Notation We will be studying several counting functions related to integer artitions. For clarity, we begin with the denitions and notation. If n is a

Transcription

1 On the Multilicity of Parts in a Random Partition Sylvie Corteel Laboratoire de Recherche en Informatique B^at 490, Universite Paris-Sud 9405 Orsay, FRANCE Sylvie.Corteel@lri.fr Carla D. Savage z Deartment of Comuter Science North Carolina State University Raleigh, NC cds@cayley.csc.ncsu.edu June 0,998 Boris Pittel y Deartment of Mathematics Ohio State University Columbus, Ohio 430 bg@math.ohio-state.edu Herbert S. Wilf Deartment of Mathematics University of Pennsylvania Philadelhia, PA wilf@math.uenn.edu Abstract Let be a artition of an integer n chosen uniformly at random among all such artitions. Let s() be a art size chosen uniformly at random from the set of all art sizes that occur in. We rove that, for every xed m, the robability that s() has multilicity m in aroaches (m(m + )) as n!. Thus, for examle, the limiting robability that a random art size in a random artition is unreeated is /. In addition, (a) for the average number of dierent art sizes, we rene an asymtotic estimate given by Wilf, (b) we derive an asymtotic estimate of the average number of arts of given multilicity m, and (c) we show that the exected multilicity of a randomly chosen art size of a random artition of n is asymtotic to (log n). The roofs of the main result and of (c) use a conditioning device of Fristedt. AMS Subject Classications 05A7, P8, P8 Suorted in art by NSF grant DMS y Suorted by thefocus on Discrete Probability Program at Dimacs Center of Rutgers University, and by NSA grant MDA z Suorted in art by NSF grant DMS977

2 Notation We will be studying several counting functions related to integer artitions. For clarity, we begin with the denitions and notation. If n is a ositive integer, then by a artition,, of n, we mean a reresentation n i in which the 's are nonnegative integers. artitions of n and let (n) j(n)j. i(i); () We use (n) to denote the set of all In () the quantity (i) isthe multilicity of the art i in the artition. If it is imortant to designate the artition exlicitly then we will use (i) for the multilicity of the art i in the artition. The number of arts in the artition in () is () P i (i) (n; k) will be the number of artitions of n with () k. The number of distinct art sizes in the artition is () jfi (i) > 0gj. (n; k) will be the number of artitions of n with () k. We will use angle brackets hi to denote averages of these quantities. In articular, hi n and hi n will denote the average values of () and () among all artitions of n. That is, hi n P k k (n; k)(n) andhi n P k k (n; k)(n). m () jfi (i) mgj will be the number of art sizes i of a given multilicity m, and (n; j; m) will be the number of artitions of n with m () j. h m i n will be the average value, over all artitions of n, of m (); that is h m i n j (n; j; m)(n). P j Finally, [x n ]fg will denote the coecient ofx n in the exression \". The number of dierent art sizes The asymtotic relation hi n n (n!) () for the average number of dierent art sizes in a artition on n is well known and can be found in [5]. (Erdos and Lehner [3] roved that the number of dierent art sizes ((), that is) lies between ( ) n, for almost all artitions.)

3 Here, following [5], we will rst rove a simle identity involving this function, and second we will show that with the aid of that identity one can nd as many terms of the asymtotic exansion for hi n as one wishes. We illustrate by dislaying one more term of its asymtotic exansion, namely hi n n 3 +, + o() (n!) (3) We claim that the sum that aears in the numerator of the comutation of the average number of dierent art sizes, namely P k k (n; k) isjust k k (n; k) (0) + () + () + + (n, ) (4) This identity is well known. It is mentioned in [], and attributed to Stanley. To rove it combinatorially, we can ma (0) [ () [ [ (n, ) ) (n) by sending each artition (i) to the artition (n) with one additional art `n, i' adjoined. We will then nd that each artition in (n) withk dierent art sizes is the image of exactly k dierent artitions from the union, namely those that one nds by removing exactly one coy ofany of its k dierent art sizes, which roves (4). 3 Asymtotic consequences In view of (4) we have thattheaverage number of dierent art sizes in a artition of the integer n is exactly (0) + () + + (n, ) hi n (5) (n) But the comlete asymtotic (and convergent) series for the artition function (n) is known, thanks to the ioneering work of Hardy and Ramanujan [5], and Rademacher []. Thus by (5), we can obtain arbitrarily many terms of the asymtotic series for hi n by a simle exercise in summation. We will sketch one ste of this rocess. For (n) itself we have j (n) 4 e 3 n, 3 n 4 e 3 n + O n 3 To rove (3), we have rst, by the Euler-MacLaurin sum formula, n e j e n (, ) j n, + + e 3 n () n + O(n,, ) (7) Now ifwe use (7) to sum () and thereby estimate (5), the claimed result (3) follows. 3

