On multiplicative conditionally free convolution

Using the combinatorics of non-crossing partitions, we construct a conditionally free analogue of the Voiculescu's S-transform. The result is applied to analytical description of conditionally free multiplicative convolution and characterization of infinite divisibility.

Two important tools in free probability theory are the R and S transforms. Those are power series with the property that if X and Y are free random variables, then R X+Y = R X + R Y and S XY = S X S Y . While a c-free version of the R-transform is constructed in [8] and used in several papers (such as [21], [16], [14]) for the study of c-free additive convolution of measures or of c-freeness with amalgamation, the literature lacked a similar development for the multiplicative case.
The present paper shows the construction of a suitable c-free version of the S-transform (in fact, as in [9], a more natural choice is its inverse, called the Ttransform), and demonstrates some of its applications in limit theorems and characterization of infinite divisibility.
The material is structured in seven sections. The second section contains notations and preliminary results used throughout the paper, but mostly in the third section. The third section presents the construction of the c T -transform and proves its multiplicative property. The argument is based on enumerative combinatorial techniques, in the spirit of [15] -the lack of a Fock space model makes difficult an analytical procedure, as in [12]. The forth section defines the multiplicative c-free convolution of two measures on the unit circle and presents its connections to free The first author was partially supported by the grant 2-CEx06-11-34 of the Romanian Government. and boolean multiplicative convolutions. The fifth and the sixth sections present applications of the multiplicative property of the c T -transform for the study of limit distributions, respectively for the characterization of infinite divisibility. These sections are using more analytical techniques, analogue to [3], [4] and [5]. The last section -the appendix -is showing an explicit combinatorial formula for computing the coefficients of the T -and c T -transforms. The computations, generalizing a result from [9], are demonstrating the use of non-crossing linked partitions in free probability.
For X ∈ A, we will write R n X and c R n X for R n (X, . . . , X), respectively c R n (X, . . . , X). The R-, respectively c R-transform of X are the formal power series Let m X (z), respectively M X (z) be the moment-generating series of X with respect to ψ and ϕ, i. e. m X (z) = n n=1 ψ(X n )z n and M X (z) = n n=1 ϕ(X n )z n . As shown in [15] and [8], the definitions of R n and c R n give Two elements, X and Y , from A are said to be conditionally free ( c-free) in (A, ϕ, ψ) if the subalgebras generated by them in A are conditionally free, as defined in Introduction. The key properties of the R and c R transforms are summarized in the following result: Proofs for the first part of (i) can be found, for example, in [12], Theorem 2.2, and [15], Theorem 16.16. Second part of (i) is proved in [8], Theorem 3.1.

2.2.
Boolean independence, the ηand B-transforms. Let now A be a unital algebra and ϕ : A −→ C be a normalized functional. Two subalgebras A 1 , A 2 are said to be boolean independent if for all x k ∈ A 1 and y k ∈ A 2 . We will say that X, Y ∈ A are boolean independent if the nonunital algebras generated in A by X, respectively Y are boolean independent.
If A = C ⊕ A 0 (direct sum of vector spaces), and ψ = Id C ⊕ 0 A0 , then conditional freeness with respect to (ϕ, ψ) is equivalent to boolean independence with respect to ϕ.
For the results in Section 4-6, we need the definitions and results below (see [18] or [11] for proves).
ϕ(X n )z n . The η-, respectively B-transforms of X are the formal power series given by the relations η X (z) = MX (z) 1+MX (z) , respectively B X (z) = 1 z η X (z). Proposition 2.3. If X and Y are two boolean independent elements of A, then: 2.3. Non-crossing partitions. By a partition on the ordered set n = {1, 2, . . . , n} we will understand a collection of mutually disjoint subsets of n , γ = (B 1 , . . . , B q ), called blocks whose union is the entire set n . A crossing is a sequence i < j < k < l from n with the property that there exist two different blocks B r and B s such that i, k ∈ B r and j, l ∈ B s . A partition that has no crossings will be called non-crossing. The set of all non-crossing partitions on n will be denoted by N C(n).
