BULLETIN DE L IFORCE

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1 ISSN : Laboratoire d Informatique Fondamentale, de Recherche Opérationnelle, de Combinatoire et d Économétrie BULLETIN DE L IFORCE Bulletin-liforce@usthb.dz Comité éditorial Prof. Abdelhafid BERRACHEDI Prof. Isma BOUCHEMAKH Prof. Sadek BOUROUBI Dr. Shehrazad SELMANE Directeur de rédaction Prof. Sadek BOUROUBI Volume N 0, Année 05

2 ISSN Présentation du Bulletin Le Bulletin du Laboratoire L IFORCE a été créé dans le but de mettre en place un système d information pour diffuser, valoriser, protéger et promouvoir les productions scientifiques des membres du Laboratoire avant leurs publications dans un journal. Il se veut être un outil de communication et d échange au sein (et éventuellement en dehors) du Laboratoire. Comité éditorial Prof. Abdelhafid BERRACHEDI Prof. Isma BOUCHEMAKH Prof. Sadek BOUROUBI Dr. Schehrazad SELMANE Directeur de rédaction Prof. Sadek BOUROUBI Contact bulletin-liforce@usthb.dz Tél : ŀbulletin du Laboratoire L IFORCE

3 Nesrine BENYAHIA TANI; Zahra YAHI; Sadek BOUROUBI Sadek BOUROUBI; Fella CHARCHALI; Nesrine BENYAHIA TANI Nesrine BENYAHIA TANI; Sadek BOUROUBI; Omar KIHEL Zahra YAHI; Nesrine BENYAHIA TANI; Sadek BOUROUBI; Farid BENCHERIF; Omar KIHEL Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular n-gon The Set Partitions: Solution for the sharing secret keys 0 An effective approach for integer partitions using exactly two distinct sizes of parts Integer partitions into Diophantine pairs 8 ŀcontenu 8 Daouya LAÏCHE; Isma BOUCHEMAKH; Eric SOPENA Packing coloring of some undirected and oriented coronae graphs 6 Isma BOUCHEMAKH; Fatma KACI On the dual König property of the order-interval hypergraph of two classes of N-free posets 6

4 Availaible on line at Bulletin du Laboratoire 0 (04) - 9 Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular n-gon Nesrine BENYAHIA TANI,ZahraYAHI, Sadek BOUROUBI Algiers University, Ahmed Waked Street, Dely Brahim, Algiers, Algeria. Abderahmane Mira University, Bejaia, Algeria. USTHB, Faculty of Mathematics P.B. El-Alia, 6, Bab Ezzouar, Algiers, Algeria. tani.nesrine@univ-alger.dz or benyahiatani@yahoo.fr, zahrayahi@yahoo.fr, sbouroubi@usthb.dz or bouroubis@yahoo.fr Abstract: Using several arguments, some authors showed that the number of non-isometric triangles inscribed in a regular n-gon equals {n /}, where {x} is the nearest integer to x. In this paper, we take back the same problem, but concerning the number of ordered and non-ordered non-isometric convex quadrilaterals, for which we give simple closed formulas, using Partition Theory. The paper is complemented by a study of two further kinds of quadrilaterals called proper and improper non-isometric convex quadrilaterals, which allows to give a connecting formula between the number of triangles and ordered quadrilaterals, which can be considered as a new combinatorial interpretation of certain identity in Partition Theory. Keywords: Ordered parallel machines, multipurpose machines, complexity, heuristic, branch and bound.

5 N. BENYAHIA TANI, Z. YAHI, S. BOUROUBI Introduction In the 98 s, Norman Anning from university of Michigan proposed the following problem [6]: From the vertices of a regular n-gon three are chosen to be the vertices of a triangle. How many essentially different possible triangles are there?. For any given positive integer n, let (n) denotes the number of such triangles. Using a geometric argument, the solution proposed by J.S. Frame, from Brown university, shown that (n) ={n /}, where{x} is the nearest integer to x. After that, other solutions were proposed by some authors, such as F. C. Auluck, from Dyal Singh college []. In 978 Richard H. Reis, from the Southeastern Massachusetts university posed the following natural general problem: From the vertices of a regular n-gon k are chosen to be the vertices of a k-gon. How many incongruent convex k-gons are there? Let us first precise that two k-gons are considered congruent if they are coincided at the rotation of one relatively other along the n-gon and (or) by reflection of one of the k-gons relatively some cord, that what we call non-isometric k-gons. For any given positive integers k n, letr (n, k) denotes the number of such k-gons. In 979 Hansraj Gupta [5] gave the solution of Reis s problem, using the Möbius inversion formula. Theorem R (n, k) = ) k ( n hk + k d/ gcd(n,k) where h k k (mod ) and ϕ(n) the Euler function. ( n d ϕ (d) ) k, d One can find the first values of R(n, k) in the Online Encyclopedia of Integer Sequences (OEIS) [7] as A00456 for k =,A0099 for k =,A005 for k =4andA079 for k =5. The immediate consequence of both Gupta s and Frame s Theorems is the following identity: { } n = n + ( ) n + χ(/n), 6 where χ(/n) =ifn 0(mod ), 0 otherwise. In 004 V.S. Shevelev gave a short proof of Theorem, using a bijection between the set of convex polygons with the tops in the n-gon splitting points and the set of all (0,)- configurations with the elements in these points [8]. The aim of this paper is to enumerate the number of two kinds of non-isometric convex

6 Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular... quadrilaterals, inscribed in a regular n-gon, the ordered ones which have the sequence of their sides s sizes ordered, denoted by R O (n, 4) and those which are non-ordered denoted by R O (n, 4), using the Partition Theory. As an example, let us consider the following figure showing three quadrilaterals inscribed in aregular -gon, the first is not convex, the second is ordered while the third is not. Observe that the second quadrilateral generates +++ as partition of 8 in four parts, that is why it is called ordered. (a) (b) (c) Figure Notations and preliminaries We denote by G n aregularn-gon and by N the set of nonnegative integers. The partition of n N into k parts is a tuple π =(π,...,π k ) N k,k N, such that n = π + + π k, π π k, where the nonnegative integers π i are called parts. We denote the number of partitions of n into k parts by p(n, k), the number of partitions of n into parts less than or equal to k by P (n, k) andbyq(n, k) we denote the number of partitions of n into k distinct parts. We sometimes write a partition of n into k parts π =(π f,...,πs fs ), where s i= f i = k, the value of f i is termed as frequency of the part π i.letm N,m k, wedenotec m (n, k) the number of partitions of n into k parts π =(π f,...,πfs s )forwhich f i m and f j = m foratleastonej {,...,s}. For example c (, 4) = 0, the such partitions are 8, 7, 46, 55, 7, 5, 44, 5, 44, 4. Let δ(n) n (mod ), so δ(n) =or0, x the integer part of x and finally {x} the nearest integer to x. Main results In this section we give the explicit formulas of R O (n, 4) and R O (n, 4). Theorem For n 4, { n R O (n, 4) = 44 + n 48 nδ(n) 6 }

