL0(n) -tL0(n) -L1(n) L1(n)\ (125)

(L0(n + 1)L,(n +^tLt(n) tL0(n) Ln(n) L0(n)). (125)

6.3.3. Image of the R in the R basis of Sym

The expansion of RI(A\tA) has been discussed in [12] and [15]. 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:

Ri (A\tA) = J2 (1 + t)v( J )tb(I'J) Rj (A), (126)

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. If one denotes 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:

(M0(n + 1)M,(n + 1)) = ( tm/nl,^ tM0n(n) M1(n) „ M0(n)Ar . (127) v 0( -T ) 1( -T n \t(M0(n) + M1(n)) 0 0 M0(n) + Mt(n^ ^ ;

For example, one can check this result on Fig. 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) alternating permutations according to their last value yields the Euler-Bernoulli triangle, sequence A010094 or A008281 of [17] 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 2m or 2m 1 according to their last value. We then get the table

n\p —6 —5 —4 —3 —2 —1 0 1 2 3 4 5 6

1 1 0 1

2 0 1 0 1 2

3 4 4 3 0 3 2 0

4 0 4 8 11 0 11 14 16 16

5 80 80 76 68 57 0 57 46 32 16 0

6 0 80 160 236 304 361 0 361 418 464 496 512 512

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 Bn ending 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.

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 [17]) 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:

n\p —6 —5 —4 —3 —2 —1 1 2 3 4 5 6

1 1 0

2 0 1 1 1

3 3 3 2 2 1 0

4 0 3 6 8 8 10 11 11

5 57 57 54 48 40 40 32 22 11 0

6 0 57 114 168 216 256 256 296 328 350 361 361

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 Eq. (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—1 i 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) = {a e S(n, p) | an—1 < —p or — 1 e a], S"(n, p) = {a e S(n, p) | an—i > — p and — 1 e a}.

Then S'(n, p) U 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) = {1342,1342,1432,1432,3142,3142,3412, 3412, 3412, 3412, 4132, 4132, 4312, 4312},

S'(4, 2) = {1342,1342,1432,1432,3142,3142,3412,3412,4132,4132, 4312},

S"(4, 2) = {3412,3412, 4312}, so that S'(4, 2) U S"(4, 2) = S(4, 2) and

S'(4, 2)/I(S"(4, 2)) = {1342,1342,1432,1432, 3142, 3142, 4132, 4132}, 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

A = A(0) u A(1) u ••• u A(r—1) = A x C,

with C = {0,..., r — 1} be an r-colored alphabet. We assume that A(i) = A x {i} is linearly ordered and that

A(0)> A(1)> ••• > A(r—1).

Colored words can be represented by pairs w = (w, u) where w e An and u e Cn. We define r-colored alternating permutations a = (a, u) by the condition

a i < a 2 > a 3 < •

hence, as permutations of shape 2n or 2n 1 as words over A. Let A¡[) be the set of such permutations. Their noncommutative generating series in FQSym(r) is then

£ £ Ga (A) = X(A) =22 R 2m (A) + R 2m 1 (A).

n>° a e Anr)

Thus, if we send A(i) to qiE, the exponential generating function of the polynomials

an(qo,...,qr—1) = £ J"^

a e A,

(r) i=1

a(z; qo,...,qr—1) =

sin(qo +-----+qr—1)z + 1

cos(qo +-----+qr—1)z

Iterating the previous constructions, we can define generalized snakes as colored alternating permutations such that a 1 e A(0), or, more generally a 1 e A(0) u ••• u A(i). Setting Bi = A(0) u ••• u A(i) and Bi = A(i+1) u ••• u A(r—1), we have for these permutations the noncommutative generating series

Y(Bi, Bi) = (cos + sin)(Bi) sec(Bi|Bi) (141)

which under the previous specialization yields the exponential generating series

u , cos(q0 + • •• + qi )t + sin(qi+1 + ••• + qr—1)t

yi (t; q0,...,qr—1) =---------. (142)

cos(q0 +-----+qr—1)t

Setting all the qi equal to 1, we recover sequences A007286 and A007289 for r = 3 and sequence A006873 for r = 4 of [17], counting what the authors of [6] called augmented alternating permutations.