For γ ∈ N C(n), a block B = (i 1 , . . . , i k ) of γ will be called interior if there exists another block D ∈ γ and i, j ∈ D such that i < i 1 , i 2 , . . . , i k < j. A block will be called exterior if is not interior. The set of all interior, respectively exterior blocks of γ will be denoted by Int(γ), respectively Ext(γ).
The following notations will also be used: , σ has only one exterior block} N C 2 (n) = {σ : σ ∈ N C(n), σ has only two exterior blocks} N C S (n) = {σ : σ ∈ N C(2n), σ the elements from the same block of σ have the same parity}.
For σ ∈ N C S (2n), denote σ + , respectively σ − the restriction of σ to the even, respectively odd, numbers from {1, 2, . . . , 2n}. Define Also, we will need to consider the mappings constructing by doubling the elements, and the juxtaposition of partitions.

the c T -transform
Let A be a unital algebra, ϕ, ψ : A −→ C be two normalized linear functionals and X be n element from A. Denote by m X (z), respectively M X (z), the moment generating power series of X with respect to ψ and ϕ, as in the formulas (1), (2).
In this section we will construct a formal power series c T X (z) such that: (i) for ϕ = ψ one has that T X (z) = c T X (z) (see 2.1 for the definition of T X (z)).
(ii) M X (z) can be obtained from T X (z) and c T X (z) via substitutional composition, substitutional inverse and algebraic operations. (iii) if X and Y are c-free elements from A, then The construction of c T X (z) is presented as a natural consequence of the combinatorial properties of the free and c-free cumulants R n and c R n .
We will focus next on free and c-free cumulants with products as entries. Particularly, Particularly, Proof. (a) is shown in [15]. The proof of (b) will be done by induction on n. For n = 1, the statement is trivial: sice the mixed c-free cumulants vanish (see 2.1).
For the induction step, we distinguish two cases: If π has more than one exterior block, then π = π 1 ⊕ π 2 for some m < n, π 1 ∈ N C(m), π 2 ∈ N C(n − m). One has that If π has exactly one exterior block, then where p is the length of the exterior block of π and π is the non-crossing partition obtained by erasing the exterior block of π. The result follows from 3.2 and the induction hypothesis.
We need to show that Before proceeding with the proof, let us take a better look at the right-hand side of 3. Since any σ ∈ N C 0 (2n) has exactly 2 exterior blocks, one containing 1 and one containing 2n, each K σ [X, Y, . . . , X, Y ] will have exactly two factors of the type c R m , namely c R p X and c R q Y , where p, q are, respectively, the length of the first and second exterior block of σ. Also, σ ∈ N C 0 (2n) implies that all other factors of On the other hand, Since all the mixed cumulants vanish, 4 becomes: one has that σ ∈ N C 0 (2n) if and only if σ 0 n = ½ 2n . Therefore: It follows that so the proof is now complete.
For stating the next result we need first a brief review of the operation ⋆ (boxed convolution), as defined in [15].

Consider the formal power series
β n z n is another formal power series, we define their boxed convo- Cf π (f ) · Cf Kr(π) (g).
We will need the following two results proved in [15]: Lemma 3.4. If X and Y are free, then Now we can proceed with the main theorems of this section. To ease the nota- As shown before, Each σ ∈ N C 0 (2n) has exactly 2 exterior blocks, one consisting on 1 = b 1 < b 2 < · · · < b p and the other one consisting on b p + 1 = d 1 < d 2 < · · · < d q = 2n. Let us denote π k the restriction of σ to (b k + 1, b k + 2, . . . , b k+1 − 1) and ω l the restriction of σ to (d l + 1, d l + 2, . . . , d l+1 − 1). Then: hence q.e.d..
Proof. From 3.5, one has: Finally, composing at the right to R −1 XY one gets the conclusion.
satisfies the properties (i)-(iii) described in the beginning of the section.
Proof. (iii) is proved in Theorem 3.6. (i) is an immediate consequence of comparing the definitions of T X (as in Lemma 2.1) and of c T X . Finally, (ii) is implied by (1), (2) and the definition of c T X .