4 4 The number of arts of multilicity m in a artition Our goal in this section is to characterize the counting functions (n; j; m) and h m i n. To this end, for xed m, we will nd the two-variable generating function of (n; j; m). Fix n and some set i < i < < i j. Following the aradigm of inclusionexclusion, we want the number of artitions of n that have at least i ;;i j as arts of multilicity m. That is to say, wewant the number of artitions of n such that i ;;i j do occur as arts of, and their multilicities are all m. But that is the number of artitions of the integer n, mi,, mi j into arts that do not include any ofi ;;i j but are otherwise unrestricted. This number is evidently [x n,mi,,mi j ] where P(x) Q k(, x k ),. Y ii ;;ii j, x i [x n,mi,,mi j ] Q j (, xi ) Q Q j [x n (, x ] xmi i ) Q i(, x i ) [x n ]P(x) 8 < jy i(, x i ) x mi (, x i ) Note that the factor in brackets contributes to the coecient oft j in the roduct Y i ( + tx mi (, x i )) If we nowsumover all sets i <i <<i j we obtain (n; j; m) [x n ]P(x) [t j ] Y i( + tx mi (, x i )) 9 ; (8) The rincile of inclusion-exclusion can, however, be stated in the following form (here we follow [4]) if h(t) is the generating function for the numbers N r, dened to be the sum over all sets of r roerties of the number of objects that have at least that set of roerties, then h(t, ) is the generating function for the number of objects that have exactly each number of roerties. Hence we have the following result. Theorem If (n; j; m) is the number of artitions of n that have exactly j arts of multilicity m, then n;j0 x n t j (n; j; m) P(x) Y i f+(t, )x mi (, x i )g (9) 4

5 We now use this generating function to nd an interesting identity. We claim that j j (n; j; m) k0 ((n, mk), (n, (m +)k)) (0) in which () is the unrestricted artition function. Indeed the identity follows at once by the usual method of logarithmic dierentiation of (9) w.r.t. t, setting t, clearing of fractions, and matching the coecients of x n on both sides. We omit the details. To rove (0) combinatorially, let i (n) be the set of artitions of n with no art equal to i and let q i (n) j i (n)j. It is easy to see that q i (n) (n), (n, i), for examle by artitioning (n) into those artitions which dohave aarti, and those which do not. Thus, (0) becomes q i (n, mi) () For the maing, take j j (n; j; m) i (n, m) [ (n, m) [ 3 (n, 3m) [ ) (n) as follows. If is a artition of n, im, in which noart`i' occurs, add m coies of i to to obtain a artition of n in which art i has multilicity m. Conversely, if is a artition of n in which art i occurs with multilicity m, delete the m coies of i to obtain a artition of n, mi in which no art has size i. Thus, each artition (n) with exactly j arts of multilicity m is the image under this maing of exactly j dierent artitions, namely those obtained by deleting all coies of one of the arts of multilicity m. This roves (). A consequence is the following. Theorem The average number of arts of multilicity m in a artition of n is h m i n j (n, jm), (n, j(m +)) (n) It is easy, with the aid of (7), to nd the asymtotic behavior. The result is that for each xed m the average number of arts of multilicity m of a artition of n is h m i n m n, (m +) n m(m +) n (n!) () 5

6 5 The total number of arts Let hi n denote the average value of () over all artitions of n, where () is the number of arts of. Then if (n; k) is the number of artitions of n with exactly k arts, The identity hi n (n) k k k (n; k) m k (n; k) j (n, mj) (3) can be established by observing that the following gives a bijection from the set counted by the right-hand-side to the set counted by the left-hand-side Given m; j and (n, mj), add m coies of art j to to obtain a artition in (n). Note that for every k, each artition of n with exactly k arts will be the image of a artition in (n, mj) for exactly k airs (m; j). A roof of (3) by generating functions can be found in [],. 9. The asymtotic behavior of hi n was studied in references [7, 8, 9] and is known to be hi n n log n (4) The reader might note the factor of log n by which this formula for the average number of arts diers from (), for the average number of art sizes, as a measure of the inuence of multilicities on the averages. The result (4) aears in [9], although after the calculations in the aer were called into question by the reviewer [0], a rigorous roof was rovided by Kessler and Livingston [8]. But in fact, an earlier exansion of hi n u to an o() remainder term can be found in the fascinating aer of Husimi [7]. (According to Husimi, (n; k) reresents the number of comlexions of a Bose gas of k articles and of energy n distributed over the energy levels ( ; ; 3;) and hi n can be interreted as the \mean number of excited articles". Husimi was motivated to investigate the asymtotic behavior of hi n in order to conrm or refute the conjecture that hi n n 3 which was suorted to within a few ercent by exerimental evidence for n 00.) Since all we need is the leading term in the formula for hi n,wegive a short derivation of that simlied formula below. In the sum for hi n, the indices m and j are subject to a restriction mj n,. Using the Hardy-Ramanujan formula () ec 4 3 ( + O(, )); (c, );