7 4 N. BENYAHIA TANI, Z. YAHI, S. BOUROUBI Proof. First of all, notice that R O (n, 4) = p(n, 4). () Indeed, each ordered convex quadrilateral ABCD inscribed in G n can be viewed as a quadruplet of integers (x, y, z, t), abbreviated for convenience, as a word xyzt, such that: { n 4=x + y + z + t; () 0 x y z t, where x, y, z and t represent the number of vertices between A and B, B and C, C and D and finally between D and A, respectively. It should be noted, that the number of solutions of System () equals p(n, 4), by setting x = x +,y = y +,z = z +and t = t +. Now, let g(z) be the known generating function of p(n, 4) []: g (z) = z 4 ( z)( z )( z )( z 4 ) From expanding g(z) in partial fractions, we obtain g(z) = ( + z) 88 ( z) 4 ( z) + 4 ( z) 4 + z 8( z 4 ) z 9( z ) Via straightforward calculations, it can be proved that g (z) = ( ( ) n ( (n +) (n +) (n +)(n +) n + n + 6 n) ) + ɛ (n) z n, n 0 where ɛ (n) { 7 7, 8, 9, 7, 0, 7, 9, 8, 7 7} Thus, we have g (z) = n 0 ( ) n 44 + n 48 + (( )n ) n + β (n) z n, where β (n) { 5 6, 4, 9 44, 6, 5 6, 8, 44, 44, 6, 6, 7, 0, 5 44, 7 44, 9, 44, 9, 7} 7 Since p (n, 4) is an integer and β (n) < /, we get { n } p (n, 4) = 44 + n 48 + (( )n ) n () Hence, the result follows. Remark G.E. Andrews and K. Eriksson said that the method used in the proof above dates back to Cayley and MacMahon [, p. 58]. Using the same method [, p. 60], they proved the following formula for P (n, 4): P (n, 4) = { (n +)(n +n + 85) 44 (n +4) n+ 8 }

8 Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular... 5 Because p(n, k) =P (n k, k) (see for example [4]), it follows: { n p (n, 4) = 44 + n n 8 n n } (4) 8 Note that the formula () seems quite simple than (4). To give an explicit formula for R O (n, 4) we need the following lemma. Lemma 4 For n 4, n c (n, 4) = p (n, 4) q (n, 4). Proof. By definition of c m (n, k) in section, it easily follows that c (n, 4) = p (n, 4) (q (n, 4) + c (n, 4) + χ(4/n)), where χ(4/n) =ifn 0(mod 4), 0 otherwise. Furthermore, c (n, 4) can be considered as the number of integer solutions of the equation: x + y = n, with y x. Since x y, thesolutionx = y = n/4, when 4 divides n, must be removed. Then, by taking y =, one can get c (n, 4) = n χ(4 n). This completes the proof. Now we can derive the following theorem. Theorem 5 For n 4, { n R O (n, 4) = 44 + n 48 nδ(n) } { (n 6) (n 6) 48 } (n 6)δ(n) 6 n. Proof. First of all, notice that q(n, k) =p(n k(k )/,k) []. Then from () we get { } (n 6) (n 6) (n 6)δ(n) q(n, 4) = p(n 6, 4) = Therefore, it is enough to prove that n R O (n, 4) = p (n, 4) + q (n, 4) (5) In fact, each non-ordered convex quadrilateral may be obtained by permuting exactly two parts of some partitions of n into four parts, which is associated from System () to a

9 6 N. BENYAHIA TANI, Z. YAHI, S. BOUROUBI unique ordered convex quadrilateral. For example, in Figure above, the ordered convex quadrilateral (b) assimilated to the solution of 8 or to the partition 44 of, generates the non-ordered convex quadrilateral (c) via the permutation. Obviously, not every partition of n can generate a non-ordered convex quadrilateral, those having three equal parts or four equal parts cannot. Also, each partition of n into four distinct parts xyzt generates two non-ordered convex quadrilaterals, each one corresponds to one of the two following permutations xytz and xzyt. On the other hand, each partition of n into two equal parts, like xxyz, withy and z both of them x, generates only one non-ordered convex quadrilateral, corresponding to the unique permutation xyxz. Thus, Hence, from Lemma 4 the theorem holds. R O (n, 4) = q (n, 4) + c (n, 4), (6) Remark 6 By substituting k =4in Theorem, we get R (n, 4) = ( n ) + ( ) n n( δ(n)) + + α, 8 6 where α = 8 8 if n 0(mod 4), if n (mod 4), 0 otherwise. Knowing furthermore that R(n, 4) = R O (n, 4) + R O (n, 4), the following identity takes place according to Theorem and Theorem 5: ( n ) + 8 { ( n ) n( δ(n)) n + + α = n 48 nδ(n) } + 6 { (n 6) (n 6) n. } (n 6)δ(n) 6 4 Connecting formula between (n) and R O (n, 4) There are two further kinds of quadrilaterals inscribed in G n, the proper ones, those which do not use the sides of G n and the improper ones, those using them. In Figure bellow,

10 Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular... 7 two quadrilaterals inscribed in G are shown, the first one is proper while the second is not. Figure Let denote by RO P (n, 4) and RP O (n, 4) respectively, the number of these two kinds of quadrilaterals. The goal of this section is to prove the following theorem. Theorem 7 For n 4, (n) =R O (n +, 4) R O (n, 4) Proof. Note first that an improper ordered quadrilateral is formed by at least one side of G n, then the concatenation of the vertices of one of such sides gives a triangle inscribed in G n, as shown in Figure. Figure Then we have RO P (n, 4) = (n ) On the other hand, it is obvious to see that R P O(n, 4) = p(n 4, 4) Then from (), we get R P O(n, 4) = R O (n 4, 4)

11 8 N. BENYAHIA TANI, Z. YAHI, S. BOUROUBI Since we obtain R O (n, 4) = RO P (n, 4) + RP O (n, 4), R O (n, 4) = R O (n 4, 4) + (n ) So, the theorem has been proved while substituting n by n +. Remark 8 The well-known recurrence relation [4, p. 7], implies by setting k =, p(n, k) =p(n +,k+) p(n k, k +), (7) p(n, ) = p(n +, 4) p(n, 4) (8) Thus, as we can see, the formula of Theorem 7 can be considered as a combinatorial interpretation of identity (8). For k n, we have the following generalization, using the same arguments to prove Theorem 7. Theorem 9 For n k, R O (n, k) =R O (n +,k+) R O (n k, k +) The formula of Theorem 9 can be considered as a combinatorial interpretation of the recurrence formula (7). References [] G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge University Press, Cambridge, United Kingdom, (004). [] N. Anning, J. S. Frame and F. C. Auluck, Problems for Solution: , The American Mathematical Monthly, Vol. 47, No. 9, (Nov., 940), [] A. Charalambos Charalambides, Enumerative Combinatorics, Chapman & Hall /CRC, (00). [4] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, Dordrecht- Holland, Boston, 974, 75.

12 Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular... 9 [5] H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure Appl. Math., 0 (979), no. 8, [6] J. R. Musselman, G. Rutledge, N. Anning, W. Leighton, V. Thebault, Problems for Solution: , The American Mathematical Monthly, Vol. 45, No. 9 (Nov., 98), 6-6. [7] OEIS, The online Encyclopedia of Integer sequences, Published electronically at njas/sequences, 008. [8] V.S. Shevelev, Necklaces and convex k-gons,indian J.Pure Appl.Math., 5,no.5,(004),

13 Availaible on line at Bulletin du Laboratoire 0 (05) 0-7 The Set Partitions: Solution for the sharing secret keys Sadek BOUROUBI, Fella CHARCHALI, Nesrine BENYAHIA TANI Faculty of Mathematics, Laboratory L IFORCE, University of Sciences and Technology Houari Boumediene (USTHB), B.P. El-Alia, Bab-Ezzouar, 6 Algiers, Algeria. Algiers University, Laboratory L IFORCE, Ahmed Waked Street, Dely Brahim, Algiers, Algeria. Abstract: Confidentiality was and will always remain a critical need in the exchanges either between persons or the official parties. Recently, cryptology has made a jump, from classical form to the quantum one, we talk about quantum cryptography. This theory, although is perfectly safe, there are still binding limits of implementation. In this paper, we developed a new cryptographic protocol, called BCB protocol, which will be used to provide random keys shared via a classical channel, using the set partitions. Each key can be long enough that the plain text in question, in purpose, for instance, to hide then to transmit the secret information using the Vernam cipher. Keywords: Cryptography, Vernam Cipher, BCB Protocol, Set partitions. Corresponding author: sbouroubi@usthb.dz.

