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Combinatorics/Algebra

r-Bell polynomials in combinatorial Hopf algebras Polynomes de r-Bell dans les algebres de Hopf combinatoires

Ali Chouria, Jean-Gabriel Luque

Laboratoire d'informatique, de traitement de l'information et des systèmes, Normandie Université, UNIROUEN, UNIHAVRE, INSA Rouen, LITIS, 76000 Rouen, France

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A R T I C L E I N F 0

Article history:

Received 12 July 2016

Accepted after revision 17 January 2017

Available online 10 February 2017

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A B S T R A C T

We introduce partial r-Bell polynomials in three combinatorial Hopf algebras. We prove a factorization formula for the generating functions which is a consequence of the Zassenhauss formula.

© 2017 Académie des sciences. Published by Elsevier Masson SAS. This is an open access

article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

R E S U M E

Nous définissons des polynômes r-Bell partiels dans trois algèbres de Hopf combinatoires. Nous prouvons une formule de factorisation pour les fonctions génératrices, qui est une conséquence de la formule de Zassenhauss. © 2017 Académie des sciences. Published by Elsevier Masson SAS. This is an open access

article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Partial multivariate Bell polynomials have been defined by E.T. Bell [2] in 1934. Their applications in Combinatorics, Analysis, Algebra, Probabilities etc. are numerous (see, e.g., [8]). They are usually defined on an infinite set of commuting variables (a1; a2,...} by the following generating function:

J2Bn,k(ai,...,ap,...)ntk = exp E am. (1)

The partial Bell polynomials are related to several combinatorial sequences. Let denotes the Stirling number of second kind, which counts the number of ways to partition a set of n objects into k nonempty subsets, and let [£] denote the Stirling number of first kind, which counts the number of permutations according to their number of cycles. We have, Bn,k(1,1, ...) = {D and Bn,k(0', 1', 2', ...) = [k].

The connection between the Bell polynomials and the combinatorial Hopf algebras has been investigated by one of the authors in [3].

E-mail addresses: ali.chouria@yahoo.fr (A. Chouria), jean-gabriel.luque@univ-rouen.fr (J.-G. Luque). http://dx.doi.org/10.1016/jxrma.2017.01.015

1631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Aiming to generalize these polynomials, Mihoubi et al. [9] defined partial r-Bell polynomials by setting

K+r,k+r(ai> a2>• • • ;bi>b2' • • ) = J2 J2 aX'Xa*i • • • aKbyr • • bK' (2)

n'+n"=n+r X'jH-----h^r=n'

XfH-----h^k'=n"

where the second sum runs over pairs of (integer) partitions (X', X"), arx, X„ is the number of set partitions n = {n', k'2, • • • , n'r, n'', 7i2', • • • , nk'} of {', 2, • • • , n} such that = X', • • • , #n'r = X'r, = X• • • , #n'k = X'k and ' e n', 2 e n22, • • • , r e rtr, and #ni denotes the cardinality of ni. Comparing our notation to those of [9], the roles of the variables ai and bi have been switched. The generating function of the r-Bell polynomials is known to be

xn yr I xj \ / xjt\

EK+rk+r ^ a2, • • • ;b1, b2, • • • )-^t« = exp £ aj+1 ^^ exp £ bjxJ^\ , (3)

n>k ■ ■ \;>0 ' W1 '

where (an; n > 1) and (bn; n > 1) are two sequences of nonnegative integers.

The aim of our paper is to show that we can define three versions of the r-Bell polynomials in three combinatorial Hopf algebras in the same way. The first algebra is Sym(2), the algebra of bisymmetric functions (or symmetric functions of level 2). The r-Bell polynomials as defined by Mihoubi belong to this algebra. The second algebra is NCSF(2), the algebra of noncommutative bisymmetric functions. In this algebra, we define non-commutative analogues of r-Bell polynomials that generalize the Munthe-Kaas polynomials. The third algebra is WSym(2) := CWSym(2, 2, • • • ), the algebra of 2-colored word symmetric functions. In this algebra, we define word analogues of r-Bell polynomials. The common feature of the three constructions is that they are based on the same algorithm, which generates 2-colored set partitions without redundance. Our main result is a factorization formula for the generating function which holds in the three algebras and which is a consequence of the Zassenhauss formula.

