Let be a partition of the positive integer n chosen uniformly at random among all such partitions. Let Ln = Ln( ) and Mn = Mn( ) be the largest part size and its multiplicity, respectively. For large n, we focus on a comparison between the partition statistics Ln and LnMn. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of LnMn Ln grows as fast as 21 log n. We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.
1 + 2 + : + k; k 1; (1) where the positive integers j satisfy 1 2 ::: k. Let (n) be the set of all partitions of n and let p(n) = j (n)j (by de nition p(0) = 1 regarding that the one partition of 0 is the empty partition). The number p(n) is determined asymptotically by the famous partition formula of Hardy and Ramanujan [HR18]: p(n) r 2n !
; 3 n ! 1: A precise asymptotic expansion for p(n) was found by Rademacher [Rad37] (see also [And76, Chapter 5]). Further on, we assume that, for xed integer n 1, a partition 2 (n) is selected uniformly at random. In other words, we assign the probability 1=p(n) to each 2 (n). We denote this probability measure on (n) by P. Let E be the expectation with respect to P. In this way, each numerical characteristic of 2 (n) can be regarded as a random variable, or, a statistic produced by the random generation of partitions of n.
In this paper, we focus on two statistics of random integer partitions 2 (n): Ln = Ln( ) = 1, which is the largest part size in representation (1) and Mn = Mn( ), equal to the multiplicity of the largest part 1 (i.e., Mn( ) = m; 1 m k 1, if 1 = ::: = m > m+1 ::: k in (1), and Mn( ) = k, if 1 = ::: = k).
Each partition 2 (n) has a unique graphical representation called Ferrers diagram [And76, Chapter 1]. It illustrates (1) by the two-dimensional array of dots, composed by 1 dots in the rst (most left) row, 2 dots in the second row,..., and so on, k dots in the kth row. Therefore, a Ferrers diagram may be considered as a union of disjoint blocks (rectangles) of dots whose side lengths represent the part sizes and their multiplicities of the partition , respectively. For instance, Figure 1 illustrates the partition 7 + 5 + 5 + 5 + 4 + 2 + 1 + 1 + 1 of n = 31 in which L31 = 7 and M31 = 1.
The earliest asymptotic results on random integer partition statistics have been obtained long ago by Husimi [Hum38] and Erdos and Lehner [EL41]. Husimi has derived an asymptotic expansion for E(Ln) in the context of a statistical physics model of a Bose gas. Erdos and Lehner were apparently the rst who have studied random partition statistics in terms of probabilistic limit theorems. In fact, they showed that lim E(Mn) = 1: n!1 where and Later on, Szekeres has studied in detail the asymptotic behavior of the number of integer partitions of n whose largest part is k and = k in the whole range of values of k = k(n). In particular, he has obtained in [Sze53] Erdos and Lehner's limiting distribution (2) using an entirely di erent method of proof. Husimi's asymptotic result was subsequently recon rmed by Kessler and Livingston [KL76]. Higher moments of Ln were studied in [Ric74]. A general method providing asymptotic expansions of expectations of integer partition statistics was recently proposed by Grabner et al. [GKW14]. Among the numerous examples, they derived a rather complete asymptotic expansion for E(Ln), namely,
E(Ln) =
1 + 2 + pn 2c 2 log c 4c2 (log n + 2
2 log c) + + O log n n ; n ! 1; log n 2c2 + where c is given by (4) and = 0:5772::: denotes the Euler's constant (see [GKW14, Proposition 4.2]). Notice that by conjunction of the Ferrers diagram the largest part and the total number of parts in a random partition of n are identically distributed for any n. The sequence fp(n)E(Ln)gn 1 is given in [Slo18] as A006128.
There are serious reasons to believe that the multiplicity Mn of the largest part of a random partition of n behaves asymptotically in a much simpler way than many other partition statistics. Grabner and Knopfmacher [GK06] used the Erdos-Lehner limit theorem (2) to establish that In addition, among many other important asymptotic results, Fristedt, in his remarkable paper [Fri93], showed that, with probability tending to 1 as n ! 1, the rst mn largest parts in a random partition of n are distinct if mn = o(n1=4). Hence it may not be that Ln constitutes the main contribution to n by a single part size and (2) (3) (4) (5) (6)
Combining the Erdos-Lehner limit theorem (2) with (8), one can easily observe that the limiting distributions of Ln and LnMn coincide under the same normalization.
(9) where H(u) and c are given by (3) and (4), respectively.
Although Ln and LnMn follow the same limiting distribution, the di erence in means E(LnMn) grows as fast as 12 log n. A complete estimate is given by the following E(Ln) Theorem 1.3. If n ! 1, then, as n ! 1,
The proofs of Theorems 1.1 and 1.3 and are based on generating function identities established in [ABBKM16] that involve products of the form P (x)G(x), where P (x) is the Euler partition generating function some smaller part sizes may occur with su cient multiplicity so that the products of these part sizes with their multiplicities could be much larger than Ln. In terms of the Ferrers diagram, this means that its rst block (the block on the highest position in the Ferrers diagram) has typically smaller area than the areas of several next blocks with larger heights (multiplicities of parts).
Our aim in this paper is to study the asymptotic behavior of the area LnMn of the rst block in the Ferrers diagram of a random partition of n. We show some similarities and di erences between the single part size Ln and its corresponding block area LnMn. As a rst step, we obtain a distributional result for Mn that con rms the limit in (6).
