Some generating functions and inequalities for the andrews–stanley partition functions

Let $\mathcal {O}(\pi )$ denote the number of odd parts in an integer partition $\pi$ . In 2005, Stanley introduced a new statistic $\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$ , where $\pi '$ is the conjugate of $\pi$ . Let $p(r,\,m;n)$ denote the number of partitions of $n$ with srank congruent to $r$ modulo $m$ . Generating function identities, congruences and inequalities for $p(0,\,4;n)$ and $p(2,\,4;n)$ were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for $p(r,\,m;n)$ with $m=16$ and $24$ . These results are refinements of some inequalities due to Swisher.

Partitions and the Maximal Excludant

For each nonempty integer partition $\pi$, we define the maximal excludant of $\pi$ as the largest nonnegative integer smaller than the largest part of $\pi$ that is not itself a part. 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, Dyson and Hickerson, and Cohen, respectively. Further, we show that, as $n\to \infty$, $\sigma\!\operatorname{maex}(n)$ is asymptotic to the sum of largest parts over 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 $n\to \infty$.

Distribution of generalized mex-related integer partitions

International audience The minimal excludant or "mex" function for an integer partition π of a positive integer n, mex(π), is the smallest positive integer that is not a part of π. Andrews and Newman introduced σmex(n) to be the sum of mex(π) taken over all partitions π of n. Ballantine and Merca generalized this combinatorial interpretation to σrmex(n), as the sum of least r-gaps in all partitions of n. In this article, we study the arithmetic density of σ_2 mex(n) and σ_3 mex(n) modulo 2^k for any positive integer k.

Rank partition functions and truncated theta identities

In 1944, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G.E. Andrews and F.G. Garvan. In this paper, we introduce truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank. As corollaries we derive new infinite families of linear inequalities for the partition function p(n). The number of Garden of Eden partitions are also considered in this context in order to provide other infinite families of linear inequalities for p(n).

Asymptotic Bounds on Graphical Partitions and Partition Comparability

An integer partition is called graphical if it is the degree sequence of a simple graph. We prove that the probability that a uniformly chosen partition of size $n$ is graphical decreases to zero faster than $n^{-.003}$, answering a question of Pittel. A lower bound of $n^{-1/2}$ was proven by Erd̋s and Richmond, meaning our work demonstrates that the probability decreases polynomially. Our proof also implies a polynomial upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.

On the largest part size and its multiplicity of a random integer partition

Let λ be a partition of the positive integer n chosen uniformly at random among all such partitions. Let Ln = Ln(λ) and Mn = Mn(λ) 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 {1 \over 2}\log n . We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.