14 The Set Partitions: Solution for the sharing secret keys Introduction Issues such as confidentiality and integrity of information have been solved by cryptography. The certificate that the Vernam cipher is unconditionally secure, has transformed the problem to ensure the confidentiality of information to a problem of distribution of the secret key used in the encryption process between two parties. Until the eighties, one way to distribute the secret key, apart from hand to hand, was to use algorithms whose security is based on the computational complexity. The keys generated by such algorithms are reasonably secret but not unconditionally secret. In the early seventies, Stephen Wiesner wrote conjugate coding [], describing the basis for a new concept that will be known to the world in the early eighty by quantum cryptography. Cryptography was attached to a quantum concept by the fact it relies on photons to transmit secret information instead of bits. Security is guaranteed not by mathematical theorems, but by the fundamental laws of physics as the Heisenberg uncertainty principle which asserts that certain quantities cannot be measured simultaneously. Charles H. Bennett (who knew about Wiesner s idea) and Gilles Brassard took the subject in 984 [4], where they show up to the world the first protocol of quantum key distribution whose security is unconditional because confidentiality is based on impossibilities imposed by the laws of physics [5]. This protocol was implemented in 989 over a distance of cm by calling efforts of F. Bessette, L. Salvail and J. Smolin, a full description of the prototype was published two years later [6]. All Quantum Key Distribution protocols consist of two phases [7]:. Initially one of the two parties sends to the other party quantum signals then perform certain measurements.. In a second time the two parties engage in classical treatment of measurement results. Concept of unconditional security - Vernam Cipher The Vernam cryptosystem, also known as the disposable mask or The One Time Pad Cipher, provides perfect security, despite its simplicity. In its classic form, it is nothing but a very long random sequence of letters, written on pages bound together to form a block. The sender uses each letter of the mask in turn to encrypt exactly one plain text character. The Vernam Cipher text C is a function of both the message M and the key K. The Vernam cipher was invented in 97 by an engineer of AT & T, Gilbert S. Vernam [9], who thought it would become widely used for automatic encryption and decryption of telegraph messages. The vernam cipher is a polyalphabetic substitution cipher belongs to secret key cryptosystems. The principle of the encryption algorithm is that if a random

15 S. BOUROUBI, F. CHARCHALI, N. BENYAHIA TANI key is added to a message, the bits of the resulting string are also random and bear no information about the message. If we use binary logic, unlike Vernam who worked with an alphabet of 6 letters, the encryption algorithm E can be written as: E K (M) = (M k, M k,..., M n k n ) mod, where M = (M, M,..., M n ) is the message to encrypt, and K = (k, k,..., k n ) is the key consisting of random bits. The message and the key are added bitwise modulo, i.e., the exclusive-or. Decryption process D of cipher text C is the same as encryption, it is given by: M = D K (C) = (C k, C k,..., C n k n ) mod. Perfect security is ensured via the concept of entropy introduced by Shannon in 949 []. Later, Vernam has been used in almost all military concerns. Vernam fits very well to the definition of a secret system [], a fact confirmed by the following theorem [8]: Theorem The Vernam cipher is unconditionally secure for any distribution of plain text. But just like any other cryptosystem, it has significant drawbacks which can cause its vulnerability such as the key must be as long as the message to encrypt; the cryptosystem becomes vulnerable if the same key is used more than once and the safest way to transport the key is the diplomatic bag which requires users from the diplomatic sector once. To remedy major drawbacks of this cipher, we propose here a new protocol, called BCB Bouroubi Charchali Benyahia 0, which is based on the set partitions concept. BCB Protocol The protocol BCB inspired from the quantum protocol BB84 Bennett Brassard 984 is based on the set partitioning problem which is NP-hard. The expected objective from the protocol is to product and to distribute a secret key via a classical channel, that will be used to ensure confidential communications between the participants by interchanging messages encrypted by the Vernam cipher. First, the two parties involved in the protocol must share π = {A, A,..., A k }, a partition of a set [n] = {,,..., n} into k-disjoint blocks (n is assumed to be large enough). Traditional protagonists who must run an exchange of information in cryptography are Alice and Bob. Both are involved in the sending and receiving secret messages and of course Eve, the intruder who wants to spy on Alice and Bob. Suppose Alice wants to send a message M to Bob, so steps to follow are:

16 The Set Partitions: Solution for the sharing secret keys. Alice calculates the number of characters L M of the message M to be encrypted.. Alice fixes the parameter m such that m = L M S, where S is a positive integer called amplification parameter, which we explain the role later.. Alice generates randomly a sequence of integers between and n of size m, and for each integer in the sequence, she sets in a list T A the index of the block to which this element belongs in the partition π. 4. Alice sends the parameter m to Bob. 5. Bob in turn generates a random sequence of integers between and n of size m, and for each integer in the sequence, he sets in a list T B the index of the block to which this element belongs in the partition π. 6. Bob sends the list T B to Alice. 7. Alice receives Bob s list, and compares it with hers. If there is correspondence, i.e, for the same index, she locates the same block in both lists, she puts a +, if not, she puts a. Doing so, she creates a new list, said T, whose elements are + and, then we have: +, if T A (i) = T B (i), T (i) =, i =,..., m., if T A (i) T B (i). 8. Alice interprets each + as the result of a function f (defined below) chosen from three functions (for example), acting on the elements of the corresponding block. The concatenation of these results provides the random secret key of length L C. To identify f, Alice takes the first + in the list T, writes in binary the block index corresponding to this +, let be j, then she considers the two first bit to the right. If the bits are: i) identical ( 00 or ) then the function f is interpreted as the sum of elements of the block A j. ii) 0 then the function f is interpreted as the product of the elements of the block A j. iii) 0 in this case, the function f is interpreted as the maximum element of the block A j. 9. Alice compares L M to L C. If L M L C, she sends the encrypted message and the sequence T to Bob, and then, Bob performs step 8 to get the same key. Otherwise, Alice must return to step and, at this level, she can keep the size m or modify it. Note that the generated keys by the protocol have been proved random, using statistical tests.