Theorem 1.1. For any n
In addition, if n ! 1, then where the constant c is given by (4). P (x) := 1 X p(n)xn = n=0 1 Y(1 j=1 xj ) 1 and G(x) is a function which is analytic in the open unit disk and does not grow too fast as x ! 1. Our asymptotic expansions in (8) and Theorem 1.3 are obtained using a general asymptotic result of Grabner et al. [GKW14] for the nth coe cient xn[P (x)G(x)]. In the proof of Theorem 1.3 we also apply a classical approach for estimating the growth of a power series around its main singularity.
We organize the paper as follows. Section 2 contains some auxiliary facts related to generating functions and the asymptotic analysis of their coe cients. In Section 3 we sketch the proofs of Theorems 1.1 and 1.3. Finally, in Section 4 we conclude with an open problem on the position of LnMn in the sequence of ordered block areas of a random Ferrers diagram.
(ii) We also have where 1 X p(n)P(Mn = m)xn = xm Y (1 n=1 j m
m 1 xj) 1 = P (x)xm Y (1
xj): j=1 1 1 X p(n)E(LnMn)xn = X n=1 k=1
kxk 1 xk k Y(1 j=1
xj) 1 = P (x)F (x);
kxk 1 xk (i) Let G(e t) = atb + O(jf (t)j) as t ! 0; <t > 0, where b have 0 is an integer and a is real number. Then, we
Preliminaries: Generating Functions and an Asymptotic Scheme
Lemma 2.1. (i) For any positive integer m, we have (10) (11) (12) for any 2 (0; (1
)=2), where and c is given by (4).
(ii) Suppose that G(x) satis es condition (11) and, for t = u + iv, let G(e t) = a log 1t + O(f (j t j)) as t ! 0, where u > 0, v = O(u1+ ) as u ! 0+ and and a are as in part (i). Then, we have
On the Method of Proof and Some Technical Details
Sketch of the Proof of Theorem 1.1. First, we set m = 1 in Lemma 2.1 (i). We have
1
X P(Mn = 1)xn = xP (x):
n=1
This implies (7) at once. The asymptotic behavior of the quotient in (7) may be found using Rademacher's
"exact-asymptotic" formula [Rad37] (see also [And76, Chapter 5]). It seems that a quicker way is to apply
the result of Lemma 2.2 (i). Here we have G(x) = x, which obviously satis es (11). Setting x = e t in (13)
and expanding e t as a Taylor series, we can take into account as many powers of t as we wish. This will be
transferred into powers of n 1=2 in the asymptotic expansion of P(Mn = 1). We decide to bound the error of
estimation up to a term of order O(n 3=2) and write
(13)
(
Sketch of the Proof of Theorem 1.3. First, we represent the function F (x) given by (10) as
The representation (
Proof of the Corollary. The total probability formula and the asymptotic estimate given by Theorem 1.1 imply that
P +P = P Grabner and Knopfmacher [GK06, formula 6.2] found a simpler alternative representation for F1(x). They showed that the right-hand side of (20) yields It is also known that 1 X p(n)E(Ln)xn = P (x)F1(x) n=1 (21) (22) (23) (see, e.g., [GKW14, p. 1059]). In addition, Grabner et al. [GKW14, p. 1084] used Mellin trasform technique to show that
log (1=t) + 1 t F1(e t) = + + O(j t j3); t ! 0:
t 4 144 From this expansion and their main result (see also both parts of Lemma 2.2) they derived asymptotic formula (5) for E(Ln). From (19) it follows that xn[F (x)] = xn[F1(x)] + xn[F2(x)], which in turn implies that
The proof of Lemma 3.1 contains lengthy technical details. We will skip them including here only a brief description.
First, we focus on an asymptotic estimate for F2(e u) as u ! 0+. We set x = e u in (21) and partition the range of summation in its right-hand side into four intervals. It turns out that the main contribution is given by the sum over all integers k 2 u1 log u1 log log u1 log 3 ; u1 log u1 + log log u1 + log 2 , while the other three sums are negligible (of maximum order O 1= log u1 as u ! 0+). Finally, we transfer the variable u into t = u+iv using Taylor's formula. For the reminder term we apply the same approach and the relationship between u and v from Lemma 2.2 (ii).
Hence, applying Lemma 2.2 (ii) with G(x) := F2(x) and f (t) := 1= log 1t , we obtain The main goal of this study is the comparison between the typical growths of the rst block area LnMn and its base length Ln in the Ferrers diagram of a random integer partition n. It turns out that the leading terms in the asymptotic expansions of the expectations of these two statistics are the same for large n; both are equal to p2cn log n. Erdos and Lehner's limit theorem (2) and Theorem 1 show that this leading terms control the weak convergence of Ln and LnMn. Both statistcs, after one and the same normalization, tend to a Gumbel distributed random variable. The expectations of Ln and LnMn are, however, di erent for large n. In fact, by Theorem 1.3 1 2 lim n!1
E(LnMn)
E(Ln) log n =
C = So, it might be interesting to determine the typical position of LnMn among all ordered areas Zn(r). If Rn denotes the smallest value of r such that Zn(r) = LnMn, then we conjecture that This claim is supported by the following non-rigorous argument. From the result of Theorem 1.3 it follows that A calculation of the expectation of the distribution in the right-hand side of (24) demonstrated in [Fri93, p. 708] shows that if r = r(n) ! 1 as n ! 1, then
E(Zn(r)) = pn 2c (log n + 2 log log log n 2 log r) + O(pn): Combining (26) with (27), one may conclude that log r(n) is of order log log log n, which supports the claim in (25). We hope to return to this question in a future study.
The full text of this work may be found in [Mut17]. [EL41] [Mut17] L. Mutafchiev. On the largest part and its multiplicity of a random integer partition. arXiv:1712.03233. [Ric74]
G. Szekeres. Some asymptotic formulae in the theory of partitions, II. Quarterly Journal of Mathematics, Oxford Academic, 4:96-111, 1953.