Multiplicative conditionally free convolution of measures
Denote by M T the family of all Borel probability measures supported on the unit circle T and by M × the moment generating function of µ, analytic in the unit disk D.
Consider now µ ∈ M T and ν ∈ M × T . To the pair (µ, ν) we associate the functions R ν (z), c R µ,ν) (z), T ν (z), c T (µ,ν) (z), analytic in some neighborhood of zero, and given by the following relations: The next two sections address some analytical properties of the operation ⊠. More precisely, Section 5 states some limit theorems and Section 6 describes ⊠ infinite divisibility.
It will be convenient to introduce a variation of the function c T (µ,ν) as follows. Namely, for the measures µ ∈ M T and ν ∈ M × T , we consider the analytic function For η ν (z) and B µ (z), defined above, one has: Let us now consider the definition of c R (µ,ν) , .
Taking into consideration (7) it follows that and composing at right with m −1 ν we get the conclusion.
Note that and the function Σ (δ λ ,ν) (z) is the constant function λ. As observed in [1], the function B µ maps D into D, and, conversely, any analytic function B : D → D is of the form B µ for a unique probability measure µ on T. As a consequence of this observation, the function Σ (µ,ν) is uniformly bounded by 1. Moreover, if ν ∈ M × T and Σ is an analytic function defined on the set η ν (D) such that the function Σ is uniformly bounded by 1, then there exists a unique probability measure µ on T such that Σ = Σ (µ,ν) .
Before starting Section 5, we will finally mention the following construction. If µ, ν are two probability measures on T, their boolean convolution µ∪ × ν is the unique measure on T given by
We would like to mention at this point that the asymptotic behavior of boolean convolution ∪ × and that of free convolution ⊠ has been studied thoroughly in [20], where the necessary and sufficient conditions for the weak convergence were found. These conditions show that the sequence δ λn ∪ × µ n1 ∪ × µ n2 ∪ × · · · ∪ × µ nkn converges weakly if and only if the sequence δ λn ⊠ µ n1 ⊠ µ n2 ⊠ · · · ⊠ µ nkn does, provided that the array {µ nk } n,k is infinitesimal. Moreover, the limit laws are proved to be infinitely divisible (see [20]), and the boolean and free limits are related in a quite explicit manner. In the sequel, we will prove the analogous results for c-free and boolean convolutions.
To our purposes, we also mention the characterization of infinite divisibility relative to boolean convolution ∪ × and that relative to free convolution ⊠. A measure ν ∈ M T is ∪ × -infinitely divisible if, for each n ∈ N, there exists ν n ∈ M T such that ν = ν n ∪ × ν n ∪ × · · · ∪ × ν n n times .
The notion of ⊠-infinite divisibility for a measure is defined analogously.
As shown in [11], a measure ν ∈ M T is ∪ × -infinitely divisible if and only if either ν is the Haar measure on T, or the function B ν can be expressed as where γ ∈ T and σ is a finite positive Borel measure on T. In other words, ν is ∪ × -infinitely divisible if and only if either the function B ν (z) = 0 for all z in D, or 0 / ∈ B ν (D). The notation ν γ,σ ∪ × will denote the ∪ × -infinitely divisible measure determined by γ and σ via the above formula.
Analogously, a measure ν ∈ M × T is ⊠-infinitely divisible if and only if the function η −1 ν can be written as for some γ ∈ T and a finite positive Borel measure σ on T. The ⊠-infinitely divisible measure ν described above will be denoted by ν γ,σ ⊠ . The Haar measure m is the only ⊠-infinitely divisible measure on T with zero first moment.
