Integer partitions into arithmetic progressions

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1 7 Integer partitions into arithmetic progressions Nesrine Benyahia Tani Saek Bouroubi ICM 01, March, Al Ain Abstract Every number not in the form k can be partitione into two or more consecutive parts. Thomas E. Mason has shown that the number of ways in which a number n may be partitione into consecutive parts, incluing the case of a single term, is the number of o ivisors of n. This result is generalize by etermining the number of partitions of n into arithmetic progressions with a common ifference r, incluing the case of a single term. Keywors: Integer partitions, Arithmetic progression, ivisors of an integer. 1 Introuction Let n be an integer. A partition of n is an integral solution of the system: { n = n1 + + n k, 1 n 1 n k. The positive integers n 1,..., n k are calle parts, an k is the length of the partition. In partition ientities, we are often intereste in the number of partitions that satisfy some conitions. For example, the number of partitions into o parts, the number of partitions into even parts, the number of partitions into even an o parts [5], the number of partitions into istinct parts, an so on. For more on integer partitions the reaer is hereby invite to see for instance [3], [4], [6], [7], [8] an [10]. Thomas E. Mason [9] was intereste in the number of ways in which a number n may be partitione into consecutive parts, incluing the case of a single term, an he has shown that this number equals the number of o ivisors of n. Let, n = α p α 1 pαr where the p s are istinct o primes an the α s are the powers for which they occur. Then the number of ways in which a number n may be partitione into consecutive parts, incluing the 1 pα case of a single term, equals r i=1 (α i + 1), except in the case n = α where the number of ways is 1. In this paper we treat the case of partitions of n into an arithmetic progression. For 1 r n, if n i n i 1 = r, i =,..., k, we say that we have a partition into an arithmetic progression with a common ifference r. Our problem can be formulate in terms of partitions: For a given positive integer n, what is the number of partitions of n into an arithmetic progression with a common ifference r? r, Both authors were supporte by LAID3 Laboratory

2 8 Integer partitions into arithmetic progressions Arithmetic progressions Partitioning n into an arithmetic progression with a common ifference r means that there exist two positive integers l 1 an m 0, such that Let us enote such a partition briefly by π(l, m). From (1), n = l + (l + r) + (l + r) + + (l + mr). (1) Hence, n = (m + 1)l + m(m + 1) r. The solution of equation () in m yiels rm + (l + r)m + l n = 0. () m = (l + r) + (l r) + 8rn (3) r Since m is an integer, (l r) + 8rn is a perfect square, i.e., Then, (l r) + 8rn = u. (4) Putting, rn = (u (l r)) (u + (l r)) A = u (l r) an B = u + (l r), we have A + B = u, an (5) B A = l r. (6) By (3), (5) an (6) we get Consequently, r ivies A. Hence r + B A an m = A r (7) r n = A r B. (8) Now we iscuss the parity of r.

3 N.Benyahia-Tani, S.Bouroubi 9.1 Arithmetic progressions with an o common ifference r By (4) u is o if r is o. Then in view of (5), A an B must have ifferent parity. Case 1. If A is even then B is o an n = A B. In this case we have r where = B, is an o ivisor of n. r + rn an m = n 1, Since l 1, must verify (n )r ( ), i.e., r + (r ) + 8rn. Case. If A is o then A r is o as well an n = A r B. In this case we have r + n r an m = 1, where = A, is an o ivisor of n. r Because l 1, must verify ( 1)r (n ), i.e., r + (r ) + 8rn. r For every such o ivisor of n there exists a partition into an arithmetic progression with an o common ifference r, an vice versa. A moment s reflection will show that an o ivisor of n cannot satisfy the inequalities (n )r ( ), ( 1)r (n ) simultaneously, otherwise ( 1)r ( ), a contraiction.. Arithmetic progressions with an even common ifference r By (4) u is even, if r is even. Then in view of (5), A an B must have the same parity. Since r ivies A, A an B are even. Hence from (8) n = A r B. From (7) we get r + an m = n 1, where = B, is a ivisor of n. Since l 1, must verify (n )r ( 1), i.e., r + (r ) + 8rn. 4

