Zhicheng Gao School of Mathematics and Statistics, Carleton University, Ottawa, Canada

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1 #A3 INTEGERS 4A (04) THE R TH SMALLEST PART SIZE OF A RANDOM INTEGER PARTITION Zhicheg Gao School of Mathematics ad Statistics, arleto Uiversity, Ottawa, aada zgao@math.carleto.ca orado Martiez Departamet de Lleguatjes i Sistemes Iformàtics, Politechical Uiversity of ataloia, Barceloa, Spai corado@lsi.upc.es Daiel Paario School of Mathematics ad Statistics, arleto Uiversity, Ottawa, aada daiel@math.carleto.ca Bruce Richmod Departmet of ombiatorics ad Optimizatio, Uiversity of Waterloo, Waterloo, aada lbrichmod@math.uwaterloo.ca Received: 3//3, Accepted: 8/4/3, Published: 5/0/4 Abstract We study the size of the rth smallest part ad the rth smallest distict part i a radom iteger partitio. This exteds the research o partitios with o small parts of Nicolas ad Sárközy. To the memory of Nicolaas Govert (Dick) de Bruij. Itroductio ombiatorial objects ca be decomposed ito simpler objects called prime, irreducible, or coected compoets. This is a combiatorial aalogue of the fact that itegers decompose ito products of primes. For example, permutatios decompose ito cycles, iteger partitios ito parts, polyomials ito irreducible factors, ad graphs ito coected compoets.

2 INTEGERS: 4A (04) The distributio of the largest ad smallest compoets i combiatorial structures has bee studied for several objects. The distributio of the largest compoet objects for several combiatorial problems has bee studied, for example, by Stepaov [9] (see also Gourdo []). Gourdo expressed his results i terms of a geeralizatio of the Dickma fuctio [6]. This fuctio uderlies the study of umbers smaller tha or equal to with o primes larger tha m; see also [3, 0]. The distributios of the smallest compoets for several combiatorial objects icludig permutatios ad polyomials over fiite fields have also bee related to umber theory [6]. I this case, the results are expressed i terms of a geeralizatio of the Buchstab fuctio [5]; this fuctio uderlies the study of umbers smaller tha or equal to with o primes smaller tha m; see also [, 0]. I this paper we focus o iteger partitios ad study the size of their smallest parts. Some of the mai cotributios to the study of radom iteger partitios are due to Fristedt [] ad to Freima ad Pitma [0]. For the cocrete problem of smallest parts that we focus o here, previous results are i [7, 5]. Our mai results are asymptotic probability estimates for the size of the rth smallest part (Theorem ) ad the rth smallest distict part (Theorem ).. Iteger Partitios with Restricted Parts I this paper we cosider iteger partitios. I particular, we are iterested i iteger partitios where some parts appear a restricted umber of times. For example, partitios with o parts smaller tha m have a restricted patter where parts,,..., m caot appear. I geeral, let A = {a, a,...} be a set of positive itegers. We let p A () be the umber of solutios of = l a + l a + + l m a m, l i 0, where there are l + + l m compoets i this partitio. It is well-kow that X p A ()x = Y = G(x). ai x 0 i 0 Moreover if q A () deotes the umber of partitios of with distict parts the X ( + x ai ). 0 q A ()x = Y i 0 The saddle poit techique [4, 9] to estimate p A () starts with a applicatio of auchy s theorem statig p A () = Z G(x) dx i x+

3 INTEGERS: 4A (04) 3 where is ay orieted couterclockwise cotour ecirclig the origi. Oe chooses the cotour to be the circle x = e, where = () is defied as the solutio of or d x dx = X j G(x) = 0, a j e aj. For q A () the aalogous procedure gives = X j a j e aj +. The asymptotic behaviour of p A () has bee studied by Richmod [8] uder the followig assumptios: () if ay fiite subset of A is deleted, the the remaiig sequece has gcd equal to oe, ad () the followig limit exists l a j lim j! l j. Let A () = A = X j a j e aj (e aj ). The followig result is cotaied i Theorem. of [8] 0 X p A () = ( A ) / l e j= a j A ( + O( )). I the followig we use 3 S = 4 Y jj j S(j) 5 to deote a patter ad p S () to deote the umber of partitios of which have exactly S(j) parts of size j for each j J (each j / J may appear ay umber of times). For a patter S satisfyig X X j = o( / ), J = o( /4 ) ad js(j) = o /, () jj jj it was show i [8] that p S () p() Y jj j p 6. ()