Let us proceed to the proof of the limit theorems for c-free convolution. We first show that weak convergence of probability measures can be translated into convergence properties of corresponding functions B and Σ.  Proof. The equivalence in (i) has been observed in [20], and it is based on the following identity: which says that the Poisson integral of the measure dµ( ζ ) is determined by the function B µ . Let us prove now (ii). Assume that {µ n } ∞ n=1 converges weakly to µ. Then the hypothesis on weak convergence of {ν n } ∞ n=1 implies that there exists a neighborhood of zero D ⊂ D such that the functions η νn (D) ⊂ D ′ for every n > N . Also, the Cauchy estimate implies that there exists K = K(D ′ ) > 0 so that the derivatives B ′ µn (z) ≤ K for every n ∈ N and z ∈ D ′ . Therefore, we have for every n > N and z ∈ D. Then (i) implies that the functions Σ (µn,νn) converge uniformly on D to the function Σ (µ,ν) . Conversely, suppose that the functions Σ (µn,νn) converge uniformly on D to the function Σ (µ,ν) . Observe that B µn (z) = Σ (µn,νn) (η νn (z)) for all z ∈ D. A similar argument as in the previous paragraph shows that there exists a positive constant K ′ such that the estimate holds in a neighborhood of zero, for sufficiently large n. Therefore, the weak convergence of {ν n } ∞ n=1 implies that the sequence B µn (z) converges uniformly on a neighborhood of zero to the function B µ (z). Moreover, this convergence is actually uniform on any compact subset of D by an easy application of Montel's theorem. We therefore conclude, from (i), that the sequence µ n converges weakly to µ.
For an infinitesimal array {µ nk } n,k , we define the complex numbers b nk ∈ T by where arg ζ denotes the principal value of the argument of ζ, and the probability measures µ • nk by dµ • nk (ζ) = dµ nk (b nk ζ). The array {µ • nk } n,k is infinitesimal and lim n→∞ max 1≤k≤kn |arg b nk | = 0. We also associate to each measure µ • nk an analytic function and observe that ℜh nk (z) > 0 for all z ∈ D unless the measure µ • nk = δ 1 . Proposition 5.2. Suppose that D ⊂ D is a disk centered at zero with radius less than 1/4. Let {λ n } ∞ n=1 be a sequence in T, and let {µ nk } n,k be an infinitesimal array in M × T . Then we have: (1)) uniformly in k and z ∈ D as n → ∞. (2) There exists a constant L = L(D) > 0 such that for every n and k we have Proof. (1) and (3) are proved in [20]. To prove (2), let us consider the analytic function for a measure µ ∈ M T . For z, w ∈ D, we have In addition, Harnack's inequality shows that there exists M = M (D) > 0 such that Therefore, we deduce that for every z ∈ D.

Lemma 5.3. For sufficiently large n, there exists a disk D ⊂ D centered at zero such that
where the limit lim n→∞ max 1≤k≤kn |u nk (z)| = 0 holds uniformly in D.

Proof. Introduce measures
, where the complex numbers b nk are defined as in (10). It was shown in [2] that the limits lim n→∞ η ν nk (z) = z and lim n→∞ η ν • nk (z) = z hold uniformly in k and on compact subsets of D. In particular, it follows that, as n tends to infinity, there exists a disk D ⊂ D centered at zero such that the functions Σ (µ nk ,ν nk ) and (1)) uniformly in k and z ∈ D.
Then the desired result follows form 5.2(i), 5.2(ii), and the following observation: As shown in [5], for every neighborhood of zero for sufficiently large n. Suppose that D ⊂ D is a neighborhood of zero. The infinitesimality of the arrays {µ nk } n,k and {ν nk } n,k implies that the functions Σ (µ nk ,ν nk ) (z) converge uniformly in k and z ∈ D to 1 as n → ∞. It follows that the principal branch of log Σ (µ nk ,ν nk ) (z) is defined in D for large n. Moreover, we have uniformly in k and z ∈ D when n is sufficiently large.
in a neighborhood of zero.
Proof. The equivalence between (2) and (3) was proved in [20]. We will show the equivalence of (1) and (2). Note first that we have in a neighborhood of zero D ⊂ D and Suppose that (1) Note that We conclude that {ρ n } ∞ n=1 converges weakly to the measure ν ∈ M × T by Proposition 5.1, and hence the limit law ν is ∪ × -infinitely divisible as we have mentioned earlier.
Consequently, the measure ν is of the form ν γ,σ ∪ × for some γ ∈ T and a finite Borel measure σ on T. Hence (2) holds.
