Two new explicit formulas for the Bernoulli Numbers

In this brief note, we give two explicit formulas for the Bernoulli Numbers in terms of the Stirling numbers of the second kind, and the Eulerian Numbers. To the best of our knowledge, these formulas are new. We also derive two more probably known formulas.Comment: Updated to give proofs of some necessary result

An identity involving Bernoulli numbers and the Stirling numbers of the second kind

Let $B_{n}$ denote the Bernoulli numbers, and $S(n,k)$ denote the Stirling numbers of the second kind. We prove the following identity $$B_{m+n}=\sum_{\substack{0\leq k \leq n \\ 0\leq l \leq m}}\frac{(-1)^{k+l}\,k!\, l!\, S(n,k)\,S(m,l)}{(k+l+1)\,\binom{k+l}{l}}.$$ To the best of our knowledge, the identity is new.Comment: 3 page

Formulas for the number of kk-colored partitions and the number of plane partitions of nn in terms of the Bell polynomials

We derive closed formulas for the number of $k$-coloured partitions and the number of plane partitions of $n$ in terms of the Bell polynomials

A formula for the rr-coloured partition function in terms of the sum of divisors function and its inverse

Let $p_{-r}(n)$ denote the $r$-coloured partition function, and $\sigma(n)=\sum_{d|n}d$ denote the sum of positive divisors of $n$. The aim of this note is to prove the following $$p_{-r}(n)=\theta(n)+\,\sum_{k=1}^{n-1}\frac{r^{k+1}}{(k+1)!} \sum_{\alpha_1\,= k}^{n-1} \, \sum_{\alpha_2\,= k-1}^{\alpha_1-1} \cdots \sum_{\alpha_k\, = 1}^{\alpha_{k-1}-1}\theta(n-\alpha_1) \theta(\alpha_1 -\alpha_2) \cdots \theta(\alpha_{k-1}-\alpha_k) \theta(\alpha_k)$$ where $\theta(n)=n^{-1}\, \sigma(n)$, and its inverse $$\sigma(n) = n\,\sum_{r=1}^n \frac{(-1)^{r-1}}{r}\, \binom{n}{r}\, p_{-r}(n). $

A formula for the number of partitions of nn in terms of the partial Bell polynomials

We derive a formula for $p(n)$ (the number of partitions of $n$) in terms of the partial Bell polynomials using Fa\`{a} di Bruno's formula and Euler's pentagonal number theorem.Comment: Accepted for publication in the Ramanujan Journa

Estimation of Driver's Gaze Region from Head Position and Orientation using Probabilistic Confidence Regions

A smart vehicle should be able to understand human behavior and predict their actions to avoid hazardous situations. Specific traits in human behavior can be automatically predicted, which can help the vehicle make decisions, increasing safety. One of the most important aspects pertaining to the driving task is the driver's visual attention. Predicting the driver's visual attention can help a vehicle understand the awareness state of the driver, providing important contextual information. While estimating the exact gaze direction is difficult in the car environment, a coarse estimation of the visual attention can be obtained by tracking the position and orientation of the head. Since the relation between head pose and gaze direction is not one-to-one, this paper proposes a formulation based on probabilistic models to create salient regions describing the visual attention of the driver. The area of the predicted region is small when the model has high confidence on the prediction, which is directly learned from the data. We use Gaussian process regression (GPR) to implement the framework, comparing the performance with different regression formulations such as linear regression and neural network based methods. We evaluate these frameworks by studying the tradeoff between spatial resolution and accuracy of the probability map using naturalistic recordings collected with the UTDrive platform. We observe that the GPR method produces the best result creating accurate predictions with localized salient regions. For example, the 95% confidence region is defined by an area that covers 3.77% region of a sphere surrounding the driver.Comment: 13 Pages, 12 figures, 2 table