2. Bi-colorations of partitions, compositions and set partitions

A bicolored partition X of n is a multiset {(Ai, ji), ..., (X^, jk)} such that Xi + • • • + Xk = n and j\,..., jk e {1, 2}. We set X I- n, û)(X) = n and ¿(X) = k. A bicolored composition I of n is a list I = [(i1, j1),..., (ik, jk)] with i1 + • • • + ik = n and ji,..., jk e{1, 2}. We set I N n, û(I) = n and ¿(I) = k. A bicolored set partition is a set n = {(n1, j1),..., (nk, jk)} such that {n1, ..., Kk} is a set partition of size n and j1,..., jk e{1, 2}. We set n lb n, û(k) = n and ¿(n) = k. We define

Srn+r,k+r = {n = {(n1,1),- • • ,(nr, 1), (rtr+1, 2), • • • ,(Kk+r, 2)}:n lb (n + r), 1 e rc^ • • , r e n-}. (4)

We have S^r = {{({1}, 1), ({2}, 1), • • • ,({r}, 1)}} and

Sh+1+r,k+r = |n u{(n + 1, 2)}: n e Srn+r,r+k_1 ) U

jn ji)} U {(%t U{n + 1}, ji)} : n = {(n1, j1), • • • ,(Kr+k, jr+k), 1 < I < r + k}e Sr++kr ^ .

The underlying recursive algorithm generates one and only one times each element of Sn+1+r k+r.

We consider also two applications: c(n) = [(#n1, j1),..., (#Kk, jk)] if n = {(n1, j1),..., (kk, jk)} with min{n1} < • • • < min{nk} and X(n) = {(#n1, j1),..., (#Kk, jk)}. We define

fXn+r,k+r(I) = #{n e Srn+1+r,k+r : c(K) = I} and grn+r,k+r(X) = #{n e Sn+1+r,k+r : X(n) = X}. 3. Three combinatorial Hopf algebras

3.1. Algebras of symmetric functions of level 2

In this section, we define three combinatorial Hopf algebras indexed by bicolored objects. The model of construction is the algebra Sym(l), which is the representation ring of a wreath product (r ; &n)n>0, r being a group with l conjugacy classes [6]. Let us recall briefly its definition for l = 2. The combinatorial Hopf algebra Sym(2) [6] is naturally realized as symmetric functions in 2 independent sets of variables Sym(2) := Sym(X(1); X(2)). It is the free commutative algebra generated by two sequences of formal symbols p' (X(1)), p2(X(1)), ... and p'(X(2)), p2(X(2)), ... , named power sums, which are primitive for its coproduct. The set of the polynomials pX := pXl (X(il)) • • • pXk(X(ik)), where X = {(X', i'),..., (Xk, iik)} is a bicolored partition, is a basis of Sym(2).

The Hopf algebra NCSF of formal noncommutative symmetric functions [5] is the free associative algebra C • • • )

generated by an infinite sequence of primitive formal variables (^)i^1. Its level l is analogous to that described in [11] as the free algebra generated by level-l complete homogeneous functions Sn. Here we set l = 2 and we use another basis. We recall that the level-2 complete homogeneous function Sn, for n e N2, is defined as a free quasi-symmetric function

of level 2 as Sn = |=ni Gi...n,u, where Ga,u denotes the dual free /-quasi-ribbon labeled by the colored permutation (a, u) [11]. Notice that Ga,u is realized as a polynomial in C(A(1) U A(2)), where A(i) denotes two disjoint copies of the same alphabet A as Ga,u = ^ w, where std is the usual standardization applied after identifying the two alphabets

we(A(1)UA(1))" std(w)=a,wi eA(i)

A(1) and A(2). Alternatively, for dimensional reasons, NCSF(2) is the minimal sub (free) algebra of C(A(1) U A(2)) containing both NCSF(A(1)) and NCSF(A(2)) as subalgebras. Hence, it is freely generated by the (primitive) power sums ^(A(j)). If 1 = [(i1, j1),..., (ik, jk)] denotes a bi-colored composition, then the set of the polynomials ^1 = ^ (A(j1)) • • • (A(jk)) is a basis of the space NCSF(2).

