Let $\overline {M}(a,c,n)$ denote the number of overpartitions of n with first residual crank congruent to a modulo c with $c\geq 3$ being odd and $0\leq a<c$. The central objective of this paper is twofold: firstly, to establish an asymptotic formula for the crank of overpartitions; and secondly, to establish several inequalities concerning $\overline {M}(a,c,n)$ that encompasses crank differences, positivity, and strict log-subadditivity.

We study the counts of smooth permutations and smooth polynomials over finite fields. For both counts we prove an estimate with an error term that matches the error term found in the integer setting by de Bruijn more than 70 years ago. The main term is the usual Dickman $\rho$ function, but with its argument shifted.

We determine the order of magnitude of $\log(p_{n,m}/\rho(n/m))$ where $p_{n,m}$ is the probability that a permutation on n elements, chosen uniformly at random, is m-smooth.

We uncover a phase transition in the polynomial setting: the probability that a polynomial of degree n in $\mathbb{F}_q$ is m-smooth changes its behaviour at $m\approx (3/2)\log_q n$.

Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$, let ${\mathcal G}(n,W)$ be the random graph obtained by independently including each edge $jk\in \binom{[n]}{2}$ with probability $W_{jk}=W_{kj}$. Given a degree sequence $\textbf{d}=(d_1,\ldots, d_n)$, let ${\mathcal G}(n,\textbf{d})$ denote a uniformly random graph with degree sequence $\textbf{d}$. We couple ${\mathcal G}(n,W)$ and ${\mathcal G}(n,\textbf{d})$ together so that asymptotically almost surely ${\mathcal G}(n,W)$ is a subgraph of ${\mathcal G}(n,\textbf{d})$, where $W$ is some function of $\textbf{d}$. Let $\Delta (\textbf{d})$ denote the maximum degree in $\textbf{d}$. Our coupling result is optimal when $\Delta (\textbf{d})^2\ll \|\textbf{d}\|_1$, that is, $W_{ij}$ is asymptotic to ${\mathbb P}(ij\in{\mathcal G}(n,\textbf{d}))$ for every $i,j\in [n]$. We also have coupling results for $\textbf{d}$ that are not constrained by the condition $\Delta (\textbf{d})^2\ll \|\textbf{d}\|_1$. For such $\textbf{d}$ our coupling result is still close to optimal, in the sense that $W_{ij}$ is asymptotic to ${\mathbb P}(ij\in{\mathcal G}(n,\textbf{d}))$ for most pairs $ij\in \binom{[n]}{2}$.

We establish some inequalities that arise from truncating Lerch sums and derive uniform asymptotic formulae for the spt-crank of ordinary partitions. The uniform asymptotic formulae improve upon a result of Mao [‘Asymptotic formulas for spt-crank of partitions’, J. Math. Anal. Appl. 460(1) (2018), 121–139].

The protection number of a vertex $v$ in a tree is the length of the shortest path from $v$ to any leaf contained in the maximal subtree where $v$ is the root. In this paper, we determine the distribution of the maximum protection number of a vertex in simply generated trees, thereby refining a recent result of Devroye, Goh, and Zhao. Two different cases can be observed: if the given family of trees allows vertices of outdegree $1$, then the maximum protection number is on average logarithmic in the tree size, with a discrete double-exponential limiting distribution. If no such vertices are allowed, the maximum protection number is doubly logarithmic in the tree size and concentrated on at most two values. These results are obtained by studying the singular behaviour of the generating functions of trees with bounded protection number. While a general distributional result by Prodinger and Wagner can be used in the first case, we prove a variant of that result in the second case.

We study the distribution of the length of longest increasing subsequences in random permutations of n integers as n grows large and establish an asymptotic expansion in powers of $n^{-1/3}$. Whilst the limit law was already shown by Baik, Deift and Johansson to be the GUE Tracy–Widom distribution F, we find explicit analytic expressions of the first few finite-size correction terms as linear combinations of higher order derivatives of F with rational polynomial coefficients. Our proof replaces Johansson’s de-Poissonization, which is based on monotonicity as a Tauberian condition, by analytic de-Poissonization of Jacquet and Szpankowski, which is based on growth conditions in the complex plane; it is subject to a tameness hypothesis concerning complex zeros of the analytically continued Poissonized length distribution. In a preparatory step an expansion of the hard-to-soft edge transition law of LUE is studied, which is lifted to an expansion of the Poissonized length distribution for large intensities. Finally, expansions of Stirling-type approximations and of the expected value and variance of the length distribution are given.