8.2. Triangles of alternating permutations with r colors

One can now count alternating permutations of shapes 2k and 2k 1 by their last value. With r = 3, the following tables present the result:

n 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1 1 1 1

2 0 1 1 2 2 3

3 9 9 8 8 7 5 5 3 0

4 0 9 18 26 26 34 41 46 46 51 54 54

5 405 405 396 378 352 352 326 292 251 205 205 159 108 54 0

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.

n 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1 1 0 0

2 0 1 1 1 1 1

3 5 5 4 4 3 2 2 1 0

4 0 5 10 14 14 18 21 23 23 25 26 26

5 205 205 200 190 176 176 162 144 123 100 100 77 52 26 0

n 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1 0 1 0

2 0 0 0 1 1 1

3 3 3 3 3 3 2 2 1 0

4 0 3 6 9 9 12 15 17 17 19 20 20

5 147 147 144 138 129 129 120 108 93 76 76 59 40 20 0

n 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1 0 0 1

2 0 0 0 0 0 1

3 1 1 1 1 1 1 1 1 0

4 0 1 2 3 3 4 5 6 6 7 8 8

5 53 53 52 50 47 47 44 40 35 29 29 23 16 8 0

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 [17]) are given by exactly the same process as for type B Springer numbers, 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 Dn and the number of snakes on type Bn is 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 Eq. (128) and of Eq. (129). We obtain

n/p -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

1 0 1

2 0 0 0 1

3 1 1 1 1 1 0

4 0 1 2 3 3 4 5 5

5 23 23 22 20 17 17 14 10 5 0

6 0 23 46 68 88 105 105 122 136 146 151 151

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

X = £ (R#m) + R#m 1)) (148)

p# I V#

and the generating series of type B snakes defined in Section 5.2.2 is

Y = 1 +£ (R 02n 1 + R 02"+1), (149)

we easily get one definition of the generating series type D snakes:

D = X — Y = J2 (R 2"1 + R 2n+1). (150)

In other words, our first sort of type D snakes corresponds to permutations of ribbon shape 2n 1 or 2n whose first letter is positive. Here are these elements for n < 4:

12, 123,132, 231,132, 231, (151)

1234,1243,1243,1324,1324,1342,1342,1423,1423,1432,1432, 2314,

2314, 2341, 2341, 2413, 2413, 2431, 2431, 3412, 3412,3421,3421. (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

dD = 1 + DX, (153)

D = G1 + B(D, X). (154)

The iteration of (154) brings up a solution close to (68) and (87):

D = 22 Bt(G0, G1), (155)

T eCBT

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 12n and 12n 1 the elements with an even number of negative signs and such that a1 I a2 < 0.

One then gets the following elements for n < 4:

1, 12, 123,132,132,231,231, (156)

1234,1243,1243,1324,1324,1342,1342,1423,1423,1432,1432, 2314, 2314,2341,2341,2413,2413,2431,2431,3412,3412,3421,3421.

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 Dn and then change all values into their opposite.

10. Tables

Figs. 1-3 give the tables of the maps A ^ tA from BSym to Sym.

All tables represent in columns the image of ribbons indexed by type-B compositions, where the first half begins with a 0 and the other half does not. So, with N = 3, compositions are in the following order:

[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.

t3 t3 -t3 t3 1 -1

t2 t2 1 1 -t3 - 2 t3 + t2 t + 1

+1 . . t +1 ) t - t3 t3 -t

^ t3 + t2 t2 + t

Fig. 1. Matrices of Rj(A, tA) on the S basis for n = 2, 3.

t3 -t3 -t3 t3 1 -1 -1 1

-t2 -1 1 \ t2 t3 -t2 -t3 -1 -t 1 t

t2 1 t ) - t t t3 -t3 -1 1 t2 -t

t t2 t2 t3 1 t t t2

Fig. 2. Matrices of Rj (A, tA) on the A basis for n = 2, 3.

t t 2 t2 t3 1 t

1t . t +1

-t - 1 t2 - 1 t +1

t2 +1 t3 +12

t2 +1 t3 +12 \ t3 +12 t2 +1

t t2 \ . t + 1 t2 + t .

. . t + 1 t2 +1

. . t2 + t t + 1 /

Fig. 3. Matrices of Rj (A, tA) on the R basis for n = 2, 3.

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