17 4 S. BOUROUBI, F. CHARCHALI, N. BENYAHIA TANI Absolute confidentiality requires a sharing of the key parameter π. The parameter π ensures that the resulting key is secret and is not known, only by legitimate users of the protocol. Therefore, the generated key by BCB offers the privacy of information transmitted, encrypted according to the Vernam cryptosystem, ensuring inability to decrypt what was encrypted by a spy. 4 Illustrative example In this section, we present an illustrative and didactic example to show how BCB protocol runs, in order to get random secret keys, which will be used for the plain texts ciphering. The first step, consists to generate a random partition of a set [n] into k-blocks (for any n and k to choose). The second consists of unrolling the BCB protocol then inject the provided key in Vernam, the adopted cryptosystem, to get out finally with the enciphered text. Suppose now, Alice wants to send an encrypted message to Bob. We consider first the following shared parameters between them: n = 0, k = and the partition π = {{}; {5}; {4}; {}; {0}; {}; {6, 8, }; {, 8}; {0}; {9, }; {5, 6, 9}; {7}; {4, 7}}. Let be it rains take the umbrella the secret message. The message written in binary form is: , with length L M = 6. Alice sets the parameter S at, it follows that m = 6 = 4. Alice generates her random sequence: {,, 4, 8, 0,, 6, 8, 8,,, 6, 5, 7, 4, 6, 8,,, 5, 7, 0,, 4, 7, 9,,, 6, 8,,, 6, 8, 0,, 5, 7,,, 5, 8, 0,, 4, 7, 9,, 4, 6, 8,,, 5, 6, 9,,, 6, 8, 4, 6, 9,,, 6, 8, 0,, 5, 7, 9,, 5, 7, 9,, 4, 6, 8,,, 6, 9, 0,, 5, 7, 9,, 4, 6, 9,,, 5, 7, 9,, 4, 6, 8, 9,, 4, 7, 9,, 4, 6, 8,,, 6, 8,,, 5, 7, 9,, 4, 6, 7, 9,,, 8, 0,, 4, 6, 8, 0,, 5, 7, 9,,, 5, 6, 8,,, 5, 7, 9,, 4, 6, 8, 0,, 5, 7, 8, 0,, 4, 7, 9,,, 5, 7, 9,,, 4, 6, 8, 9,,, 7,,,, 5, 8, 0, 6, 8, 0,, 5,7, 9,,, 6, 9,, 4, 6, 9,,, 6, 8,, 5, 7, 0,, 4, 6, 9,, 4, 6, 8,,, 5, 8, 0,, 5,,, 5, 8,,, 5, 8, 0,, 5, 7, 0,, 5, 7, 9,, 4, 7, 9,, 4, 6, 9,,, 6,, 5, 7, 0,, 4, 7, 9,, 6, 8, 0,, 5, 7, 9,, 4, 7, 9,, 4, 6, 9,, 4, 7, 9,,, 6, 7, 0,, 4, 6, 9,, 4, 7, 9,, 4, 6, 9,, 4, 0,, 5, 7, 0,, 4, 7, 9,, 6, 8, 0,, 5, 7, 9,, 4, 6, 8,,, 6, 8,, 4, 6, 9,, 4, 6, 8,,, 5, 8, 0,, 5, 7, 0,, 4, 7, 9,, 8, 0,, 5, 7, 0,, 4, 7,, 4, 6, 9,, 4, 6, 8, 0,, 5, 7, 0,, 5, 7, 0,, 5,, 4, 7, 9,, 4, 6, 9,,, 6, 8, 0,, 5, 8, 0,, 4, 7, 9,, 4, 6, 8, 0,, 5, 8,,, 6, 8, 0,, 5, 7, 9,, 4, 6, 8,,, 4, 8, 0,, 5, 7, 9,, 4, 6, 9,,, 8, 0,, 4}. For each integer in the sequence, she sets in a list T A the block index to which this element belongs in the partition π. T A = {7,,, 7, 5, 8,, 7, 0, 8,,,,,, 7,, 8,,, 5, 6,,,, 0, 4,, 7,, 8, 7, 8, 5, 6,,, 0, 4,, 7, 9, 7,,, 0,,, 7, 8, 0, 4,, 7,, 0, 4,, 7,,, 0,, 8, 7, 8, 5, 6,,,, 7,,, 0, 6,, 7, 8, 7, 4,, 0, 9, 8,,, 0, 6,, 7,, 0, 4,,,, 7,,, 7,, 0,,, 0, 6,, 7, 8, 0, 4,, 7,, 8,,, 0, 6, 7,, 0,, 8, 7, 9, 7,,, 7, 9, 7,,, 0,, 8,,, 7,, 8,,, 0, 6,, 7, 8, 5, 4,,, 8, 5, 6,,,, 0, 4,,,, 0, 4,, 7, 8, 0,, 8,, 0, 4, 6,, 7, 9,, 7, 9, 8,,, 0, 6, 8, 7,, 7,,, 0,, 8, 7, 8, 8,,,

18 The Set Partitions: Solution for the sharing secret keys 5 5, 6,, 7,, 0,,, 7,, 8,, 8, 5, 6,, 0, 4,, 7,, 8,, 8, 5, 6,,, 9, 7,,, 0, 6,,,, 7,,, 0,, 8, 7, 7,,, 9, 7,,, 0,,, 7, 9, 7,,, 0, 6,,,, 0,,, 0,,,,, 0, 4,,, 9, 7,,, 0, 6,,,, 7,,, 0,,, 5, 4,,, 9, 9,,, 0, 4,, 7, 9, 8,,, 0, 6,, 7, 8, 0, 4,, 7, 7,,, 7,, 0,,, 7,, 8,, 8, 5, 6,,, 9, 7,,, 0, 6, 8, 5, 4,,, 9, 7,,, 0,,, 0,,, 7, 8, 5, 4,,, 9, 8,,, 5, 4,, 7,,, 0, 6,, 7,, 0, 7,, 7, 9, 8,, 8, 5, 6,,,, 0,,, 7, 9, 8,, 8, 0, 4,, 7, 9, 8,,, 0, 6,, 7, 8, 0, 4,, 7, 9, 7,,, 0,,, 7,, 0, 4, 8, 5, 6, }. Alice sends m to Bob. Bob in turn generates his random sequence of length m: {6, 5, 0,, 6,,, 6, 8, 0, 8, 0, 4,,, 6, 8, 0,, 4, 5, 7, 9,,, 5, 6, 8, 0,, 4, 5, 7, 9,,, 4, 6, 8, 0,,, 5, 7, 9,,, 5, 7, 9,,, 5, 7, 8,, 4, 5, 8, 0,, 4, 6,, 4, 6, 9,,, 6, 8, 0,, 4, 7, 8, 0, 4, 9,,, 4, 7, 0,, 5, 7, 9,, 4, 6, 8,,, 5, 7, 0,, 4, 7, 9,,, 6, 6, 4, 7, 9,,, 6, 8, 0,, 5, 8,,, 5, 8, 0,, 5, 7, 0,, 4, 6, 9,,, 5, 8, 0,, 5, 7, 0,, 5, 7, 9,, 4, 6, 9,, 5, 7, 0,, 5, 7, 0,, 5, 7, 0,, 4, 7, 8,, 4, 9,, 4, 7,,, 7, 0,, 5, 8,,, 6, 9, 6, 8,, 9, 4, 7, 0,, 5, 8, 0,, 5, 7, 0,, 4, 6, 8,,, 5, 8, 0,, 4, 6, 8,,, 5, 8, 0,, 5, 7, 9,, 5, 7, 9,,, 6, 8, 0,, 5, 8, 0,, 4, 7, 9,, 4, 6, 8,,, 5, 7, 0,, 5, 7, 9,,, 5, 8, 9,, 4, 6, 8,,, 4, 6, 9,,, 6, 8, 0,, 4, 7, 9,,, 5, 7, 0,, 5, 7, 9, 0,, 4, 7, 9,,, 5, 7, 0,, 4, 6, 9,,, 5, 7, 9,,, 5, 7, 9,,, 6, 7, 0,, 4, 6, 8,,, 5, 7, 0,, 4, 6, 8, 0,, 6, 8, 0,, 5, 7, 9,,, 5, 8, 9,, 4, 7, 0,, 5, 7, 9,,, 5, 7, 0,, 4, 7, 8,,, 5, 7, 9,, 4, 6, 8, 0,, 5, 7, 9, 0,, 4, 6, 9,,, 6, 8, 0,, 4, 6, 8,,, 4, 7, 0,, 5, 7, 8,,, 5, 7, 0,, 4, 6, 8, 0,, 5, 7, 9,,, 6, 7, 9,,, 5, 8, 0,, 5, 7, 9,,, 4, 7, 9,, 4, 7, 8,,, 5, 7, 0,, 4}. For each integer in the sequence, Bob sets in a list T B the block index to which this element belongs in the partition π: T B = {7,, 9, 4, 7, 0, 8,, 8, 9, 7, 5,, 7, 4,, 7, 9, 0,,,,, 0, 4,, 7, 8, 5, 6,,,, 0,, 8,,, 7, 9, 7, 4,,,, 0, 6,,,, 0, 4,,, 8, 0,,, 7, 9, 8,,, 6,, 7,, 0, 4,, 7, 9, 7,,, 7, 9,, 0,, 8,,, 5, 6,,,, 0,,, 7,, 8,,, 5, 6,,,, 0, 4,, 7,,, 0,, 8, 7, 8, 5, 4,, 7,, 7,, 8, 5, 4,,, 9, 7,,, 0, 8,, 8, 5, 6,,, 9, 7,,, 0, 6,, 7,, 0,,, 5, 4,,, 9, 7,,, 5,,,, 8, 0,,, 0,,, 4, 8,, 9, 7,, 8, 0, 4,, 0, 7, 8, 7,,,, 9, 8,, 7, 9, 8,,, 5, 6,, 7, 8, 0, 4,, 7, 9, 7,,, 7,, 8,, 8, 5, 4,,,, 7,,, 0,, 8, 7, 8, 5, 4,, 7, 9, 7,,, 0,,, 7, 8, 0, 4,,, 9, 7,,, 0,, 8,, 8, 0, 6,, 7, 8, 0, 6,, 7,, 0, 4,, 7, 9, 7,,, 0,, 8,,, 5, 6,,,, 5, 4,,,, 0, 4,,, 9, 7,,, 0,, 8,,, 0,, 8,,, 0,, 8, 7,, 5, 6,, 7, 8, 0, 4,,, 9, 7,,, 7, 9, 8, 7, 8, 5, 4,,,, 0, 4,, 7,, 7,,, 5, 6,,,, 0, 4,,, 9, 0,,, 7,, 8,,, 0, 6,, 7, 8, 5, 6,,,, 5, 6,, 7,, 0, 4,, 7, 9, 7,,, 7,, 7,,, 5,,,, 8, 0, 4,,, 9, 7,,, 7, 9, 8,,, 0,, 8, 7,, 0,, 8,, 8, 5, 4,,,, 0, 4,,,, 7,,, 7,, 8,,, 5, 6, }. The first integer obtained by Alice is which belongs to the block A 7, where the second one is which belongs to the block A. While, the first integer obtained by Bob is 6 belongs to the block A 7, and the second is 5 belongs to the block A, and so on. Since we have the same index for the first integer in both sequences, Alice obtains the first +. By comparing the second integer obtained in both sequences, we can see that we have not the same index, so Alice put a in the second position. Doing so, Alice establishes the list T : T = {+, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, -, +, +, +, -, -, +, +, -, +, +, +, +, -, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, -, -, -, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, -, -, +, -, +, -, -, +, +, +, -, -, -, -, -, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -,

