We show that certain sums of partition numbers are divisible by multiples of 2 and 3. For example, if $p(n)$ denotes the number of unrestricted partitions of a positive integer n (and $p(0)=1$, $p(n)=0$ for $n<0$), then for all nonnegative integers m,

$$ \begin{align*}\sum_{k=0}^\infty p(24m+23-\omega(-2k))+\sum_{k=1}^\infty p(24m+23-\omega(2k))\equiv 0~ (\text{mod}~144),\end{align*} $$

where $\omega (k)=k(3k+1)/2$.

Let $p_t(a,b;n)$ denote the number of partitions of n such that the number of t-hooks is congruent to $a \bmod {b}$ . For $t\in \{2, 3\}$ , arithmetic progressions $r_1 \bmod {m_1}$ and $r_2 \bmod {m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$ vanishes were established in recent work by Bringmann, Craig, Males and Ono [‘Distributions on partitions arising from Hilbert schemes and hook lengths’, Forum Math. Sigma 10 (2022), Article no. e49] using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of t-cores and t-quotients.

Let p(n) denote the partition function. In this paper, we will prove that for $n\ges 222$,

$$\left| {\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \cr {p(n-1)} & {p(n)} & {p(n + 1)} \cr {p(n-2)} & {p(n-1)} & {p(n)} \cr } } \right| > 0.{\rm }$$

$n\ges 222$

$$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$

A point process is R-dependent if it behaves independently beyond the minimum distance R. In this paper we investigate uniform positive lower bounds on the avoidance functions of R-dependent simple point processes with a common intensity. Intensities with such bounds are characterised by the existence of Shearer's point process, the unique R-dependent and R-hard-core point process with a given intensity. We also present several extensions of the Lovász local lemma, a sufficient condition on the intensity and R to guarantee the existence of Shearer's point process and exponential lower bounds. Shearer's point process shares a combinatorial structure with the hard-sphere model with radius R, the unique R-hard-core Markov point process. Bounds from the Lovász local lemma convert into lower bounds on the radius of convergence of a high-temperature cluster expansion of the hard-sphere model. This recovers a classic result of Ruelle (1969) on the uniqueness of the Gibbs measure of the hard-sphere model via an inductive approach of Dobrushin (1996).

The sequence of random probability measures ν n that gives a path of length n, $\frac{1}{n}$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment.Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.

The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts, such as continuous time finance models and one-dimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.