Assume now (2). Then we have uniformly on compact subsets of D. Theorem 5.5 shows that uniformly in a neighborhood of zero. Observe now there exists a measure µ ∈ M × T such that Therefore the function B ν γ,σ ∪ × (z) has the form Σ " µ,ν γ ′ ,σ ′ ⊠ " (z) in a neighborhood of zero, and (1) follows from Proposition 5.1.
Next, we address the case that the measures µ n converge to m, for m the Haar measure on T.
(2) The sequence ρ n converges weakly to m.
Since the sequence µ n converges weakly to m , the above product converges to zero as n tends to infinity. As shown in [20], Theorem 4.3, this condition implies that (2) holds.
Conversely, assume that (2) holds. Then the limit lim n→∞ B ρn (z) = 0 holds uniformly on the compact subsets of D, and therefore Theorem 5.5 shows that lim n→∞ Σ (µn,νn) (z) = 0 uniformly in a neighborhood of zero D ⊂ D. Note first that the measure ν must be ⊠-infinitely divisible (see [2]). If the measure ν has nonzero first moment, then we conclude, by Proposition 5.1, that (1) holds since the zero function is of the form Σ (m,ν) . If the first moment of the measure ν is zero, then ν = m and the sequence {η νn (z)} ∞ n=1 converges uniformly on D to zero. In particular, the set η νn (D) is contained in D when n is sufficiently large. We conclude in this case that the functions B µn (z) = Σ (µn,νn) (η νn (z)) converge uniformly on D to zero as well, and therefore the sequence µ n converges weakly to m by Proposition 5.1.
Remark 5.8. We remark here for a further use that the proof of Theorem 5.7 gives a convenient criterion of determining whether a limit law is the Haar measure m or not. Namely, if the free convolutions ν n converge weakly to ν, and the c-free convolutions µ n converge to a measure µ with T ζ dµ(ζ) = 0, then the measure µ must be the Haar measure m.

Infinite Divisibility
In this section, we give a complete characterization of ⊠-infinite divisibility for a pair (µ, ν) ∈ M T × M T . We begin with the case when µ has zero first moment. We will see that these are the pairs (m, ν), where m is the Haar measure on T and the measure ν is ⊠-infinitely divisible. Proposition 6.2. Suppose that the pair (µ, ν) is ⊠-infinitely divisible and that T ζ dµ(ζ) = 0. Then µ = m. Proof. We first assume that T ζ dν(ζ) = 0. In this case, for every n ∈ N, there exist measures µ n ∈ M T and ν n ∈ M × T such that (µ, ν) = (µ n , ν n ) ⊠ · · · ⊠ (µ n , ν n ) n times . Thus we have Σ (µ,ν) = Σ (µn,νn) n . Or, equivalently, we have In [1], Proposition 3.3, it is shown that the function η −1 νn (η ν (z)) extends analytically to the entire disk D. Therefore the function B µ is the n-th power of an analytic function in D for any n ∈ N. This happens if and only if either B µ is identically zero in D or 0 / ∈ B µ (D). Hence the measure µ is the ∪ × -infinitely divisible measure with first moment zero, that is, µ = m.
Suppose now T ζ dν(ζ) = 0. Then we have that (µ, ν) = (µ, m) = (µ 1 , m) ⊠ (µ 1 , m) for some measure µ 1 ∈ M T . In this case we can view the pair (µ, m) as the distribution of XY in a noncommutative C * -probability space (A, ϕ, ψ), where X and Y are boolean independent random variables with the common distribution (µ 1 , m). Then, for any n ∈ N, the n-th moment of the measure µ is given by Therefore the measure µ is the Haar measure m.
We focus next on the ⊠-infinitely divisible pair (µ, m) such that µ ∈ M × T . Proposition 6.3. The above measure µ is either a point mass δ λ for some λ ∈ T or the harmonic measure for the unit disc relative to some point α ∈ D \ {0}, i.e., dµ = P α dm, where P α is the Poisson kernel at α.
