# Asymptotic analysis of expectations of plane partition statistics

## Abstract

Assuming that a plane partition of the positive integer *n* is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as *n* becomes large. The generating functions that arise in this study are of the form *Q*(*x*)*F*(*x*), where \(Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}\) is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function *F*(*x*) around \(x=1\). The representation of a plane partition as a solid diagram of volume *n* allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.

## Keywords

Plane partition statistic Asymptotic behavior Expectation## Mathematics Subject Classification

05A17 05A16 11P82## 1 Introduction

*n*is an array of non-negative integers

*h*th row, so that for some

*l*, \(\lambda _1\ge \lambda _2\ge ...\ge \lambda _l >\lambda _{l+1}=0\), then the (linear) partition \(\lambda =(\lambda _1,\lambda _2,...,\lambda _l)\) of the integer \(m=\lambda _1+\lambda _2+\cdots +\lambda _l\) is called the shape of \(\omega \), denoted by \(\lambda \). We also say that \(\omega \) has

*l*rows and

*m*parts. Sometimes, for the sake of brevity, the zeroes in array (1) are deleted. For instance, the abbreviation

Any plane partition \(\omega \) has an associated solid diagram \(\Delta =\Delta (\omega )\) of volume *n*. It is defined as a set of *n* integer lattice points \({\mathbf {x}}=(x_1,x_2,x_3)\in {\mathbb {N}}^3\), such that if \({\mathbf {x}}\in \Delta \) and \(x_j^\prime \le x_j, j=1,2,3\), then \({\mathbf {x}}^\prime =(x_1^\prime ,x_2^\prime ,x_3^\prime )\in \Delta \) too. (Here \({\mathbb {N}}\) denotes the set of all positive integers.) Indeed the entry \(\omega _{h,j}\) can be interpreted as the height of the column of unit cubes stacked along the vertical line \(x_1=h, x_2=j\), and the solid diagram is the union of all such columns. Figure 1 represents the solid diagram of the plane partition in the example (2).

*q*(

*n*) denote the total number of plane partitions of the positive integer

*n*(or, the total number of solid diagrams of volume

*n*). It turns out that (3) implies the following generating function identity:

*q*(

*n*), as \(n\rightarrow \infty \), has been obtained by Wright [28] (see also [20] for a little correction). It is given by the following formula:

### Remark 1

In fact, Wright [28] has obtained an asymptotic expansion for *q*(*n*) using the circle method.

Next, we introduce the uniform probability measure \({\mathbb {P}}\) on the set of plane partitions of *n*, assuming that the probability 1 / *q*(*n*) is assigned to each plane partition. In this way, each numerical characteristic of a plane partition of *n* becomes a random variable (a statistic in the sense of the random generation of plane partition of *n*). In the following, we will discuss several different instances of plane partition statistics. Our goal is to develop a general asymptotic scheme that allows us to derive an asymptotic formula for the *n*th coefficient \([x^n]Q(x)F(x)\) of the product *Q*(*x*)*F*(*x*), where *Q*(*x*) is defined by (4) and the power series *F*(*x*) is suitably restricted on its behavior in a neighborhood of \(x=1\). We will show further that expectations of plane partition statistics we will consider lead to generating functions of this form. From one side, our study is motivated by the asymptotic results of Grabner et al. [9] on linear partition statistics. Their study is based on general asymptotic formulae for the *n*th coefficient of a similar product of generating functions with *Q*(*x*) replaced by the Euler partition generating function \(P(x)=\prod _{j=1}^\infty (1-x^j)^{-1}.\) The second factor *F*(*x*) satisfies similar general analytic conditions around \(x=1\). In addition, our interest to study plane partitions statistics was also attracted by several investigations during the last two decades on shape parameters of random solid diagrams of volume *n* as \(n\rightarrow \infty \). Below we present a brief account on this subject.

