Scholarly article on topic 'r -Bell polynomials in combinatorial Hopf algebras'

r -Bell polynomials in combinatorial Hopf algebras Academic research paper on "Physical sciences"

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Abstract of research paper on Physical sciences, author of scientific article — Ali Chouria, Jean-Gabriel Luque

Abstract 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.

Academic research paper on topic "r -Bell polynomials in combinatorial Hopf algebras"

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C. R. Acad. Sci. Paris, Ser. I


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


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

Presented by the Editorial Board


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 (


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 (

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: (A. Chouria), (J.-G. Luque).

1631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (

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'


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)


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


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)


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)


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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