19 6 S. BOUROUBI, F. CHARCHALI, N. BENYAHIA TANI -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, -, +, +, +, +, -, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -,-, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, -, -, -, -, +, +, +}. As the first + refers to the block A 7 and 7 = in binary form, with the two first bit to the right are, then the function f will be the sum of elements of blocks, hence: f(a 7 ) = = 6. Therefore the provided key is: {6, 50,, 50,, 50,,, 0, 6,,, 50, 6,, 5, 6, 0,, 5,, 0,7, 6, 0,,, 6, 0, 4, 6,, 0, 50, 7, 6, 0,, 4}. Here the key is written in binary form, with length L C = : Since L C > L M, Alice has to encrypt the message. She gets the following encrypted text: Then, she sends the list T and the enciphered message to Bob. Using the list T, Bob gets the key, therefore, he obtains the plain text by performing the Xor operation (operating principle of Vernam) between the enciphered message and the key. 5 Conclusion Quantum cryptography ensures that the secret key is shared in confidential way and an unauthorized party has not copy, the Vernam Cipher, under restriction of eliminating its major drawbacks mentioned above, offers unconditional security of the encrypted message with assurance that without the possession the encryption key, it is impossible to decipher what has been encrypted. The BCB protocol, carries out two objectives: The first objective being the production of a random key at least as long as the message to be encrypted with assurance of the synchronization between the transmitter and the receiver. So, the constraint mentioned above will be removed. The second is the inability of a third person, said Eve, to determine the secret key generated in a reasonable time. This is because if Eve intercepts all data exchanged between Alice and Bob, she has no information on the partition π and the secret key. To determine π in order to find the key, Eve is opposite to the following problem: Find all the k-blocks of a set in the following form [k + i] = {,,..., k + i} i =,,..., for each obtained partition, unroll the protocol in order to generate all possible random keys and then, lead a exhaustive key search, to find the right key, which is not feasible in a reasonable time, at least during the lifetime of the shared secret between Alice and Bob.

20 The Set Partitions: Solution for the sharing secret keys 7 References [] C.E. Shannon. Communication Theory of Secrecy Systems. Bell System Technical Journal, Volume: 8, Page: , 949. [] D. Stebila. Classical Authenticated Key Exchange and Quantum Cryptography. Thesis for degree of doctor of philosophy in Combinatorics and Optimisation, University of Waterloo, Ontario, Canada, 009. [] S. Wiesner. Conjugate coding. unpublished manuscript circa 970; subsequently made available in SIGACTNews, Volume: 5, Number:, Pages: 78-88, 98. [4] G. Brassard. A Bibliography of Quantum Cryptography. Département IRO, Université de Montréal, Canada, December 99. [5] C.H. Bennett, G. Brassard. Quantum Cryptography: Public Key Distribution And Coin Tossing. Proccedings of the IEEE International Conference on Computers, Systems and Signal Processing, Pages: 75-79, Bangalore, India, 984. [6] C.H. Bennett, F. Bessette, G. Brassard, L. Salvail, J.A. Smolin. Experimental Quantum Cryptography. Journal of Cryptology, Volume:5, Numéro: Pages: -8, 99. [7] E. Karpov, T. Durt, F.V. Berge, N.J. Cerf, T. D Hondt. Cryptographie Quantique. Publication de Cryptax, Phase, cryptax.vub.ac.be, [8] S. Vaudenay. A Classical Introduction to Cryptography: Applications for Communications Security. Swiss Federal Institute of Technologies, Springer Science+Bussiness Media, 006. [9] G. S. Vernam. Cipher Printing Systems for Secret Wire and Radio Telegraphic Communications. J. AIEE 45, Pages: 09-5, 96.

21 Availaible on line at Bulletin du Laboratoire 0 (05) 8-7 An effective approach for integer partitions using exactly two distinct sizes of parts Nesrine BENYAHIA TANI, Sadek BOUROUBI, Omar KIHEL Algiers University, Ahmed Waked Street, Dely Brahim, Algiers, Algeria. USTHB, Faculty of Mathematics P.B. El-Alia, 6, Bab Ezzouar, Algiers, Algeria. Brock University Department of Mathematics and Statistics, 500 Glenridge Avenue St. Catharines, Ontario, Canada. tani.nesrine@univ-alger.dz or benyahiatani@yahoo.fr, sbouroubi@usthb.dz or bouroubis@yahoo.fr, okihel@brocku.ca Abstract: In this paper we consider the number of partitions of a positive integer n into parts of a specified number of distinct sizes. We give a method for constructing all partitions of n into parts of two sizes, as well as an explicit formula to count them with a new self-contained proof. As a side effect, by using the möbius function we also give a formula for the number of partitions of n into coprime parts. Keywords: Integer partitions, partitions into parts of different sizes, partitions into parts of two sizes, divisors number, Möbius function.