Cerf and Kenyon [4] have determined the asymptotic shape of the random solid diagram, while Cohn et al. [5] have studied a similar problem whenever a solid diagram is chosen uniformly at random among all diagrams boxed in *B*(*l*, *s*, *t*), for large *l*, *s* and *t*, all of the same order of magnitude. Okounkov and Reshetikhin [21] rediscovered Cerf and Kenyon’s limiting shape result and studied asymptotic correlations in the bulk of the random solid diagram. Their analysis is based on a deterministic formula for the correlation functions of the Schur process. The joint limiting distribution of the height [the largest part in (1)], depth [the number of columns in (1)] and width [the number of rows in (1)] in a random solid diagram was obtained by Pittel [23]. The one-dimensional marginal limiting distributions of this random vector were found in [17]. The trace of a plane partition is defined as the sum of the diagonal parts in (1). Its limiting distribution is determined in [12]. Bodini et al. [3] studied random generators of plane partitions according to the size of their solid diagrams. They obtained random samplers that are of complexity \(O(n\log ^3{n})\) in an approximate-size sampling and of complexity \(O(n^{4/3})\) in exact-size sampling. These random samplers allow to perform simulations in order to confirm the known results about the limiting shape of the plane partitions.

In the proof of our main asymptotic result for the coefficient \([x^n]Q(x)F(x)\) we use the saddle point method. In contrast to [9], we base our study on a theorem due to Hayman [11] for estimating coefficients of admissible power series (see also [7, Section VIII.5]). We show that *Q*(*x*) (see (4)) is a Hayman admissible function and impose conditions on *F*(*x*) which are given in terms of Hayman’s theorem. In the examples we present, we demonstrate two different but classical approaches for estimating power series around their main singularity.

*n*th coefficient \([x^n]Q(x)F(x)\) under certain relatively mild conditions on

*F*(

*x*). The proof of Theorem 1 is given in Sect. 3. Section 4 contains some examples of plane partition statistics that lead to generating functions of the form

*Q*(

*x*)

*F*(

*x*). We apply Theorem 1 to obtain the asymptotic behavior of expectations of the underlying statistics. For the sake of completeness, in the Appendix we show how Wright’s formula (5) follows from Hayman’s theorem.

### Remark 2

An extended abstract of this work was presented at EUROCOMB 2017 (European Conference on Combinatorics, Graph Theory and Applications, Vienna, Austria, August–September, 2017) and published in [19].

## 2 Some remarks on Hayman admissible functions and Meinardus theorem on weighted partitions; statement of the main result

Our first goal in this section is to give a brief introduction to the analytic combinatorics background that we will use in the proof of our main result. Clearly, \(x^n[Q(x)F(x)]\) with *Q*(*x*) given by (4) can be represented by a Cauchy integral whose integrand includes the product *Q*(*x*)*F*(*x*) (the conditions that *F*(*x*) should satisfy will be specified later). Its asymptotic behavior heavily depends on the analytic properties of *Q*(*x*) whose infinite product representation (4) shows that the unit circle is a natural boundary and its main singularity is at \(x=1\). The main tools for the asymptotic analysis of \(q(n)=x^n[Q(x)]\) are either based on the circle method (see [28]) or on the saddle-point method (see [3]). Both yield Wright’s asymptotic formula (5). An asymptotic formula in a more general framework [see (7)] was obtained by Meinardus [16] (see also [10] for some extensions). A proof of formula (5) that combines Meinardus approach with Hayman’s theorem for admissible power series is started in Sect. 3 and completed in the Appendix.

*D*(

*z*) has an analytic continuation. The second one \((M_2)\) is related to the asymptotic behavior of

*D*(

*z*) whenever \(|v|\rightarrow \infty \). A function of the complex variable

*z*which is bounded by \(O(|\mathfrak {I}{(z)}|^{C_1}), 0<C_1<\infty \), in a certain domain in the complex plane is called function of finite order. Meinardus’ second condition \((M_2)\) requires that

*D*(

*z*) is of finite order in the whole domain \({\mathcal {H}}\). Finally, Meinardus’ third condition \((M_3)\) implies a bound on the ordinary generating function of the sequence \(\{b_j\}_{j\ge 1}\). It can be stated in a way, simpler than Meinardus’ original expression, by the inequality:

The infinite product representation (4) for *Q*(*x*) implies that \(b_j=j, j\ge 1\), and therefore, \(D(z)=\zeta (z-1)\). It is known that this sequence satisfies the Meinardus’ scheme of conditions (see, e.g., [20] and [10, p. 312]).

Now, we proceed to Hayman admissibility method [11, 7, Section VIII.5]. To present the idea and show how it can be applied to the proof of our main result, we need to introduce some auxiliary notations.

