Relatively Prime Uniform Partitions

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1 Gen. Math. Notes, Vol. 13, No., December, 01, pp.1-1 ISSN ; Copyright c ICSRS Publication, 01 Available free online at Relatively Prime Uniform Partitions A. Davi Christopher 1 an M. Davamani Christober 1, Department of Mathematics The American College Tamilnau, Inia 1 avchrame@yahoo.co.in jothichristopher@yahoo.com (Receive: / Accepte: Abstract A partition of a positive integer n is a sequence of non increasing positive integers say λ = (λ 1,, λ 1 (f 1 times,, λ k,, λ k (f k times, with λ i > λ i+1, whose sum equals n. If gc(λ 1,, λ k = 1, we say that λ is a relatively prime partition. If f i = 1 i = 1,,, k, we say that λ is a istinct partition. If f i = f j i j, then we say that λ is an uniform partition. In this note, the class of partitions which are both uniform an relatively prime are analyze. They are foun to have connections to istinct partitions. Furthermore, we term a partition of a certain kin as Conjugate Close (abbreviate CC if its conjugate is also of the same kin. Enumeration of CCuniform partitions, CC-relatively prime partitions, an CC-uniform relatively prime partitions are stuie; these enumeration formulas involves functions from multiplicative number theory such as ivisors counting function an Lioville s function. Keywors: Relatively prime partitions, uniform partitions, conjugate, conjugate close partitions. 1 Introuction an Definitions The stuy of partition ientities was initiate by the mathematician L. Euler []. Much of the work one in partition theory since its beginnings has centere on the problem of fining asymptotic estimate, congruence ientities,

2 A. Davi Christopher et al. reccurence relations, parity results an equinumerous results for the counting function of a efine class of partitions. In this sequence, we introuce a new kin of partitions namely uniform partitions, which are partitions with ientical number of occurences of each part, an we establish its connection to earlier efine partitions. The enumeration of uniform partitions in finite sets was stuie by Karen Meagher an Lucia Moura [4]. Present efinition of uniform integer partition is quite an analogue to the efinition of uniform set partition. The motivation of the present work stems from the work of Mohame El Bachraoui. In [6], Mohame El Bachraoui iscuss a kin of partitions calle relatively prime partitions, which are partitions whose parts form a relatively prime set. The iea of relatively prime set was foun by M. B. Nathanson [7]. We intertwine the efinition of relatively prime partition with uniform partition, an fin new results. We stuy the enumeration of relatively prime partitions that are uniform an we fin that: number of relatively prime partitions that are uniform is equal to the number of istinct partitions. Moreover, we term a partition of a certain kin as conjugate close if its conjugate is also of the same kin. In the latter part of this article, we are concerne with the enumeration of conjugate close uniform partitions, conjugate close uniform partitions that are relatively prime, conjugate close uniform partitions that are self conjugate, etc., Now, we state some basic efinitions that are require for our stuy. Let n an k be integers with k n. By a partition of n, we mean a sequence of non increasing positive integers say λ = (λ 1,, λ 1 (f 1 times,, λ k,, λ k (f k times whose sum equals n, where λ i s are istinct integers. We sometimes write λ = (λ f 1 1 λ f λ f k ( k. Each λ i is calle part of the partition λ. If λ = λ f 1 1 λ f k k, then we say that λ has precisely k iferent sizes namely λ 1,, λ k. The value of f i is terme as frequency of the part λ i. We efine p(n to be the number of partitions of n. Definition 1. Let λ = (λ 1 f 1 λ f λ k f k be a partition of n. We say that λ is uniform if all f i s are ientical. We efine u(n to be the number of uniform partitions of n. Example. Among the partitions of 4, the partitions (1 4, (, ( , an (4 1 are uniform partitions, whereas the partition (1 1 is not an uniform partition of 4. Definition 3. Let λ be a partition of a positive integer n. We say that λ is relatively prime if its parts form a relatively prime set. We efine p ψ (n to be the number of relatively prime partitions of n. Example 4. Among the partitions of 4, the partitions (1 4, ( , an (1 1 are relatively prime partitions, whereas the partitions(4 1, an ( are not relatively prime partitions of 4.