Proof. By the virtue of (8), it suffices to show that the function B µ is a constant function. Set c = T ζ dµ(ζ). Then the same argument as in the proof of Proposition 6.2 shows that the n-th moment of the measure µ is c n . We conclude that the function m µ (z) = cz + c 2 z 2 + · · · = cz 1 − cz in a neighborhood of zero. This fact implies that B µ (z) = c for all z ∈ D as desired.
It is easy to see that if the measure ν is ⊠-infinitely divisible, a pair of measures of the form (m, ν), (δ λ , m) or (P α dm, m), is also ⊠-infinitely divisible.
where µ n ∈ M T and ν n ∈ M × T . The ⊠-infinite divisibility of the measure ν implies that there exists an analytic function u(z) in D such that the function η −1 ν (z) = z exp(u(z)) and ℜu(z) ≥ 0 for all z ∈ D (see [4]). It follows that η −1 νn (z) = z exp(u(z)/n), and hence we deduce that the measures ν n converge weakly to δ 1 . On the other hand, the identity (B µn (z)) n = Σ (µ,ν) (η νn (z)) and Proposition 5.1 imply that the measures µ n converge weakly to δ 1 as well. Define two infinitesimal arrays {µ nk } n,k and {ν nk } n,k by setting µ nk = µ n and ν nk = ν n , where 1 ≤ k ≤ n. Then the measures (µ, ν) can be viewed as the weak limit of c-free convolutions (µ n1 , µ n1 ) ⊠ · · · ⊠ (µ nn , , ν nn ). Hence (2) follows from Theorem 5.6. Now, assume that (2) holds. It was also proved in [4] that there exists a weakly continuous semigroup {ν t } t≥0 relative to ⊠ so that ν 0 = δ 1 and ν 1 = ν. Note that, for every t ≥ 0, there exists a unique probability measure µ t on T such that B µt (z) = Σ (µ,ν) (η νt (z)) t for all z ∈ D, where µ 0 = δ 1 . Then it is easy to see that the convolution semigroup {(µ t , ν t )} t≥0 has the desired properties. The implication from (3) to (1) is obvious. To conclude, we only need to show the assertions about the measure σ and the number γ. Assume that the pair (µ, ν) is ⊠-infinitely divisible, and let {(µ t , ν t )} t≥0 be the corresponding convolution semigroup as in (3). Let {t n } ∞ n=1 be a sequence of positive real numbers such that lim n→∞ t n = 0. Let k n = [1/t n ] for every n ∈ N, where [x] denotes the largest integer that is no greater than the real number x. Observe that 1 − t n < t n k n ≤ 1, n ∈ N.
Hence we have lim n→∞ t n k n = 1, and further the properties of the semigroup {(µ t , ν t )} t≥0 show that the c-free convolutions (µ tn , ν tn ) ⊠ (µ tn , ν tn ) ⊠ · · · ⊠ (µ tn , ν tn ) kn times = (µ tnkn , ν tnkn ) converge weakly to (µ 1 , ν 1 ) = (µ, ν) as n → ∞. Then Theorem 5.6 implies that the converge weakly to the measure σ and where the centered measures dµ • tn (ζ) = dµ tn (b n ζ) and the numbers b n are defined as in (10). The desired result follows immediately from the fact that lim n→∞ b n = 1, and that the topology on the set M T determined by the weak convergence of measures is actually metrizable.
Proof. Note first that the measure ν is ⊠-infinitely divisible as we have seen earlier.