7 (uniformly for ), after simle algebra we get (n, mj) mj +O + O(n, mj), (n) n 3 ex, cmj n ( ( + O(n, )) ex(,cmjn, ); if mj n; O(ex(,cn )); if mj > n Thus hi n n j n j Z n mn(j) ex(,cmjn, )+O(ex(,c 0 n )) ex(,cjn, ), ex(,cjn, ) + O(ex(,c0 n )) ex(,cxn, ), ex(,cxn, ) dx + O(n ) n c, log + O(n, e ),cn, n (c), log n + O(n ) (We have used here the Maclaurin formula with the remainder term.) (8 c 0 <c) The robability that a art size has multilicity m The fact that the average number of arts of multilicity m and the average number of distinct art sizes are both roortional to n make lausible the following conjecture. Consider a two-ste samling rocedure, in which we rst samle uniformly at random (uar) a artition of n and second samle uar one of the dierent art sizes in. Then the unconditional robability that the chosen art size has multilicity m aroaches a universal constant, m,asn tends to innity. We rove this with m (m(m + )) in Theorem 3 below. Let j be the multilicity of the art j in a random artition of n, (that is, j () (j)); let I j be the indicator of the event f j g, and let I j;m be the indicator of the event f j mg; j. Then D n P j I j is the total number of dierent art sizes in the random artition. (Of course, E D n hi n n see Sections and 3. Goh and Schmutz [] roved the asymtotic normality of D n, from which it follows that D n is asymtotic to n in robability as well.) Likewise, D n;m P j I j;m is the total number of art sizes of multilicity m, and ED n;m h m i n 7

8 is sharly estimated in Section 4. We see that, given the random variables ( ; ;), the conditional robability () that the randomly selected size has multilicity m is given by Now n, the robability in question, is () D n;m D n n E() E Dn;m D n that is, it equals the exected value of (). ; (5) Theorem 3 The robability that, for a xed m, a randomly chosen art size of a random artition of n occurs with multilicity m aroaches (m(m + )) as n!. In short, lim n! n (m(m + )). Proof. We know from () and () that ED n n c ; ED n;m n cm(m +) ; c ; () and that D n is tyically close to E D n. So, intuitively, one is justied in relacing D n in (5) by n c,. To do this rigorously though, we need to know howunlikely is the event A n D n n c,, " Such an estimate does not follow from the results in [3], [], and [5]. Instead, we get a good bound by using the conditioning device, suggested for the integer artitions by Fristedt [4] (see also []) and atterned after the analogous treatment of random ermutations by She and Lloyd [3]. Namely, introduce the sequence of indeendent random variables Y (Y ;Y ;), where Y j is geometrically distributed with a arameter q j, Pr fy j kg (, q j )q jk Then, for every xed q, the sequence has the same distribution as the sequence Y, conditioned on the event B n 8 < j jy j n (see [4] for a roof). It is natural to ick q for which Pr (B n ) is as large as ossible, and Fristedt's almost otimal choice was to set q e,cn,. For this q, 9 ; Pr (B n ) const n, ;

9 Let D n jfj Y j gj j I j ; Then I j A n ( ; if Yj ; 0; if Y j 0; D n n c,, " Pr(A n ) Pr(A n jb n ) Pr(A n \ B n ) Pr(B n ) Pr(A n) Pr(B n ) O n 3 4 Pr(A n ) (7) So we need to bound Pr(A n ). This is easy since Y ;Y ; are indeendent. Here is a standard argument. Let u R be given. Then E(e udn ) Y j E(e ui j ) Y j So, for every u>0, by Chebyshev's inequality, Pr (, q j + e u q j (e u, ) j ex ex (e u q, ), q D n n c ( + ") ) ex q j A (e u, ) q,q ex(u n c ( + ")) ; and an almost otimal u (that minimizes the bound) is log( + "), which gives an uer bound ex,n a ; a a (") c, [( + ") log( + "), "] > 0 Analogously, using u log(, ") < 0, we obtain ( ) Pr D n n c (, ") ex,n a ; a a (") c, [" +(, ") log(, ")] 9