3 Relatively Prime Uniform Partitions 3 Definition 5. A partition say λ of n is sai to be relatively prime uniform partition if it is both uniform an relatively prime. We efine u ψ (n to be the number of relatively prime uniform partitions of n. Example 6. Among the partitions of 4, the partitions (1 4, an ( are relatively prime uniform partitions, whereas the partition (1 1 is relatively prime but not uniform, an the partitions ( an (4 1 are uniform but not relatively prime partitions. Definition 7. Let λ = ( λ f 1 1 λ f k k be a partition of n. If gc(f 1,, f k = 1, then we say that λ has relatively prime frequencies. We efine f ψ (n to be the number of partitions of n with relatively prime frequencies. Example 8. Among the partitions of 4, the partitions ( , (1 1, an (4 1 have relatively prime frequencies, whereas the partitions (1 4, an ( are partitions with non relatively prime frequencies. Given a partition λ = ( λ f 1 1 λ f k k of n. There is a simple way of representing λ geometrically by using isplay of n lattice points calle a graph. The construction of the graph of λ goes as follows: first f 1 rows shoul have λ 1 points; next f rows shoul have λ points, an so on. For example, the partition of 15 given by ( can be represente by using 15 lattice points arrange in five rows as follows: If we rea this graph vertically, we get The corresponing partition to this graph is λ = ( The partition λ is sai to be conjugate of λ. Definition 9. Let A be a set of partitions of n. We say that λ A is conjugate close in A (abbreviate CC if λ A. We say that λ is self conjugate if, λ = λ.

4 4 A. Davi Christopher et al. Base on the above mentione efinitions, we erive several convolution ientities, parity results, an equinumerous results. This is the core of section. In section 3, we confine to the enumeration of CC-partitions of the above mentione kins of partitions. Throughout this paper, we enote the number of istinct partitions of n (partitions with istinct parts by q(n; number of istinct partitions of n with k parts by q(n, k; number of istinct relatively prime partitions of n by q ψ (n; number of relatively prime istinct partitions of n with exactly k parts by q ψ (n, k, an the number of positive ivisors of n by τ(n. Main Results.1 Convolution Ientities In this section, we erive several convolution ientites relating the functions u(n; q(n; u ψ (n, an q ψ (n. Theorem 10. Let n be a positive integer. Then we have: (i u(n = n q (1 (ii q(n = n µ(u ( (iii u ψ (n = n q ψ (3 (iv q ψ (n = n µ(u ψ (4 (v u(n = n τ(q ψ (5 where µ is the möbius function.

5 Relatively Prime Uniform Partitions 5 Proof. Let U( n be the set of all uniform partitions of n. We efine F (λ = f when λ = λ f 1λ f λ f k. In U n, efine a relation say R in the following way: λ 1 Rλ if, an only if, F (λ 1 = F (λ. Clearly, R is an equivalence relation. Further, we observe that, each equivalence class of R equals the set {λ U n : F (λ = f} for some f n; we enote this set by F f. We notice that, the mapping ( λ f 1λ f λ f k (λ 1 λ λ k is a bijection. Conse- from the set F f to the set of all istinct partitions of ( n f quently, we have F f = q n f. Furthermore, u(n = U n = f n F f ; hence, (i follows. As an application of möbius inversion formula, one can get (ii from (i, an (iv from (iii. Similar proof goes well to (iii. To prove (v, following ientity is require: q(n = n q ψ (6 Now, we establish the ientity 6; to that en, we enote the set of all istinct partitions of n by Q n, an let λ = (λ 1,, λ k Q n. Define gc(λ = gc(λ 1,, λ k. Further, we efine a relation say R p in Q n in the following way: λ 1 R p λ if, an only if, gc(λ 1 = gc(λ. Clearly, R p is an equivalence relation, an each equivalence class of R p equals the set {λ Q n : gc(λ = } for some n; we enote this set by R p. Then, from the equation: λ i = λ i =, where λ i s are istinct an gc(λ 1, = 1, it follows that R p = q n ψ. Since n R p = Q n = q(n; (6 follows. As an application of the following well known ientity: one can get (v from (6 an (i. f(k = τ(f n k n P. A. MacMahon [5] stuie the partition function t(n, k, s which counts the number of partitions of n into k parts of precisely s ifferent sizes. Recently, Nesrine Benyahia Tani an Saek Bouroubi [8] bring out an exciting expression for t(n, k,. Here, similar enumeration theorems were prove. Following result is concerne with uniform partitions of n into parts of precisely s ifferent sizes. Total number of such partitions is enote by u(n, s. Theorem 11. Let n an s be two positive integers with n s(s+1. Then we have: (7