*Capture condition.* \(lim_{r\rightarrow \rho } a(r)=\infty \) and \(\lim _{r\rightarrow \rho } b(r)=\infty \).

*Locality condition.*For some function \(\delta =\delta (r)\) defined over \((R_0,\rho )\) and satisfying \(0<\delta <\pi \), one has

*Decay condition.*

**Hayman Theorem.**Let

*G*(

*x*) be a Hayman admissible function and \(r=r_n\) be the unique solution of the equation

*G*(

*x*) satisfy, as \(n\rightarrow \infty \),

In the next section we will show that *Q*(*x*), given by (4), is admissible in the sense of Hayman and apply Hayman’s Theorem setting \(G(x):=Q(x)\) (and \(\rho :=1\)).

Furthermore, as in [9], we will introduce two rather mild conditions on the second factor *F*(*x*). Since we will employ Hayman’s method, the first one is given in terms of the solution \(r_n\) of Eq. (13). The second one requires that *F*(*x*) does not grow too fast as \(|x|\rightarrow 1\).

*Condition A.*Let \(r=r_n\) be the solution of (13). We assume that

*Condition B.*There exist two constants \(C>0\) and \(\eta \in (0,2/3)\), such that, as \(|x|\rightarrow 1\),

### Remark 3

Revisiting Hayman’s locality condition, we conclude that the function \(\delta (r)\) defines an arc on the circle \(|x|=r\) that is very close to the main singularity of the underlying function. The MacMahon generating function (4) has infinitely many singularities in any neighborhood of the point \(x=1\) and thus in the case at hand we have \(\delta (r)\rightarrow 0\) as \(r=|x|\rightarrow 1\). Note that from (11) and (12) it follows that the asymptotic behavior of \(Q(r e^{i\theta })\) significantly changes when \(\theta \) leaves the interval \((-\delta (r),\delta (r))\). So, an important detail in the analysis of the asymptotic behavior of the product *Q*(*x*)*F*(*x*) around the point \(x=1\) is to determine the function \(\delta (r)\) explicitly. We will find an expression for \(\delta (r)\) in the next section. Condition A for the second factor *F*(*x*) is rather technical. It avoids pathological examples for *F*(*x*) (e.g., zeros or oscillations of \(F(r e^{i\theta })\) which provide the ratio in (15) with points of accumulation that are \(\ne 1\) as \(r\rightarrow 1\) whenever \(\theta \in (-\delta (r),\delta (r))\)).

### Theorem 1

*F*(

*x*) satisfies conditions (A) and (B) and

*Q*(

*x*) is the infinite product given by (4). Then, there is a constant \(c>0\) such that, as \(n\rightarrow \infty \),

*q*(

*n*) is the

*n*th coefficient in the Taylor expansion of

*Q*(

*x*).

- (i)
Proof of Hayman admissibility for

*Q*(*x*). - (ii)
Obtaining an asymptotic estimate for the Cauchy integral (17).

## 3 Proof of Theorem 1

*Part (i).*

We will essentially use some more general observations established in [10, 18].

First, we set in (9) and (10) \(G(x):=Q(x)\) and \(r=r_n:=e^{-d_n}\). The next lemma is a particular case of a more general result due to Granovsky et al. [10, Lemma 2].

### Lemma 1

*n*, the unique solution of the equation

### Proof

Lemma 1 shows that \(a(e^{-d_n})\rightarrow \infty \) and \(b(e^{-d_n})\rightarrow \infty \) as \(n\rightarrow \infty \), that is Hayman’s “capture” condition is satisfied with \(r=r_n=e^{-d_n}\).

*Q*(

*x*), we also set

### Lemma 2

*n*, we have

### Proof

*Q*(

*x*) is bounded in the same way with \(\epsilon _1=2/3\). To prove this, we follow an argument from [10] and [18]. Thus, setting \(\theta =2\pi u\) and taking logarithms of the infinite product (4) twice, for \(d_n/2\pi \le |u|=|\theta |/2\pi <1/2\), we get

*y*, we define \(\Vert y\Vert \) to be the distance from

*y*to the nearest integer. To obtain a lower bound for \(S_n\) in this region, we will also use the inequality:

*u*|

*j*. Recalling that \(|u|\ge \delta _n/2\pi \) and applying (22) and (19), we conclude that there is a constant \(C_3^{\prime \prime }>0\) such that, for large enough

*n*,

This lemma, in combination with (18) and (16), implies that \(\mid Q(e^{-d_n+i\theta })\mid =o(Q(e^{-d_n})/\sqrt{b(e^{-d_n})})\) uniformly for \(\delta _n\le \mid \theta \mid \le \pi \), which is just Hayman’s ”decay” condition.

