Partitions and the maximal excludant

For each nonempty integer partition $\pi$, we define the maximal excludant of $\pi$ to be the largest nonnegative integer smaller than the largest part of $\pi$ that is not a part of $\pi$. Let $\sigma\!\operatorname{maex}(n)$ be the sum of maximal excludants over all partitions of $n$. We show that the generating function of $\sigma\!\operatorname{maex}(n)$ is closely related to a mock theta function studied by Andrews \textit{et al.} and Cohen. Further, we show that, as $n\to \infty$, $\sigma\!\operatorname{maex}(n)$ is asymptotic to the sum of largest parts of all partitions of $n$. Finally, the expectation of the difference of the largest part and the maximal excludant over all partitions of $n$ is shown to converge to $1$ as $n\to \infty$.


Introduction
In a recent paper [3], Andrews and Newman studied the minimal excludant of an integer partition π, which is the smallest positive integer that is not a part of π. Since a nonempty partition π is a finite sequence of positive integers, we may also study the maximal excludant of π, by which we mean the largest nonnegative integer smaller than the largest part of π that is not itself a part. For example, 5 has seven partitions: 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1 and 1 + 1 + 1 + 1 + 1, the maximal excludants of which are, respectively, 4, 3, 1, 2, 0, 0 and 0.
Let mex(π) and maex(π) denote, respectively, the minimal and maximal excludant of π. Andrews and Newman further investigated the function σmex(n) := π n mex (π) in which the summation is over all partitions of n. They proved that the generating function of σmex(n) satisfies where we adopt the conventional q-Pochhammer symbol for n ∈ N ∪ {∞}: Likewise, we may define another function where, again, the summation runs over all partitions of n. In this paper, we are to study the generating function of σmaex(n). As we shall see in Theorem 1, unlike the generating function of σmex(n), which is modular, the generating function of σmaex(n) is closely related to a mock theta function studied in two side-by-side papers of Andrews, Dyson and Hickerson [2] and Cohen [4].
Remark 2. Using a formula due to Andrews, Dyson and Hickerson [2], we may give an explicit expression of σmaex(n). This will be discussed in Section 2.
Now recall that if L(π) denotes the largest part of a partition π and denotes the sum of largest parts over all partitions of n, a standard result tells us that In light of (3) and (5), we have the following corollary.  σL(n) − σmaex(n) q n = 1 (q; q) ∞ n 1 q n (q 2 ; q 2 ) n−1 .
It was shown by Kessler and Livingston [6] that σL(n) satisfies the asymptotic formula where γ is the Euler-Mascheroni constant. Now we shall show asymptotic relations as follows.
Further, if E n denotes the expectation of the difference of the largest part and the maximal excludant over all partitions of n, then Remark 5. Notice that for any nonempty partition π, we always have L(π)−maex(π) 1. Hence, for all n 1, we have E n 1. Further, for all n 3, it is always able to find a partition π of n with L(π) − maex(π) > 1 (if n is odd, then such a partition could be ((n + 1)/2, (n − 1)/2); if n is even, then such a partition could be (n/2, (n − 2)/2, 1)). This implies that E n > 1 for n 3.

A mock theta function
In his paper [4], Cohen observed the following identity Hence, (3) and (4) are equivalent. It is worth mentioning that Cohen's identity (11) can be generalized to a trivariate identity as follows.
Proof of Proposition 6. Both sides of (12) can be treated as the generating function of partitions in which the largest part appears only once and all the remaining distinct parts appear exactly twice. Here, the exponent of x represents the number of distinct parts in this partition and the exponent of y represents the largest part minus the number of distinct parts.
Remark 7. Let us denote It is also necessary to introduce its companion The two q-hypergeometric functions are of substantial research interest along the following lines. First, Andrews, Dyson and Hickerson [2] showed that the coefficients in the expansions of σ(q) and σ * (q) are very small. In fact, these coefficients are related with the arithmetic of the field Q( √ 6). Second, let us define a sequence {T (n)} n∈24Z+1 by Cohen [4] proved that the function (in which K 0 (x) is the Bessel function) φ 0 (τ ) := y 1/2 n∈24Z+1 T (n)K 0 (2π|n|y/24)e 2πinx/24 is a Maass wave form on the congruence group Γ 0 (2). This, in turn, explains the modularity nature of the identity qσ(q 24 ) = a,b∈Z a>6|b| Third, by noticing the following relation due to Cohen [4]: whenever q is a root of unity (here the definitions of σ(q) and σ * (q) at roots of unity are valid since the second summations in both (13) and (14) are finite), Zagier [8] is able to construct a quantum modular form f : Q → C by f (x) := q 1/24 σ(q) = −q 1/24 σ * (q −1 ) where q = e 2πix .
Theorem 8. For n 1, we have where p(n) denotes the number of partitions of n.

Proof of Theorem 1
The equivalence of (3) and (4) has already been shown in Section 2. It suffices to prove (2) and (3). Given a partition with maximal excludant k, it can be split into two components: the first component is a partition with parts not exceeding k − 1 and the second component is a gap-free partition (i.e. a partition in which the difference between any consecutive parts is at most 1) with smallest part k + 1. Further, by considering the conjugate, there is a bijection between gap-free partitions with smallest part k + 1 and partitions in which the largest part repeats k + 1 times and all remaining parts are distinct. Hence, if g(k, n) counts the number of partitions of n with maximal excludant k, we have the generating function identity Now applying the operator [∂/∂z] z=1 directly to G(z, q) implies (2). Next, we prove (3). Recall Euler's first summation [1,Eq. (2.2.5) In light of (18), we have An easy combinatorial argument implies that, for all n 1, n m=1 q m (q; q) m−1 = 1 − (q; q) n .
This completes the proof of (3).