4 30 Integer partitions into arithmetic progressions Now, we are able to formulate our results as follows: Theorem 1 Let n 3 be a positive integer an let 1 r n be an o integer. Then the number of partitions of n into an arithmetic progression with an o common ifference r, incluing the case of a single term, is the number of o ivisors of n, satisfying ( n ) r ( ) or ( 1)r (n ) an for every such o ivisor, the partition π(l, m) is given by: r + n r an m = 1 if ( 1)r (n ), r + rn an m = n 1 if (n )r ( ). Example An example illustrating the above theorem is the following: Let n = 15 an r = 3. The o ivisors of 15 satisfying (n )r ( ) or ( 1)r (n ) are 1, 3 an 15, so 15 amits three partitions into an arithmetic progression of the common ifference 3, each one is associate with one of these ivisors an this can be shown as follows: =1 satisfies ( 1)r (n ). We have r + n r = 15 an m = 1 = 0. Hence π(l, m) = 15. =3 satisfies ( 1)r (n ). We have r + n r Hence π(l, m) = =15 satisfies ( n ) r ( ). We have r + rn Hence π(l, m) = 6+9. = an m = 1 =. = 6 an m = n 1 = 1. Theorem 3 Let n 3 be a positive integer an let 1 r n be an even integer. Then the number of partitions of n into an arithmetic progression with an even common ifference r, incluing the case of a single term, is the number of ivisors of n, satisfying (n )r ( 1) an for every such ivisor, the partition π(l, m) is given by: r + an m = n 1. Example 4 An example illustrating the above theorem is the following: Let n = 30 an r = 4. The ivisors of 30 satisfying (n )r ( 1) are 10, 15 an 30, so 30 amits three partitions into an arithmetic progression of the common ifference 4, each one is associate with one of these ivisors an this can be shown as follows: =10 =15 =30 r + r + r + = 6 an m = n 1 =, hence π(l, m) = = 13 an m = n 1 = 1, hence π(l, m) = = 30 an m = n 1 = 0, hence π(l, m) = 30.

5 N.Benyahia-Tani, S.Bouroubi 31 3 Why oes Theorem1 generalizes Mason s theorem? Theorem 1 is an extension of Masons theorem [9]. Inee, for r = 1 an for every o ivisor of n, one an only one of the two inequalities ( 1)r (n ), (n ) r ( ) necessarily hols. In fact, for r = 1, = p + 1 an n = k, imply ( 1) (n ) p k 1, an Hence, if p k 1, we get n ( ) p k. otherwise 1 + k 1 + k an m = 1, an m = k 1. Example 5 For example, n = 15 has four o ivisors: 1, 3, 5 an 15: =1 p = 0 an k = 15, then 1 + k =3 p = 1 an k = 5, then 1 + k =5 p = an k = 3, then 1 + k =15 p = 7 an k = 1, then 1 + k = 15 an m = 1 = 0, hence π(l, m) =15. = 4 an m = 1 =, hence π(l, m) = = 1 an m = 1 = 4, hence π(l, m) = = 7 an m = k 1 = 1, hence π(l, m) =7+8. References [1] S. BOUROUBI, N.BENYAHIA-TANI, Integer Partitions into Arithmetic Progressions with an O Common Difference, INEGERS, Electronic Journal of Combinatorial, Number Theory, Vol. 09, (009). [] S. BOUROUBI, N.BENYAHIA-TANI, Integer Partitions into Arithmetic Progressions, RO- MAKO, Rostock. Math. Kolloq. 64, (009). [3] G. E. Aews, K. Eriksson, Integer partitions. Cambrige University Press, Cambrige, (004). [4] A. Charalambos Charalambies, Enumerative Combinatorics, Chapman & Hall/ CRC, (00). [5] Y. Guo an X. Zhang, Partitions of a positive integer as the sum of even an o numbers, J. Univ. Electron. Sci. Technol. China 35, (006).

6 3 Integer partitions into arithmetic progressions [6] I. Pak, Partition bijections, a survey, Ramanujan Journal, 1, 5-75 (006). [7] H. Raemacher, On the partition function p(n), Proc. Lonon, Math. Soc. 43, (1937). [8] R. P. Stanley, Enumerative Combinatorics, Vol. 1, Wasworth, (1986). [9] Thomas E. Mason, On the Representation of an Integer as the Sum of Consecutive Integers, The American Mathematical Monthly, Vol. 19, N. 3, (191). [10] H. S. Wilf, Lectures on integer partitions. (unpublishe), available at upenn. eu / wilf. Nesrine Benyahia Tani Algiers 3 University, Faculty of Economics, Commercial an Management Sciences, Ahme Wake Street, Dely Brahim, Algiers, Algeria benyahia tani.nesrine@univ-alger3.z or benyahiatani@yahoo.fr Saek Bouroubi University of Science an Technology Houari Boumeiene, Faculty of Mathematics, P.Box El-Alia, Bab-Ezzouar, Algiers, Algeria sbouroubi@usthb.z or bouroubis@yahoo.fr