The case µ, ν ∈ M × T is an application of Theorems 5.6 and 6.4. Remark 5.8 shows that µ = m when T ζ dµ(ζ) = 0. Only the case µ ∈ M × T and ν = m requires a proof. In this case we set (µ n , ν n ) = (δ λn , δ λ ′ n ) ⊠ (µ n1 , ν n1 ) ⊠ (µ n2 , ν n2 ) ⊠ · · · ⊠ (µ nkn , ν nkn ) Observe that B µn (z) = Σ (µn,νn) (η νn (z)), and that the family {Σ (µn,νn) (z)} ∞ n=1 is uniformly Lipschitz in a neighborhood of zero D with a Lipschitz constant K > 0. We have It follows that the functions B µn converge uniformly on compact subsets of D to the constant function T ζ dµ(ζ), and hence the function B µ is the same constant function. The measure µ in this case is either a point mass concentrated at a point on T or the harmonic measure for D relative to a point in D \ {0}. Therefore (µ, m) is ⊠-infinitely divisible. 7. Appendix:Non-crossing linked partitions and a formula for the coefficients of the T -and c T -transforms 7.1. Non-crossing linked partitions. The notion of non-crossing linked partitions, that we will largely use in Section 4, is a generalization of non-crossing partitions. It was first discussed in [9], in connection to the T -transform. By a non-crossing linked partition γ of the ordered set {1, 2, . . . , n} we will understand a collection B 1 , . . . , B k of subsets of {1, 2, . . . , n}, called blocks, with the following properties: (2) B 1 , . . . , B k are non-crossing, in the sense that there are no two blocks B l , B s and i < k < p < q such that i, p ∈ B l and k, q ∈ B s . (3) for any 1 ≤ l, s ≤ k, the intersection B l B s is either void or contains only one element. If {j} = B i B s , then |B s |, |B l | ≥ 2 and j is the minimal element of only one of the blocks B l and B s . An element will be said to be singly, respectively doubly covered by γ if it is contained in exactly one, respectively exactly two blocks of γ. The set of all noncrossing linked partitions on {1, . . . , n} will be denoted by N CL(n). If γ ∈ N CL(n) and B = i 1 < i 2 < · · · < i p is a block of γ, the element i 1 will be denoted min(B). The block B will be called exterior if there is no other block D of γ containing two elements l, s such that l ≤ i 1 < i p < s. The set of all exterior blocks of γ will be denoted by ext(γ); the set of all blocks of γ which are not exterior will be denoted by int(γ). We will use the notation N CL 1 (n) for the set of all elements in N CL(n) with only one exterior block. we have again q.e.d.. Proof. Part (i) of the result is also shown in [9]. The proof presented below is shorter and employs a different approach. First, for γ ∈ N CL(n), let us denote Note that if γ = γ 1 ⊕ γ 2 , then E(γ) = E(γ 1 )E(γ 2 ) and c E(γ) = c E(γ 1 ) c E(γ 2 ). Fix γ ∈ N CL(n) and let F = 1 < i 2 < · · · < i k be the first block of γ. Particularly, F ∈ ext(γ). Let F be the smallest subset of {1, 2, . . . , n} with the following properties: (i) F ⊂ F (ii) if j ∈ F , then {1, 2, . . . , j} ⊂ F . (iii) if j ∈ F and j ∈ D ∈ γ, then D ∈ F . Since γ = γ |F ⊕ γ |{1,...,n}\F , it follows that N CL(n) = Applying the induction hypothesis, the above equality becomes Say now σ ∈ N CL 1 (p) and F = 1 < i 2 < · · · < i k is the exterior block of σ. For 2 ≤ l ≤ k − 1 define σ(l) = σ |{i l ,i l +1,...,i l+1 −1} if i l is singly covered σ |{i l ,i l +1,...,i l+1 −1} \ (i l ) if i l is doubly covered also let σ ′ (k) = σ |{i k ,i k +1,...,p} if i k is singly covered σ |{i k ,i k +1,...,p} \ (i k ) if i k is doubly covered and σ(k) = σ ′ (k) ⊕ σ |{2,3,...,i2−1} Example: The partition σ = (1,4,5,9), (2, 3), (5, 6, 7), (8), (9,10) from N CL 1 (10) is represented graphically below: Note that, since σ ∈ N CL 1 (p), we have that σ ′ (k) ∈ N CL(p − k + 1). Furthermore, the knowledge of σ(2), . . . , σ(k) determines σ uniquely and c E(σ) = c t k−1 k l=2 E (σ(l)) .
Combining the above result with (10)  The inductive step for m n is analogous (for ϕ = ψ, the sequences {m n } n and {M n } n coincide).