10 So Pr(A n ) e,an ; and, combining this bound with (7), we have a>0; The rest is short and easy. We write Pr(A n ) e,bn ; 0 <b<a (8) n E (()) E, ()I A c n + E (())IAn ) E + E By the denition of () andtheevent A c n, Furthermore, by (8), E ( + O(")) c n ( + O(")) c n E(D n;m I A c n ) [E(D n;m ), E (D n;m I An )] (9) E(D n;m I An ) O(nPr(A n )) O(ne,bn ) o() Combining this estimate with (9) and (), we have It remains to notice that E + O(")+o(); as n! m(m +) E Pr(A n )o(); as n!; and we conclude, letting n!and then "! 0, that lim n! n m(m +) ; m Notes.. Analogously to D n, we could have roved that the distribution of D n;m is concentrated around ED n;m. Consequently, the emirical robability () D n;m D n that the random art size has multilicity m, converges, in robability, to (m(m + )). This also imlies that n! (m(m +)). 0

11 . How willthemultilicity distribution change if we samle a art from the set of all arts with the (conditional) robability roortional to a art's size? An answer is not that obvious, since in a tyical artition the small arts i have multilicities of order n, while the middle range arts are of order n, but of low multilicity. In this case the conditional robability that the selected art's size has multilicity m is given by () m n j j m and the unconditional robability is therefore given by n E() m n We leave it to the reader to show that j j; jpr( j m) lim n! n m + m(m +) ; m The sum of the limits is again! And so, as before, the multilicity is bounded in robability. 3. What if a art is chosen uar among all arts without any size bias? Intuitively, one should exect the multilicity to be higher in robability. For this samling, the conditional robability of multilicity m is () md n;m ; P n where P n P () is the total number of arts in the random artition. Erdos and Lehner [3] roved that, in robability, P n is asymtotic to EP n ( hi n ). Extending our argument in the roof of Theorem 3, we can also show, for m o(n ), that md n;m is asymtotic in robability to Therefore, in robability, whence me j Pr(u n m) I j;m A () m (, q j )q jm j n c(m +) (m +)logn ; (m +)logn ; m o(n ) Here u n denotes the random multilicity of the chosen art. Consequently Pr log un log n x! x; for every x<. Thus log u n is tyically of order log n.

12 7 The exected multilicity of a art We now consider, for a random artition, the average multilicity of a art size selected at random from the set of all art sizes occurring in. Equations () and (4) suggest the following. Theorem 4 If M n is the exected multilicity of a randomly chosen art size in a random artition of n then M n log n Proof. With j, I j,andd n as dened in the revious section, clearly M n E P! j j D n Here P j j n; so it can be shown, analogously to the revious argument, that M n is asymtotic to ~M n E (P j j) ED n Note that from (4), and from (), so E j j hi n E D n n log n n ; M n ~ Mn log n In contrast, for size-unbiased samling from the set of all arts discussed in Note 3 of the revious section, it is the logarithm of the multilicity which is of order log n. Acknowledgement. We gratefully acknowledge our debt to a referee who carefully read the aer and rovided us with very useful comments and thoughtful suggestions.

13 References [] Cohen, Daniel I. A., Basic Techniques of Combinatorial Theory, John Wiley & Sons, New York, 978. [] Comtet, L., Advanced Combinatorics, Reidel, 974. [3] Erd}os, P., and Lehner, J., The distribution of the number of summands in the artitions of a ositive integer, Duke Math. J. 8 (94), [4] Fristedt, B., The structure of random artitions of large integers, Trans. Amer. Math. Soc. 337 (993), [5] Hardy, G. H., and Ramanujan, S., Asymtotic formul in combinatory analysis, Proc. London Math. Soc. 7 (98), [] Goh, W. M. Y. and Schmutz, E., The number of distinct art sizes in a random integer artition, J. Combin. Theory Ser. A 9 (995), [7] Husimi, K., Partitio numerorum as occurring in a roblem of nuclear hysics, Proceedings of the Physico-Mathematical Society of Jaan 0 (938), [8] Kessler, I. and Livingston, M., The exected number of arts in a artition of n, Monatshefte fur Mathematik 8 (97), 03-. [9] Luthra, S. M., On the average number of summands in a artition of n, Proc. Nat. Inst. Sci. India, Part A 3 (957), 483{498. [0] Newman, M., Review #, Math. Rev. 9 (958). [] Pittel, B., On a likely shae of the random Ferrers diagram, Adv. Al. Math.8 (997), [] Rademacher, H., On the artition function (n), Proc. London Math. Soc. 43 (937), [3] She, L. A. and Lloyd, S. P., Ordered cycle lengths in a random ermutation, Trans. Amer. Math. Soc. (9), [4] Wilf, Herbert S., generatingfunctionology, Academic Press, 994. [5] Wilf, Herbert S., Three roblems in combinatorial asymtotics, J. Combin. Theory Ser. A 35 (983),