22 An effective approach for integer partitions using exactly two distinct sizes of parts 9 Introduction A partition of a positive integer n is a sequence of non increasing positive integers n (a times), n (a times),..., n s (a s times), with n i > n i+, that sum to n. We sometimes write the such partition π = (n a n a n as s ), each n i is called part of the partition π and a i its frequency. The partition function p(n) counts the partitions of n. If we ignore some unpublished work of G.W.V. Leibniz, the theory of integer partitions can find its origin in the work of L. Euler [6]. In fact, he made a sustained study of partitions and partition identities, and exploited them to establish a huge number of results in Analysis in 748. An excellent introduction to this subject can be found in the book of G. E. Andrews []. Definition Let π = (n a n a n a s s ) be a partition of n. We say that π is a partition into k parts with s distinct sizes if n = a n + + a s n s ; n > n > > n s ; a + + a s = k; a,..., a s. () Let t(n, k, s) be the number of solutions of system () and t(n, s) the total number of partitions of n into s distinct sizes. Then we have t(n, s) = n s(s ) k=s t(n, k, s). () Example Among 7 partitions of n = into distinct sizes, the partitions (7 4 ), (4 ), ( 4 ) and ( ) are the only ones which are into 5 parts. This kind of partitions appeared for the first time in the work of P. A. MacMahon [7]. Next, E. Deutsch presented the number of partitions of n into exactly two odd sizes of parts and the number of partitions of n into exactly two sizes of parts, one odd and one even. One can find these values in the Online Encyclopedia of Integer Sequences (OEIS) [8] as A7955 for the first number, A7956 for the second one and A00 for the number of partitions of n using only types of parts. In the work of Benyahia-Tani and Bouroubi [], we can find proof of effective and non-effective finiteness theorems on t(n, k, s). We can cite for example the following results:

23 0 N. BENYAHIA TANI, Z. YAHI, S. BOUROUBI Theorem For k s, n k + s(s ) and n max{k, s(s+) }, we have t(n, k, s) = n s(s ) k k s+ i= t(n ki, k j, s ), () j= t(n, k, ) = n k i= τ k (n ki), (4) where τ d (k) denotes the number of positive divisors of k less than or equal to d. Main Results One of the aim of this paper is to give an explicit formula for t(n, k, ) using an effective new approach. Thus, let consider the system: n = a n + a n ; a + a = k; n > n ; a, a. (5) and let m = n n throughout the remainder of the paper. First of all, we introduce the following lemma to prepare the main theorem. Lemma System (5) has integral solutions if and only if the following conditions are satisfied: (i) n n k(mod m), (ii) max (, ) n k m + χ(k n) n n k χ(k n), where χ(k n) = if k divides n, and 0 otherwise. Proof. From system (5), we have ( n n ) ( a ) ( n ) a = k

24 An effective approach for integer partitions using exactly two distinct sizes of parts Since m > 0, we can write ( a ) ( n n k ) a = m n + n k Then, system (5) has integral solutions if and only if m divides n n k, n n k > 0 and n + n k > 0. That is, n n k(mod m) and n k m < n < n k Since k can divide n, and n, the result holds. From this lemma, we can now derive the following theorem. Theorem For k, n max{k, }, d = gcd(n, k) and e d, let I e be the set of pairs (α, β) N, such that: α ( ) n k e and gcd α, k e =, β ( n e ) ( k e ) (mod α) and 0 β min ( α, n k χ(k n) ). Then t(n, k, ) = e d ( n ( k χ(k n) β max, n ) ) k + χ(k n) αe β +. α α (α,β) I e Proof. Put e = gcd(m, k) and let α = m, that is α ( ) n k e e and gcd α, k e =. By Lemma, case (i), we can see that e divides d, and n ( ) ( n k ) e e (mod α). Let 0 β < α, such that β ( n e ) ( k e ) (mod α). Then n = β + tα, t Z. Since 0 β < α and β n > 0, then t N and 0 β min It follows from Lemma, case (ii), that ( n ) n max, + χ(k n) m β + tα χ(k n) k k ( n ) α, χ(k n). k Finally, t(n, k, ) equals the number of positive integers t, such that ( max, n ) k + χ(k n) m β n k χ(k n) β t α α This completes the proof.

25 N. BENYAHIA TANI, Z. YAHI, S. BOUROUBI Remark 4 One nice application of Theorem concerns the following algorithm which allows us to generate all partitions of n using exactly two distinct sizes of parts. Algorithm Partitions into k parts with exactly two distinct sizes of parts Require: k, n max{k, } Ensure: Set of quadruple (n, a, n, a ), d gcd(n, k) for each divisor e of d do for α from to n k e do if gcd ( ) α, k e = then β ( ) ( n k ) e e (mod α) if β min ( α, ) n k χ(k n) then max(, t n k +χ(k n) αe) β n t α k χ(k n) β α for t from t to t do n β + tα n αe + n n n a k n n a k a end for end if end if end for end for This algorithm runs in O(n). Example Let n = and k = 8, then d = gcd(, 8) =. So, e = is the only one divisor of d. The values of α that satisfies α and gcd(α, 8) = are or.. For α =, we get β.8 (mod ) = 0, which is min(0, ). The pair (α, β) = (, 0) is then accepted and gives only one value of t: 0 max(, ) 0 t = + =. Therefore, we have only one partition corresponding to the pair (α, β) = (, 0). By applying Algorithm 4, we get: n =, n =, a = 5 and a =,

26 An effective approach for integer partitions using exactly two distinct sizes of parts and thus the partition ( 5 ).. For α =, we get β = min(, ), then the pair (α, β) = (, ) is accepted and gives the values: Thus the associated partition is (4 7 ). t =, n =, n = 4, a = 7 and a =. We get, finally t(, 8, ) =. Example Let n = and k = 8, then d = gcd(, 8) =. So, we have two divisors of d, e = and e =. Case: e =. The values of α that satisfies α 4 and gcd(α, 8) = are,, 5, 7, 9, or.. For α =, we get β = 0. The pair (, 0) is accepted and gives the values: and then the partition ( 6 ). t =, n =, n =, a = and a = 6,. For α =, we get β =. The pair (, ) is accepted and gives the values: and then the partition (5 6 ). t =, n =, n = 5, a = 6 and a =,. For α = 5, we get β = 4 > min(4, ), then the pair (5, 4) is rejected. 4. For α = 7, we get β =. The pair (7, ) is accepted and gives the values: and then the partition (8 6 ). t =, n =, n = 8, a = 6 and a =, 5. For α = 9, we get β = 5 > min(8, ), then the pair (9, 5) is rejected. 6. For α =, we get β = > min(0, ), then the pair (, ) is rejected. 7. For α =, we have β = 6 > min(, ), then the pair (, 6) is rejected. Case: e =. The values of α that satisfies α 7 and gcd(α, 8) = are,, 5 or 7.. For α =, we have β = 0. The pair (, 0) is accepted and gives t. Applying Algorithm 4, we obtain two partitions corresponding to the pair (, 0); the first one is ( 7 ) for t = and the second one is (4 5 ) for t =.