6 6 A. Davi Christopher et al. (i u(n, s = n q, s (8 (ii q(n, s = n µ(u, s (9 (iii u(n, s = n τ(q ψ, s (10 (iv u(n, = n n 1 (11 Proof. Define the relation R as it is in the proof of theorem 10 on the set of all uniform partitions of n with precisely s ifferent sizes. Then, it is clear that, each equivalence class of R contains precisely q, s elements for some k n, k provie n s(s+1. Thus, (i follows. As an application of möbius inversion k formula, one can get (ii from (i. Since n k f(k = n τ(f ( n for any arithmetical function f, an q(n, s = n q ψ, s, one can get (iii from (i. Since q(n, = n 1, one can get (iv from (i. An aitional constraint on uniform partitions with exactly k parts yiels new results. Definition 1. Let n, k an be three positive integers such that n k. We efine the function, u(n, k,, to be the number of uniform partitions of n with exactly k parts an least part greater than or equal to. Theorem 13. Let n be a positive integer. Then we have: (i u(n, k, 1 = ( f n,f k q n, k f f (ii u(n, k, = u(n ( 1k, k, 1 for > 1 (iii u(n, k, = ( f n ( 1k,f k q n ( 1k, k f f Proof. Let U,k, 1 be the set of all uniform partitions of n with exactly k parts, an let λ = λ f 1λ f λ f r U n,k, 1. Then eviently, r f = k therefore, f k. Since fλ i = n, we have: f n. Define relation R as it is in the proof of theorem 10 on the set U n,k, 1. Then eviently, R is an equivalence relation,

7 Relatively Prime Uniform Partitions 7 ( n an each equivalence class of R contains q, k elements where f is some f f ivisor ( of both n an k. This enumeration can be ha from the equation f λ 1 + λ + + λ k = n; λ i s being istinct positive integers. Thus, (i f follows. Let U n,k, be the set of all uniform partitions of n with exactly k parts an least part greater than or equal to, where > 1 is a positive integer. If (λ 1, λ,, λ k U n,k,, then (λ 1 1,, λ k 1 U n k,k, 1. Furthermore, if (λ 1,, λ k U n k,k, 1, then (λ 1 + 1,, λ k + 1 U n,k,.thus, the mapping f : U n,k, U n k,k, 1 efine by f((λ 1,, λ k = (λ 1 1,, λ k 1 has inverse function. Consequently, u(n, k, = U n,k, = U n k,k, 1 = u(n k, k, 1. Repeate application of this equality for 1 times gives (ii, an an application of (i in (ii gives (iii. Corollary 14. Let n an k be two coprime integers with k n. Then u(n, k, 1 = q(n, k an u(n, k, = q(n ( 1k, k. Parity Results Theorem 15. Let n be a positive integer. Then we have: { 1( mo if n,= u(n 3k ±k I( is o 0( mo if n,= 3k ±k I( is even (1 where unit function I is efine by I(n = 1 for every positive integer n. Proof. Partition theoretic version of Euler s pentagonal number theorem [] (for proof see [3] states that e (n o (n = { ( 1 k if n = 3k ±k, 0 otherwise (13 where e (n (resp. o (n enotes the number of istinct partitions of n with even (resp. o number of parts. Consequently, we have q(n { 1( mo if n = 3k ±k, 0( mo otherwise (14 Since u(n = n q(; the result follows.