4 INTEGERS: 4A (04) 4 We are iterested i the rth smallest part size of a radom partitio. I this case the relevat sequece is A = {m, m +,...}; that is, partitios with o parts of size smaller tha m. Dixmier ad Nicholas [7] ad Nicolas ad Sárközy [5] defied r(, m) to be the umber of partitios of such that each part is bigger tha or equal to m. They obtai a asymptotic formula for r(, m), where apple m apple c / log 7k, for ay k 3. Whe m = o( /3 ), their formula simplifies to m r(, m) p() p (m )! exp + m p, (3) 4 where = p /3, ad we have the well-kow [] estimate for the total umber of partitios p() 4 p 3 exp p. (We observe that the formula for r(, m) o page 3 of [5] has a missig factor m+ that is correctly icluded i [7].) We ote that the formula i Equatio (3) coicides with Nicolas ad Sárközy s formula whe m = o( /4 ). Ideed i this case, their formula simplifies to m r(, m) p() p (m )!. (4) It seems possible to relax the rage for the patter S; however, as i [5], the asymptotic expressio is more complicated ad will cotai a parameter which is defied by the saddle poit equatio. 3. The Size of the r th Smallest Part The result of Dixmier ad Nicolas [7] ad of Nicolas ad Sárközy [5] shows that the probability that the smallest part of a radom partitio of has size at least m, whe m = o( /3 ), is give by r(, m) p() m p (m )! exp + m p. 4 We ca use Equatio (3) to derive the probability that the rth smallest part of a radom partitio of has size at least m. I the followig, p [r] (, m) is the umber of partitios of such that the size of its rth smallest part is at least m. Let X [r] deote the size of the rth smallest part i a radom partitio of size.

5 INTEGERS: 4A (04) 5 Theorem. The probability that the rth smallest part of a radom partitio of has size at least m = o( /3 ) satisfies P (X [r] > m) = p[r] (, m) p() m + r r (m )! (5) m p exp + m p. 4 Proof. First we observe that p( j) p() for 0 apple j < m, where m = o( /3 ). osiderig the relatio betwee the secod smallest part ad the first smallest part leads to mx p [] (, m) = p [] (, m) + p [] ( j, m) mx j=0 p() j= p( j) p j m p (m )! exp mp [] (, m), m (m )! exp p where we used the fact that i the rage 0 apple j apple m = o( p ), p j/! m. m + m p! m X j=0 j! m p j/ The above argumet ca be exteded to geeral r. Let p r (, m) be the umber of partitios of cotaiig exactly r parts (allowig repetitio) i {,,..., m }. The we have p [r] (, m) = p [r ] (, m) + p r (, m). Let S(),..., S(m ) be a sequece of o-egative itegers satisfyig S() + S()+ +S(m ) = r. There are m+r 3 r such sequeces. We have, provided that P j js(j) = o(p ), p r (, m) = X X m + r 3 p [] ( js(j), m) p [] (, m). r S j The coditio P j js(j) = o(p ) clearly holds whe r is a costat ad m is i the rage of the theorem. Ideed, r ca be a fuctio of such that r = O(log ). Now usig iductio o r, we obtai p [r] (, m) f r (m)p [] (, m), where f r (m) satisfies the followig recursio m + r 3 f r (m) = f r (m) +, f (m) =. r

6 INTEGERS: 4A (04) 6 We have the solutio ad hece p [r] (, m) m + r f r (m) =, r m + r p [] (, m) r m m + r (m )! r p exp + m p p(), 4 ad Equatio (5) follows. We observe that f r (m) m r /(r )! as m! for ay fixed r. Fristedt [] lets X k ( ) be the umber of parts of the partitio that equal k, k =,,.... Thus, kx k ( ) is the cotributio of the part k to the sum of the parts of, that is, kx k ( ) = l k k i our otatio. Theorem. i [] states that if k = o( / ) the lim P p k X k apple v = e v.! 6 Here P deotes the distributio of the kth part. Sice d( e v ) dv that e v is the probability that the kth part is v. Therefore Z 0 e v dv = is the expected value of the kth part of a partitio of for k = o( / ). = e v we have 4. The Size of the r th Smallest Part i Partitios with Distict Parts I this sectio we cosider partitios with distict parts. Let q [r] (, m) be the umber of partitios of with distict parts such that the size of the rth smallest part is at least m. Also, let Y [r] deote the size of the rth smallest part i a radom partitio with distict parts. Freima ad Pitma [0] obtaied a asymptotic formula for q [] (, m) whe m = o( log 9 ). Whe m = o( /3 ), their formula simplifies to q [] (, m) = m 3 /4 3/4 exp (/3) / m(m ) + (3) /. (6) 8 Hardy ad Ramauja [3] ad Hua [4] give the followig estimate for q(), the umber of partitios of ito distict parts q() 3 /4 exp (/3) /, 3/4