27 4 N. BENYAHIA TANI, Z. YAHI, S. BOUROUBI. For α =, we get β =. The pair (, ) is accepted and gives the values: t =, n =, n = 8, a = 7 and a =, and then the partition (8 7 ).. For α = 5, we get β = 4 > min(4, ), the pair (5, 4) is rejected. 4. For α = 7, we get β =. The pair (7, ) is accepted and gives the values: t =, n =, n = 5, a = 7 and a =, and then the partition (5 7 ). We get, finally t(, 8, ) = 7. After having counting the number t(n, k, s), it would be of considerable interest to explore the number of partitions of n into k parts with exactly s distinct coprime sizes, which we denote by g(n, k, s). Thus, let set g(n, s) = n s(s ) k=s g(n, k, s). (6) Theorem 5 For k s and n max{k, s(s+) }, we have g(n, k, s) = ( n ) µ t(d, k, s), (7) d d n where µ(.) denotes Möbius function. Proof. Let T (n, k, s) be the set of partitions of n into k parts with s distinct sizes and G(n, k, s) the subset of the such partitions but with s distinct coprimes sizes. We notice that, the mapping from the set T (n, k, s) to d n G(d, k, s) defined by: (n a n a n a s s ) (( n is a bijection, where δ = gcd(n, n,..., n s ). Consequently, we have δ ) a ( n δ ) a ( ns ) as ), δ t(n, k, s) = d n g(d, k, s). (8) Hence, the result follows by using the Möbius inversion formula.

28 An effective approach for integer partitions using exactly two distinct sizes of parts 5 { } Remark 6 Since t(d, k, s) = 0 if d < max k, s(s+) { extended only over all divisors d of n such that n max d take n = and k = 8, then, the summation in (7) can be }. For example, if we k, s(s+) g(, 8, ) = µ () t(, 8, ) + µ () t(, 8, ), and, according to Examples and, we get g(, 8, ) = 7 = 5. These partitions are: ( 7 ), ( 6 ), (5 6 ), (8 6 ) and (5 7 ). Using Theorems 5 and, we can construct the following table: n\k g(n, ) Table : g(n, k, ), k < n 0. From identity (7) we can see that if k n, then t(n, k, ) = g(n, k, ). In the present theorem we present this observation in a more explicit form. Theorem 7 For n max{, k} and k max{, n }, we have t(n, k, ) = g(n, k, ) = τ(n k) χ(n = k), where τ(n) denotes the number of positive divisors of n and χ(n = k) = if n = k, 0 otherwise.

29 6 N. BENYAHIA TANI, Z. YAHI, S. BOUROUBI Proof. Let us first notice that if k + max{, n n+ }, then k, and by Identity (4) the result yields (see [], Corollary ). Let now k = max{, n }. Since the result is true for n =, we can assume k = n. Let π = (na n a ) be a partition of n into k parts with two distinct sizes. If n is even, then n =, else n > (a +a ) n = k n n = n, a contradiction. Hence, n k = (n ) a, in which case n divides n k. So, for each divisor d of n k, we get n = d +, a = n k > 0 and a d = k n k > 0, except for d d =, where a = k n k = 0. Thus, the result yields. d Now, if n is odd, then n = or (n, n ) = (, ). Indeed, if (n = and n 4) or (n ), then n > a +a = k +a n + = n, a contradiction. In case of n =, by the same argument above, we get for each divisor d of n k, n = d +, a = n k > 0 d and a = k n k > 0, except for d =, where a d = k n k < 0, which is completed by d the partition ( n k k n ). This completes the proof. Remark 8 As shown in the proof above, the t(n, k, ) s partitions have been generated explicitly. Acknowledgements This research was partly supported by L IFORCE Laboratory. References [] T. M. Apostol, Introduction to Analytic Number Theory. Springer-Verlag, New York (976). [] G. E. Andrews and K. Eriksson, Integer Partitions. Cambridge University Press, Cambridge (004). [] N.B. Tani and S. Bouroubi, Enumeration of the partitions of an integer into parts of a specified number of different sizes and especially two sizes, Journal of Integer Sequences, 4, Article..6 (0). [4] A. Charalambos Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC (00). [5] L. Comtet, Advanced Combinatorics, pp. 75. D. Reidel Publishing Company, Dordrecht-Holland, Boston, (974). [6] E. Knobloch, Leibniz on combinatorics, pp Historia Math, (974).

30 An effective approach for integer partitions using exactly two distinct sizes of parts 7 [7] P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc.,, pp. 75 (99). [8] OEIS : The online Encyclopedia of Integer sequences. Published electronically at (008). [9] I. Pak, Partition bijections, a survey, Ramanujan Journal,, 5 75 (006). [0] H. Rademacher, On the partition function p(n), Proc. London, Math. Soc., 4, 4 54 (97). [] H. S. Wilf, Lectures on integer partitions, unpublished, edu / wilf.

31 Availaible on line at Bulletin du Laboratoire 04 (05) 8-5 Integer partitions into Diophantine pairs Zahra YAHI, Nesrine BENYAHIA TANI, Sadek BOUROUBI, Farid BENCHERIF 4 and Omar KIHEL 5 Abderahmane Mira University, Algiers University, Algiers, Algeria,, 4 USTHB University, Algiers, Algeria, 5 Brock University, Ontario, Canada. zahrayahi@yahoo.fr, tani.nesrine@univ-alger.dz or benyahiatani@yahoo.fr, sbouroubi@usthb.dz or bouroubis@yahoo.fr, fbencherif@usthb.dz, okihel@brocku.ca Abstract: Let n, a and b be positive integers. The pair (a, b) is called integer partition of n into Diophantine pair if n = a + b, ab + is a perfect square and a > b. In this paper we give, for any positive integer n, a closed formula of the number of integer partitions into Diophantine pairs, denoted by q D (n, ). Keywords: Integer partitions, Diophantine pairs, τ(n), q D (n, )

32 Integer partitions into Diophantine pairs 9 Introduction Let n be an integer, a partition of n is a non increasing sequence of positive integers n, n,..., n k whose sum is n, that is n = n + + n k, with n n k. Each n i is called a part of the partition. The function p(n) denotes the number of partitions of n. The study of partitions has fascinated several great mathematicians such as Leibniz, Euler, Legendre, Ramanujan, Hardy, Rademacher, Sylvester, Selberg and Dyson, who are interested in many, many number of partitions that satisfy some conditions, denoted p(n [condition]), such as Eulers s identity: p(n [odd parts]) = p(n [distinct parts]). Many other interesting problems in the theory of partitions remain unsolved up till now, for example despite a good deal of effort, is to find a simple criterion for deciding whether p(n) is even or odd. There is a vast literature on integer partitions, for more details see for instance [], [], [4], [5], [6], [0], [], [] and []. Now let s move on to another combinatorial concept. A set {a, a,..., a m } of m positive integers is called a Diophantine m-tuple if a i a j + is a perfect square for all i, j with i < j m. The set {,, 8, 0} was the first example of a Diophantine quadruple found by Fermat. A folklore conjecture says that there does not exist a Diophantine quintuple. Arkin, Hoggatt and Strauss [] shown in general, that for a given Diophantine triple {a, b, c}, the set {a, b, c, d + } is always a Diophantine quadruple, where d + = a+b+c+ abc+rst, and r, s, t are the positive integers satisfying ab+ = r, ac+ = s, bc+ = t. The such Diophantine quadruple is called regular. Dujella [8] proved that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. For more on Diophantine m-tuples results and its history, see for instance Dujella s webpage [9]. The purpose of this paper is to make a link between integer partitions and Diophantine m tuples, more precisely the main goal of this paper is to find the number of partitions of n into two distinct parts, forming a Diophantine pairs. Definition A partition of n into exactly two distinct parts, forming a Diophantine pair, is an integral solution of the following system: n = a + b, ab + = s, a > b. ()

