PARTITIONS WITH PARTS IN A FINITE SET
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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volue 28, Nuber 5, Pages S ( Artcle electroncally publshe on February 7, 2000 PARTITIONS WITH PARTS IN A FINITE SET MELVYN B. NATHANSON (Councate by Dav E. Rohrlch Abstract. Let A be a nonepty fnte set of relatvely pre postve ntegers, an let p A (n enote the nuber of parttons of n wth parts n A. An eleentary arthetc arguent s use to prove the asyptotc forula n k ( p A (n a A a (k! + O n. Let A be a nonepty set of postve ntegers. A partton of a postve nteger n wth parts n A s a representaton of n as a su of not necessarly stnct eleents of A. Two parttons are consere the sae f they ffer only n the orer of ther suans. The partton functon of the set A, enote p A (n, counts the nuber of parttons of n wth parts n A. If A s a fnte set of postve ntegers wth no coon factor greater than, then every suffcently large nteger can be wrtten as a su of eleents of A (see Nathanson [3] an Han, Krfel, an Nathanson [2], an so p A (n for all n n 0. In the specal case that A s the set of the frst k ntegers, t s known that p A (n nk k!(k!. Erős an Lehner [] prove that ths asyptotc forula hols unforly for k o(n /3. If A s an arbtrary fnte set of relatvely pre postve ntegers, then n k ( p A (n a A a (k!. The usual proof of ths result (Netto [4], Pólya Szegö [5, Proble 27] s base on the partal fracton ecoposton of the generatng functon for p A (n. The purpose of ths note s to gve a sple, purely arthetc proof of (. We efne p A (0. Theore. Let A {a,..., } be a set of k relatvely pre postve ntegers, that s, gc(a (a,...,. Receve by the etors June 5, Matheatcs Subect Classfcaton. Prary P8; Seconary 05A7, B34. Key wors an phrases. Partton functons, asyptotcs of parttons, atve nuber theory. Ths work was supporte n part by grants fro the PSC CUNY Research Awar Progra an the NSA Matheatcal Scences Progra. 269 c 2000 Aercan Matheatcal Socety Lcense or copyrght restrctons ay apply to restrbuton; see
2 270 MELVYN B. NATHANSON Let p A (n enote the nuber of parttons of n nto parts belongng to A. n k p A (n a A a (k! + O ( n. Proof. Let k A. The proof s by nucton on k. Ifk,thenA {} an p A (n, snce every postve nteger has a unque partton nto a su of s. Let k 2, an assue that the theore hols for k. Let For,...,k, we set (a,...,. (,. a a. A {a,...,a k } s a set of k relatvely pre postve ntegers, that s, gc(a. Snce the nucton assupton hols for A,wehave n p A (n a (! + O ( n k 3 for all nonnegatve ntegers n. Let n (. Snce (,, there exsts a unque nteger u such that 0 u an n u (o. s a nonnegatve nteger, an n u If v s any nonnegatve nteger such that O(n. n v (o, then v u (o Lcense or copyrght restrctons ay apply to restrbuton; see
3 PARTITIONS WITH PARTS IN A FINITE SET 27 an so v u (o, that s, v u + l for soe nonnegatve nteger l. If n v n (u + l 0, then 0 l [ n u ] [ ] r. We note that r O(n. Let π be a partton of n nto parts belongng to A. If π contans exactly v parts equal to,thenn v 0ann v 0(o, snce n v s a su of eleents n {a,..., }, an each of the eleents n ths set s vsble by. Therefore, v u + l, where 0 l r. Consequently, we can ve the parttons of n wth parts n A nto r + classes, where, for each l 0,,...,r, a partton belongs to class l f t contans exactly u + l parts equal to. The nuber of parttons of n wth exactly u+l parts equal to s exactly the nuber of parttons of n (u + l nto parts belongng to the set {a,..., },or, equvalently, the nuber of parttons of n (u + l nto parts belongng to A,whchsexactly n (u + lak p A p A ( l. Therefore, p A (n r p A ( l l0 r ( ( lak + O( k 3 a (! l0 r k ( l a + O(n. (! l0 To evaluate the nner su, we note that r l l0 r+ ( + + O(r an ( k + 0 k ( ( k. Lcense or copyrght restrctons ay apply to restrbuton; see
4 272 MELVYN B. NATHANSON r l0 ( l (! (! (! (! k k k (! r l0 0 ( ( l r l0 ( r ( a k + ( + + O(r ( ( ( l ( (!( + + O( ( (!!( + + O( ( (k ( +!( +! + O( k ( k + O( (k! + 0 k (k! + O(. + a + k ( + + O( Therefore, p A (n r k ( l a + O(n (! l0 ( k k a (k! + O(n + O(n ( k (n a (k! ua k k + O(n ( k (n k + O(n a (k! n k k a (k! + O(n. Ths copletes the proof. Lcense or copyrght restrctons ay apply to restrbuton; see
5 PARTITIONS WITH PARTS IN A FINITE SET 273 References [] P. Erős an J. Lehner. The strbuton of the nuber of suans n the parttons of a postve nteger. Duke Math. J., 8: , 94. MR 3:69a [2] S. Han, C. Krfel, an M. B. Nathanson. Lnear fors n fnte sets of ntegers. Raanuan J., 2:27 28, 998. MR 99h:0 [3] M. B. Nathanson. Sus of fnte sets of ntegers. Aer. Math. Monthly, 79:00 02, 972. MR 46:3440 [4] E. Netto. Lehrbuch er Cobnatork. Teubner, Lepzg, 927. [5] G. Pólya an G. Szegö. Aufgaben un Lehrsätze aus er Analyss. Sprnger Verlag, Berln, 925. Englsh translaton: Probles an Theores n Analyss, Sprnger Verlag, New York, 972. MR 49:8782 Departent of Matheatcs, Lehan College (CUNY, Bronx, New York 0468 E-al aress: nathansn@alpha.lehan.cuny.eu Current aress: School of Matheatcs, Insttute for Avance Stuy, Prnceton, New Jersey E-al aress: nathansn@as.eu Lcense or copyrght restrctons ay apply to restrbuton; see
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