Finally, since \(\zeta (z-1)\) satisfies Meinardus’ conditions \((M_1)\) and \((M_2)\), Lemma 2.3 from [18], implies Hayman’s ”locality” condition for *Q*(*x*).

### Lemma 3

### Remark 4

The fact that MacMahon’s generating function *Q*(*x*) given by (4) is admissible in the sense of Hayman is a particular case of a more general result established in [18] and related to the infinite products \(f_b(x)\) of the form (7). It turns out that the Meinardus scheme of assumptions on \(\{b_j\}_{j\ge 1}\) implies that \(f_b(x)\) is admissible in the sense of Hayman.

*Part (ii)*

*F*(

*x*). We have

*F*(

*x*). It implies that, for certain constants \(c_0, c_1>0\), we have

## 4 Examples

### 4.1 The trace of a plane partition

*u*yields

Combining (39) with (16) and applying the result of Theorem 1 to (34), we obtain the following asymptotic equivalence for \({\mathbb {E}}(T_n)\).

### Proposition 1

### Remark 5

One can compare this asymptotic result with the limit theorem for \(T_n\) obtained in [12], where it is shown that \(T_n\), appropriately normalized, converges weakly to the standard Gaussian distribution.

### 4.2 The largest part, the number of rows and the number of columns of a plane partition

Let \(X_n\), \(Y_n\) and \(Z_n\) denote the size of the largest part, the number of rows and number of columns in a random plane partition of *n*, respectively. Using the solid diagram interpretation \(\Delta (\omega )\) of a plane partition \(\omega \), one can interpret \(X_n\), \(Y_n\) and \(Z_n\) as the height, width and depth of \(\Delta (\omega )\). Any permutation \(\sigma \) of the coordinate axes in \({\mathbb {N}}^3\), different from the identical one, transforms \(\Delta (\omega )\) into a diagram that uniquely determines another plane partition \(\sigma \circ \omega \). The permutation \(\sigma \) also permutes the three statistics \((X_n,Y_n,Z_n)\). So, if one of these statistics is restricted by an inequality, the same restriction occurs on the statistic permuted by \(\sigma \). The one to one correspondence between \(\omega \) and \(\sigma \circ \omega \) implies that \(X_n\), \(Y_n\) and \(Z_n\) are identically distributed for every fixed *n* with respect to the probability measure \({\mathbb {P}}\). (More details may be found in [26, p. 371].) Hence, in the context of the expected value \({\mathbb {E}}\) with respect to the probability measure \({\mathbb {P}}\), we will use the common notation \({\mathbb {E}}(W_n)\) for \(W_n=X_n,Y_n,Z_n\).

*l*and

*m*fixed, setting the other one equal to infinity, we obtain

*m*. Hence, for all \(m\le N_1\),

*O*-term tends to 0 as \(n\rightarrow \infty \) if we set

*n*. First, we analyze the \(\log {\log }\)-term in (52). We have

### Proposition 2

### Remark 6

In [17] is shown that all three dimensions \(X_n\), \(Y_n\) and \(Z_n\) of the random solid diagram with volume *n*, appropriately normalized, converge weakly to the doubly exponential (extreme value) distribution as \(n\rightarrow \infty \).

### Remark 7

It is possible to obtain more precise asymptotic estimates (expansions) using the circle method. A kind of this method was applied by Wright [28] who obtained an asymptotic expansion for the numbers *q*(*n*) as \(n\rightarrow \infty \). His asymptotic expansion together with a suitable expansion for \(F(e^{-d_n})\) would certainly lead to better asymptotic estimates for the expectations of various plane partition statistics.

## Notes

### Acknowledgements

I am grateful the referee for the carefully reading the paper and for her/his helpful comments.

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