In 2007, Andrews introduced Durfee symbols and k-marked Durfee symbols so as to give a combinatorial interpretation for the symmetrized moment function $\eta _{2k}(n)$ of ranks of partitions. He also considered the relations between odd Durfee symbols and the mock theta function $\omega (q)$, and proved that the $2k$th moment function $\eta _{2k}^0(n)$ of odd ranks of odd Durfee symbols counts $(k+1)$-marked odd Durfee symbols of n. In this paper, we first introduce the definition of symmetrized positive odd rank moments $\eta _k^{0+}(n)$ and prove that for all $1\leq i\leq k+1$, $\eta _{2k-1}^{0+}(n)$ is equal to the number of $(k+1)$-marked odd Durfee symbols of n with the ith odd rank equal to zero and $\eta _{2k}^{0+}(n)$ is equal to the number of $(k+1)$-marked Durfee symbols of n with the ith odd rank being positive. Then we calculate the generating functions of $\eta _{k}^{0+}(n)$ and study its asymptotic behavior. Finally, we use Wright’s variant of the Hardy–Ramanujan circle method to obtain an asymptotic formula for $\eta _{k}^{0+}(n)$.

Let $k \geqslant 2$ be an integer. We prove that factorisation of integers into k parts follows the Dirichlet distribution $\mathrm{Dir}\left({1}/{k},\ldots,{1}/{k}\right)$ by multidimensional contour integration, thereby generalising the Deshouillers–Dress–Tenenbaum (DDT) arcsine law on divisors where $k=2$. The same holds for factorisation of polynomials or permutations. Dirichlet distribution with arbitrary parameters can be modelled similarly.

Given a connected graph $H$ which is not a star, we show that the number of copies of $H$ in a dense uniformly random regular graph is asymptotically Gaussian, which was not known even for $H$ being a triangle. This addresses a question of McKay from the 2010 International Congress of Mathematicians. In fact, we prove that the behavior of the variance of the number of copies of $H$ depends in a delicate manner on the occurrence and number of cycles of $3,4,5$ edges as well as paths of $3$ edges in $H$. More generally, we provide control of the asymptotic distribution of certain statistics of bounded degree which are invariant under vertex permutations, including moments of the spectrum of a random regular graph. Our techniques are based on combining complex-analytic methods due to McKay and Wormald used to enumerate regular graphs with the notion of graph factors developed by Janson in the context of studying subgraph counts in $\mathbb {G}(n,p)$.

Suppose that m drivers each choose a preferred parking space in a linear car park with n spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case $m=n$.

We investigate various probabilistic properties of a uniform parking function. Through a combinatorial construction termed a parking function multi-shuffle, we give a formula for the law of multiple coordinates in the generic situation $m \lesssim n$. We further deduce all possible covariances: between two coordinates, between a coordinate and an unattempted spot, and between two unattempted spots. This asymptotic scenario in the generic situation $m \lesssim n$ is in sharp contrast with that of the special situation $m=n$.

A generalization of parking functions called interval parking functions is also studied, in which each driver is willing to park only in a fixed interval of spots. We construct a family of bijections between interval parking functions with n cars and n spots and edge-labeled spanning trees with $n+1$ vertices and a specified root.

We derive three critical exponents for Bernoulli site percolation on the uniform infinite planar triangulation (UIPT). First, we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is $\beta = 1/2$. Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least n vertices decays like $n^{-1/7}$. Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to n decays like $n^{-4/3}$. Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, and analytic combinatorics.

We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$-uniform hypergraph with a given number of edges. More specifically, we compare the degree sequence of a random $r$-uniform hypergraph with a given number edges to certain models involving sequences of binomial or hypergeometric random variables conditioned on their sum.

We prove and generalise a conjecture in [MPP4] about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$, where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.

Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define the asymptotic density

$${c_t} = \mathop {\lim }\limits_{N \to \infty } {1 \over N}|\{ 0 \le n < N:s(n + t) \ge s(n)\} |.$$

We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d1, …, dn) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes dj= λn + O(n1/2+ε) for some λ bounded away from 0 and 1.

We study random unlabelled k-trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle-pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov–Hausdorff–Prokhorov and Benjamini–Schramm limits that describe their asymptotic geometric shape on a global and local scale as the number of (k + 1)-cliques tends to infinity.

In this paper, we investigate $\unicode[STIX]{x1D70B}(m,n)$, the number of partitions of the bipartite number$(m,n)$ into steadily decreasing parts, introduced by Carlitz [‘A problem in partitions’, Duke Math. J.30 (1963), 203–213]. We give a relation between $\unicode[STIX]{x1D70B}(m,n)$ and the crank statistic $M(m,n)$ for integer partitions. Using this relation, we establish some uniform asymptotic formulas for $\unicode[STIX]{x1D70B}(m,n)$.

Bevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result using only Stirling's formula, the method of Lagrange multipliers, and the singular value decomposition of matrices. Our proof relies on showing that the maximum over the space of n × n matrices with non-negative entries summing to one of a certain function of those entries, parametrized by the entries of another matrix Γ of non-negative real numbers, is equal to the square of the largest singular value of Γ and that the maximizing point can be expressed as a Hadamard product of Γ with the tensor product of singular vectors for its greatest singular value.

A strongly concave composition of $n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7(9) (2013), 2103–2139].

We prove a generalmulti-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.