8 8 A. Davi Christopher et al. Theorem 16. Let p be a prime number. Then we have q ψ (p m 0( mo m 3 (15 Proof. From Euler s pentagonal number theorem it follows that q(n { 1( mo if n = 3k ±k, 0( mo otherwise (16 We claim that, q(p m 0( mo for all positive integer m. For otherwise, we have: p m = k(3k ± 1 for some positive integer k. When p =, we have k = α for some positive integer α with α m; this gives 3k ± 1 = 3. α ± 1. Consequently, 3k ± 1 oes not ivie m+1 ; a contraiction. If p is an o prime number, then k = 1 is not an actual possibility. Therefore, ivies either k or 3k ± 1. Also, since m, the case k = will be rule out. Therefore, if ivies k, then we have k = p α for some positive integer α with α < m, which in turn implies that, 3k ±1 = 3 p α ±1. Consequently, 3k ± 1 oes not ivie p β for any positive integer β > 1; a contraiction. In case if, ivies 3k ± 1, then we have 3k ± 1 = p α for some positive integer α with α < m, an k = p β for some positive integer β > 1; this is absur. Thus, claim follows. Since q ψ (p m = q(p m q(p m 1 ; the result follows..3 Equinumerous Theorems In [], L. Euler prove that: number of partitions of n with istinct parts is equal to the number of partitions of n with o parts. In this section, we give two such equinumerous results; these results are the irect consequence of previously prove convolution ientities. Theorem 17. Let n be a positive integer. Then we have: (i (ii p ψ (n = f ψ (n (17 q(n = u ψ (n (18 Proof. In [6], Mohame El Bachraoui prove that: p(n = p ψ n (19

9 Relatively Prime Uniform Partitions 9 Then from möbius inversion formula it follows that: p ψ (n = µ(p n (0 We efine f ψ (n to be the number of partitions of n with greatest ( common ivisor of its frequencies equals. Then, we have f ψ (n = f n ψ. The valiity of this equality follows from the equation: (f i λ i = ( f i λ i = n where gc(f 1, f, = 1. Moreover, we have: n f ψ(n = p(n. Thus, p(n = n f ψ. By möbius inversion formula, we get that f ψ (n = n µ(p (1 Then from (0 an (1, we get (i. To prove (18, we nee the following ientity u(n = n u ψ ( ( Proof of the above ientity is similar to the proof of the ientity (6. Application of möbius inversion formula in ( gives u ψ (n = n µ(u (3 Then comparison of (3 with (ii of theorem 10 gives (ii. 3 Enumeration of Conjugate Close Partitions In the following result, we give a beautiful expression for enumeration of CC uniform partitions of n. Further, we show that relatively prime uniform partition of a positive integer n is conjugate close only when n is a triangular number. Theorem 18. Let A be the class of uniform partitions of a positive integer n. Then the number of CC-uniform partitions in A is equal to ( n τ T k n T k where T k enote the k th triangular number.

10 10 A. Davi Christopher et al. Proof. We consier a uniform partition λ of n. If frequency of each part of λ is k, that is, λ = ( a k 1a k ar k, then the graph of λ got the following form: first k rows has a 1 points; next k rows has a points;, an last k rows has a r points. An conjugate of λ got the following form: first a r rows has rk points; next a r 1 a r rows has (r 1k points; next a r a r 1 rows has (r k points, an so on, finally, a 1 a rows has k points. If conjugate of λ is uniform, then we have: a 1 a = a a 3 = a 3 a 4 = = a r 1 a r = a r = s (say. This implies: a 1 = rs, a = (r 1s,, a r 1 = s, a r = s. Thus, n = k s T r, where T r is the r th triangular number. Conversely, if n is of the form n = k s T r, then we can fin a uniform partition of n namely λ = ( (sr k (s(r 1 k s k. Clearly λ is conjugate close. Thus enumeration of uniform partitions of n that are conjugate close equals the enumeration of (k, s, r with n = k s T r. We notice that enumeration of the triplets (k, s, r with n = k s T r equals ( T τ n r n T r. Thus, the result follows. Corollary 19. Let n be a positive integer. If no triangular number ivies n other than 1, then number of uniform partitions of n which are conjugate close is equal to τ(n Theorem 0. If n is a positive integer, then relatively prime uniform partition of n will be conjugate close only when n is a triangular number. In particular, if n is a triangular number, then there exist only one relatively prime uniform partition which is conjugate close, an if n is not a triangular number, then there will be no conjugate close relatively prime uniform partition of n. Proof. Let λ = ( a k 1a k a k r be a conjugate close relatively prime uniform partition. Aopt the notation as it is in the first paragraph of the proof of theorem 18. Since λ is relatively prime; we get s = 1. An, since λ is conjugate close; we get k = 1. Thus, the result follows. To present the next theorem we nee Liouville s function λ(n which is efine as follow: λ(n = ( 1 g(n where g(n is the number of prime factors of n counte with multiplicity. Theorem 1. Let A be the set of all uniform partitions of n. Then the number of CC- partitions in A that are self conjugate is equal to ( n τ(λ T r T r n n Tr where T r enote the r th triangular number.