The last algebra, WSym(2), is a level 2 analogue of the algebra of word symmetric functions introduced by M.C. Wolf [12] in 1936. We construct it as a special case of the Hopf algebras CWSym(a) of colored set partitions introduced in [1] for a = (2, 2, ..., 2, ...). As a space CWSym(a) is generated by the set where n denotes a bicolored set partition. Its product is defined by

= $nUn', (6)

where U denotes the shifted union obtained by shifting first the elements of n' by the weight of n and hence compute the union, and its coproduct is

A($n) = $std(n1) ® $std(n2), (7)

7T1 Un 2 =n

n 1nft 2=0

where the standardized std(n) of n is defined as the unique colored set partition obtained by replacing the ith smallest integer in the nj by i.

The algebra Sym(2) (resp. NCSF(2), WSym(2)) is naturally bigradued Sym(2) = 0nk Symn2k (resp. NCSF(2) = 0nk NCSFn2k, WSym(2) = 0n k WSymfk) where Sym^k = span{pX : ¿(X) = k, w(X) = n} (resp. NCSFn2k = spanj^1 : ¿(1) = k, co( 1) = n}, WSymn2k = span{$n : ¿(n) = k, a>(%) = n}). We denote by R the subalgebra of Sym(2) (resp. NCSF(2), WSym(2)) spanned by the polynomials p{(X1 '2)'...'(Xk-2)> (resp. ^[(i1>2)>...>(ik>2)], ${(n1,2), . ,(nk,2)>), which is isomorphic to Sym (resp. NCSF, WSym). Notice also that R = 0n k Rn,k is naturally bigraded.

In the rest of the paper, when there is no ambiguity, we use ai to refer to pi(X(1)), ^(A(1)) or ${({1'. .'n>'1)} and bi to refer to pi(X(2)), (A(2)) or ${({1' . 'n>'2)>. Notice that with this notation all the ai and the bi are primitive elements. We define the natural linear maps S : WSym(2) ^ NCSF(2) and % : WSym(2) ^ Sym(2) by S($n) = ^c(n) and = pX(n).

Notice that these maps are morphisms of Hopf algebras.

3.2. r-Be// polynomials and (commutative/noncommutative/word) symmetric functions

In Sym(2) and NCSF(2), we define the operator d as the unique derivation acting on the right and satisfying ai d = ai+1 and bi d = bi+1. In WSym(2), we define d as the operator acting linearly on the right by 1d = 0 and

^{[TC1,i1],...,[TCk,ik ]} d = $({[n1.'1].....[nk,'k]}\[nj ,ij ])U{[n j U{n + 1}, ij ]}. (g)

In the three algebras, we define r-Bell polynomials in a similar way to Ebrahimi-Fard et al., who defined Munthe-Kaas polynomials, that is by the use of the operator d. More precisely, the polynomial Brn+r k+r is the coefficient of tk in ar1(tb1 + d)n. In WSym(2), from (5), we have

Brn+r,k+r = E (9)

neSn+r,k+r

Hence, using the maps S and %, we obtain

Brn+r,k+r = E pX(K) = E grn+r,k+r (X)pX (10)

neSn+r,k+r X

in Sym(2) and

Brn+r,k+r = E = E fn+rk+r(1 )*1 (11)

neSn+r,k+r 1

in NCSF(2). Notice that in Sym(2), Bn+r k+r is nothing but the classical r-Bell polynomial and in NCSF(2), it is a r-version of the Munthe-Kaas polynomial [4,10].

Example 1. In WSym(2), we have

B2 3 = ${({1,3},1),({2},1),({4},2)} + ${({1,4},1),({2},1),({3},2)} + ${({1},1),({2,3},1),({4},2)} + ${({1},1),({2,4},1),({3},2)} + ${({1},1),({2},1),({3,4},2)}.

In NCSF(2), we have

B4,3 = 2* K^MUMUH + 2* [(1.1).(2.1).(1.2)] + y [(1,1),(1,1),(2,2)] = 2a2a1b2 + 2^1 + a1a1b2. In Sym(2), B2,3 = 4pf(2'1)'(1'1)'(1'2)l + p{C,i),(i,i),(2,2)} = 4a2a1b2 + a^b,.