7 INTEGERS: 4A (04) 7 ad so i the rage m = o( /3 ) we get q [] (, m) m exp We give ext the mai result of this sectio. m(m ) (3) / q(). (7) 8 Theorem. The probability that the rth smallest part i a radom partitio with ito distict parts has size at least m = o( /3 ) satisfies P (Y [r] > m) = q[r] (, m) m r q() m (r )! exp m(m ) (3) /. (8) 8 Proof. We start by cosiderig q [] (, m) = mx j=0 q [] ( j, m) m X m+ 3 /4 mq [] (, m). j=0 ( j) 3/4 exp (( j)/3) / m(m ) + 8(3( j)) / Usig a similar argumet to that i the proof of Theorem, ad observig that all parts are distict here, we obtai q [r] (, m) g r (m)q [] (, m), where g r (m) satisfies the followig recursio m g r (m) = g r (m) +, g (m) =. r We observe that g r (m) does ot have a sum-free closed form expressio (easily verified usig Gosper s algorithm [7]). We also ote m m g r (m) = + + +, r ad g r (m) m r /(r )! as m! for ay fixed r. Therefore, puttig all pieces together, we obtai Equatio (8). Refereces [] G. E. Adrews, The Theory of Partitios, Ecycl. Math. Appl., Addiso-Wesley, Readig, 976; reprited ambridge Uiversity Press, 998.

8 INTEGERS: 4A (04) 8 [] N. de Bruij, O the umber of ucacelled elemets i the sieve of Eratosthees, Idag. Math. (950), [3] N. de Bruij, O the umber of positive itegers apple x ad free of prime factors > y, Idag. Math. 3 (95),. [4] N. de Bruij, Asymptotic Methods i Aalysis, 3rd editio, Dover Publicatios Ic., New York, 98. [5] A. Buchstab, Asymptotic estimates of a geeral umber theoretic fuctio, Mat. Sborik 44 (937), [6] K. Dickma, O the frequecy of umbers cotaiig prime factors of a certai relative magitude, Ark. Mat. Astr. Fys. (930), 4. [7] J. Dixmier ad J. L. Nicolas, Partitios sas petits sommats, i A Tribute to Paul Erdös, ambridge Uiversity Press, ambridge, 5, 990. [8] L. Dog, Z. Gao, D. Paario ad B. Richmod, Asymptotics of smallest compoet sizes i decomposable combiatorial structures of alg-log type, Discrete Math. Theor. omput. Sci. (00), 97. [9] Ph. Flajolet ad R. Sedgewick, Aalytic ombiatorics, ambridge Uiversity Press, ambridge, 009. [0] G. Freima ad J. Pitma, Partitios ito distict large parts, J. Aust. Math. Soc., Series A 57 (994), [] B. Fristedt, The structure of radom partitios of large itegers, Tras. Amer. Math. Soc. 337 (993), [] X. Gourdo, ombiatoire, algorithmique et géométrie des polyômes, PhD thesis, École Polytechique, 996. [3] G. H. Hardy ad S. Ramauja, Asymptotic formulae i combiatory aalysis, Proc. Lod. Math. Soc. 7 (98), [4] L. K. Hua, O the umber of partitios of a umber ito uequal parts, Tras. Amer. Math. Soc. 5 (94), [5] J. L. Nicolas ad A. Sárközy, O partitios without small parts, J. Théor. des Nombres Bordeaux (000), [6] D. Paario ad B. Richmod, Smallest compoets i decomposable structures: exp-log class, Algorithmica 9 (00), [7] M. Petkovšek, H. Wilf ad D. Zeilberger, A = B, A. K. Peters Ltd., 996. [8] L. B. Richmod, Asymptotic relatios for partitios, J. Number Theory 7 (975), [9] S. Stepaov, Limit distributios of certai characteristics of radom mappigs. Theory Probab. Appl. 4 (969), [0] G. Teebaum, Itroductio to Aalytic ad Probabilistic Number Theory, ambridge Uiversity Press, 995.