33 0 Z.YAHI, N. BENYAHIA TANI, S. BOUROUBI, F. BENCHERIF, O. KIHEL Throughout this paper, q D (n, ) denotes the number of solutions of system (). Example Among 49 partitions of n = 00 into distinct parts, the partitions: 00 = , = , = 99 +, are the only ones which form a Diophantine pairs: = 50, = 49, = 0. The organization of this paper is as follows: In Sect. we give an upper bound for q D (n, ), the number of integer partitions into Diophantine pairs. In Sect. we collect some results to prepare the main result. Section 4 combines the lemmas from Sect to prove the main theorem. Finally, Sect. 5 gives some effective calculus with some examples. Upper bound for q D (n, ) Theorem For n, we have n + 4 q D (n, ) n +, where w and w denotes the floor and the ceiling of w respectively. Proof. From (), we have, b + nb + = s. () Since [ b < n/, and the function f : t t + nt + is increasing on the interval, n [, it follows n b + nb + < n Thus, we have Hence, the result follows. n + 4 n s <

34 Integer partitions into Diophantine pairs Some Lemmas We now present some preliminary results to prepare the main theorem. Let S n be the set of all solutions of the following Diophantine equation: where (x, y) N N with n x > 0 (n x) + (y) = n + 4, Lemma We have q D (n, ) = card(s n ). Proof. From () we get (n b) + (s) = n + 4. This shows that we have a - correspondence between S n and the set of all solutions of System (). Let define r N (n) and r Z (n) to be the number of solutions of the equation x + y = n, in N and Z respectively. It is clear that if n is not a perfect square, then we have: r N (n) = rz (n) () 4 In his work on number theory, C. G. J. Jacobi established the following result [7]: r Z (n) = 4(τ [4] (n) τ [4] (n)), (4) where, τ [4] (n) = d/n d (mod 4) and τ [4] (n) = d/n d (mod 4). Since n + 4 is never a perfect square, from () and (4) we get r N (n + 4) = τ [4] (n + 4) τ [4] (n + 4). (5) The next Lemma, shows that τ [4] (n + 4) = 0, for n. Lemma All odd divisors of n + 4 are congruent to modulo 4, for n.

35 Z.YAHI, N. BENYAHIA TANI, S. BOUROUBI, F. BENCHERIF, O. KIHEL Proof. Let v (n) denotes the -adic order of n. Then n + 4 = v (n +4) M, with M odd. The result holds if M =. Suppose M and let p be an odd prime divisor of M. Since p and are coprime, it exists u Z, such that u (mod p), and so Since n (mod p), we obtain from (6) 4u (mod p). (6) (un) (mod p). By using the first supplement to quadratic reciprocity, we get finally p (mod 4), which completes the proof. The following corollary is direct consequence from (5) and Lemma. Corollary 4 For any positive integer n, we have r N (n + 4) = τ(n + 4) + v (n + 4) 4 Main result We are now ready to formulate the main result as follows: Theorem 5 For n, we have q D (n, ) = τ(n + 4) + ( ) n+ + v (n + 4). Proof. Let T n = {(a, b) N : a + b = n + 4}. We distingue two cases: Case. If n is odd, let Tn = {(a, b) N : a + b = n + 4, a odd and b even}, Tn = {(a, b) N : a + b = n + 4, a even and b odd}. From corollary 4, we have card(t n ) = τ(n + 4). (7) It is clear that Tn Tn = and T n = Tn Tn. Then, card(t n ) = card(tn). (8) Notice that (x, y) S n if and only if (x, y) Tn \ {(n, )}. Which implies card(s n ) = card(tn). (9)

36 Integer partitions into Diophantine pairs It follows from (7), (8) and (9) card(s n ) = τ(n + 4) Case. If n is even, so let n = m, S n becomes the set of all solutions of the following Diophantine equation: (m x) + y = m +, where (x, y) N N with m x > 0. Let then, T m = {(a, b) N : a + b = m + }. Since m + is never a perfect square, we have. card(t m ) = τ(m + ) + v (m + ) Notice that (x, y) S n if and only if (x, y) T m \ {(m, )}. Then, card(s n ) = card(t m ) = τ(m + ) + v (m + ). Since n + 4 = 4(m + ), it exists M a positive odd integer such that: Then, Thus n + 4 = v (n +4) M and m + = v (m +) M. τ(m + ) + v (m + ) = τ(n + 4) + v (n + 4) card(s n ) = τ(n + 4) + v (n + 4). Finally, the Theorem holds by virtue of Lemma. Remark 6 If we note that Theorem 5 can be reformulated as follows: ( v n + 4 ) 0 if n, = if n, if 4 n, Theorem 7 For n, we have q D (n, ) = τ (n + 4) if n, τ (n + 4) 4 if n, τ (n + 4) if 4 n,

37 4 Z.YAHI, N. BENYAHIA TANI, S. BOUROUBI, F. BENCHERIF, O. KIHEL As an immediate consequence of Theorem 7, we obtain the following corollary: Corollary 8 For any positive integer n, we have τ ( n + 4 ) (mod ) if n, 0 (mod 4) if n, (mod ) if 4 n. 5 Effective calculus and some examples Example Let n = 000. We have n +4 =.5.89, so τ(n +4) = 8. From Theorem 7, we get q D (000, ) = 5. The such partitions are: 000 = , = , = , = , = , of course, all of these partitions verify the Diophantine condition: = 500, = 449, = 44, = 65, = 0. Example Let n = 09, a prime number. Since n +4 = 5.660, we get τ(n +4) = 6. Therefore, from Theorem 7, we obtain q D (09, ) =. The such partitions are: and 09 = , = , = 979, = 85. By using a computer algebra package, Theorem 7 allows us to obtain q D (n, ) for large values of n. The following table is introduced to illustrate a few:

38 Integer partitions into Diophantine pairs 5 n q D (n, ) 7 7 References [] G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge University Press, Cambridge, 004. [] J. Arkin, V. E. Hoggatt and E. G. Straus, On Euler s solution of a problem of Diophantus, Fibonacci Quart. 7 (979), -9. [] S. Bouroubi, Integer partitions and convexity, J. Integer Seq. 0 (007), Article [4] S. Bouroubi and N. Benyahia Tani, A new identity for complete Bell polynomials based on a formula of Ramanujan, J. Integer Seq. (009), Article [5] A. Charalambos Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, 00. [6] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, Dordrecht- Holland, Boston, 974, pp. 75. [7] L. E. Dickson, Introduction to the Theory of Numbers, Dover Publications, Inc., New York, 99. [8] A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (004) 8-4. [9] A. Dujella, Diophantine m-tuples, duje/ dtuples.html. [0] H. Rademacher, On the partition function p(n), Proc. London, Math. Soc., 4 (97), [] I. Pak, Partition bijections, a survey, Ramanujan J., (006), [] R.P. Stanley, Enumerative Combinatorics, Vol., Wadsworth, 986. [] H. S. Wilf, Lectures on integer partitions, wilf/pims /PIMSLectures.pdf.

39 Availaible on line at Bulletin du Laboratoire 05 (05) 6-6 Packing coloring of some undirected and oriented coronae graphs Daouya LAÏCHE,IsmaBOUCHEMAKH, Eric SOPENA, Faculty of Mathematics, Laboratory L IFORCE, USTHB, B.P. El-Alia, Bab-Ezzouar, 6 Algiers, Algeria, Univ. Bordeaux, LaBRI, UMR5800, F-400 Talence, France. Abstract: The packing chromatic number χ ρ (G) of a graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V,..., V k, in such a way that every two distinct vertices in V i are at distance greater than i in G for every i, i k. For a given integer p, the generalized corona G pk of a graph G is the graph obtained from G by adding p degree-one neighbors to every vertex of G. In this paper, we determine the packing chromatic number of generalized coronae of paths and cycles. Moreover, by considering digraphs, and the (weak) directed distance between vertices, we get a natural extension of the notion of packing coloring to digraphs. We then determine the packing chromatic number of orientations of generalized coronae of paths and cycles. Keywords: Path, Cycle Packing coloring, Packing chromatic number, Corona graph, Corresponding author: Eric.Sopena@labri.fr.