We consider also the polynomials B n+k+rk+r = ar1bk1dn. Notice that in WSym(2), we have

B n+k+r,k+r = ^ ${(n1,1),...,(nr ,1),(nr+1,2),...,(nr+k ,2)}. (12)

{(n1,1),...,(nk+r ,1)}eSS+rk+r,k+r

1eK1,...,reKr

4. Generating functions

We consider the following generating functions:

S(t, x, y) = J2 Bn+rk+x/^t1 = exp (a1 y) exp (x(b + 9)), (13)

n, r, k

S °(t, x) = S (t, x, 0) = V Bnk—tk = 1. exp (x(tb1 + d)), (14)

—' ' n I

xn yr tk

s '(t, x y) = j2B rn+k+r,k+rn r! ki=exp (a1 y) exp (tb1) exp (x9), (15)

n,r,k ! ! !

S *(x, v) = V Bl,„.,

S*(x, y) = £ B'n+r.r^ = exp (yb1 ) exp (xd). (16)

Theorem 4.1. The generating functions S (t, x, y) and S °(t, x) satisfy the following factorization

S (t, x, y) = S'(xt, x, y) Z (x, t) and S °(t, x) = S * (x, xt ) Z (x, t), (17)

where Z (x, t) = ^>2 exp (xn £k tkCn,k), CnM = k!(n-1k-1)! adna-k-1adkb1 d, and adx is the derivation adxP = [x, P ] = xP - Px.

In Sym(2) and NCSF(2) the operator Cn,k is the multiplication by a primitive polynomial belonging to the subalgebra Rn,k.

Proof. Equalities (17) are obtained from (13) and (14) by using Zassenhaus formula [7]. In Sym(2) and NCSF(2), since d is a derivation, ad'aadj^ d is primitive. Furthermore, remarking that [bi, 9] = bi+1, we prove that adgadj d e Rn,k. □

Example 2. In NCSF(2), consider the coefficient of 3 V^tt in the left equality of (17). In the left-hand side, we find B5 3 = 3a2a1b\ + 3a1a2b\ + 2a21b2b1 + a1b1b2. The same coefficient in the right-hand sides is 3B2 4 — 3B3 3C2,1 + 3!B2 2C3,2. Since B2,4 = a2a1b\ + a1a2b2 + a\b2b1 + a2b1b2, B2,3 = a2b1, B2,2 = a\, C2,1 = — 5b2, and C3,2 = ^№1, b2], we check that 3B2,4 — 3B3 3C2,1 + 3!BB2 2C3 2 = B23 as expected by Theorem 4.1.

In NCSF(2), we compute explicitly the polynomial Cn,k

( 1 )k 1 n k 1

C",k = LnL k!(n — k — 1)! E (i1 — 1,...,ik—1 — 1, ik — 2) [bn, [bi2, ••• , [bik—1, bik—1 ]•••". (18)

i1,...,ik

Example 3. Consider for instance the polynomial C5,2 in NCSF(2)

Cs,2 = -48 adX, 9 = -48 ad4[bi, 1

= -^[[№1, b2],3],3],3],3] 48

= --L (2[b3, b4] + 3[b2, b5] + [b1, b6]) 48

= -48 ([b5, b2] + 4[b4, b3] + 6[b3, b4] + 4[b2, b5] + [b1, b6]).

Remark 1. If we set ai = bi for each i, then we have S*(t, x, y) = S*(y + t, x), and so S(t, x, y) = S*(y + xt, x)Z(x, t). In Sym(2), the series Z (x, t) has a nice closed form

Z(x, t) = exp I - E ^^bA . (19)

Indeed, since the algebra is commutative ad'gadb d is nonzero only if j = 1 and when j = 1 formula (18) gives [9, bi] = -bi+1.

As a consequence, using equality (19) together with Theorem 4.1 and Formula (3), we find

S '(xt, x, y) = exp I ^ aj+1 j I exp I ^ jbjj I . (20)

\j>0 h J h J

In other words, equating the coefficients in the left- and the right-hand sides of (20), we find

K+k+r,k+r = (n + k) K+k+r,k+r ^ a2-..; 2b2, 3b3,...). (21)

In the case where r = 0, we obtain

B 0+Kk (ai, a2,... ; bi,...) = Bkn+Kk(bi, b2,...; bi, b2,...) = + ^ Bn+Kk (bi, 2b2, 3bs,...). (22)

Acknowledgements

The authors thank the referee for his valuable comments. This paper is partially supported by the PHC MAGHREB project IThèM (14MDU929M), the ANR project CARMA and the GRR project MOUSTIC.

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