11 Relatively Prime Uniform Partitions 11 Proof. Let λ = ( a k 1a k ar k be a uniform partition which is self conjugate. Aopt the notations as it is in the first paragraph of the proof of theorem 18. Since λ is self conjugate; we get s = k. Thus, enumeration of self conjugate uniform partitions of n equals the enumeration of (k, r with n = k T r. We notice that, enumeration of (k, r with n = k T r equals ( T τ n r n T r, where τ (m counts the number of square ivisors of m. Also, it is well known that, τ (m = m k λ(k = m τ(λ ( m. Thus, the result follows. Theorem.. Let A be the set of all partitions of n with relatively prime frequencies. Then the number of CC-partitions in A is equal to µ(p ψ n ( Proof. Let λ = λ f 1 1 λ f λ fr r be a partition of n with relatively prime frequencies. Then the graph of λ has f 1 rows with λ 1 points; f rows with λ points, an so on. Further, we observe that, conjugate of λ has λ r rows with (f 1 + f + + f r points; λ r 1 λ r rows with (f 1 + f + + f r 1 points, an so on, finally λ 1 λ rows have f 1 points. That is, frequencies of conjugate of λ are λ r, λ r 1 λ r,, λ 1 λ. It is not har to see that, gc(λ 1, λ,, λ r = 1 if, an only if, gc(λ r, λ r 1 λ r,, λ 1 λ = 1. Accoringly, we have the following equivalence: conjugate of λ will have relatively prime frequencies if, an only if λ is a relatively prime partition of n. Thus, enumeration of conjugate close partitions of n with relatively prime frequencies equals enumeration of relatively prime partitions with relatively prime frequencies. Now, we count the number of relatively prime partitions with relatively prime frequencies; towars that en, we efine rf (n to be the number of relatively prime partitions of n with greatest common ivisor of its frequencies equals. This gives p ψ (n = n rf (n = n rf 1. The valiity of this equality can be seen from the following equation: (f i λ i = ( f i λ i = n, where gc(f 1, f, = 1 Then from möbius inversion formula, it follows that, rf 1 (n = n µ(p ψ. Since rf1 (n counts the relatively prime partitions of n with relatively prime frequencies; the result follows. Similar arguments gives the following result. Theorem 3. Let A be the set of all relatively prime partitions of n. Then the number of CC- partitions in A is equal to n µ(p ψ 4 Further Work we note that, one may get new enumeration formula for CC partitions in various kins of partitions other than what is mentione in this note.

12 1 A. Davi Christopher et al. The other matters of question which has to be taken into account are: (i Can a bijection be efine between set of all istinct partitions of n an set of all relatively prime uniform partitions of n? (ii Does Anrew s partition ieal [1] or similar tool be wiele to answer the equinumerous results foun in this article? References [1] G.E. Anrews, The Theory of Partitions, Cambrige University Press, (1998. [] L. Euler, Introuctio in Analysin Infinatorum, Lausanne 1(1748, [3] F. Franklin, Surle évelopment u prouit infini (1 x(1 x (1 x 3 (1 x 4..., Comptes Renus Aca.Sci(Paris, 9(1881, [4] K. Meagher an L. Moura, Erös-Ko-Rao theorems for uniform setpartition systems, The Electronic Journal of Combinatorics, 1(005, R40. [5] P.A. MacMahon, Divisors of numbers an their continuations in the theory of partitions, Proc. Lonon Math. Soc., (1919, 75113, Coll. Papers II, [6] M.El Bachraoui, On the parity of p(n, 3 an p ψ (n, 3, Contributions to Discrete Mathematics, 5( (010, [7] M.B. Nathanson, Affine invariants, relatively prime sets an a Phi function for subsets of {1,,..., n}, Integers, 7(007, A01. [8] N.B. Tani an S. Bouroubi, Enumeration of the partitions of an integer into parts of a specifie number of ifferent sizes an especially two sizes, Journal of Integer Sequences, 14(Article (011.