Anti-lecture Hall Compositions and Overpartitions

We show that the number of anti-lecture hall compositions of n with the first entry not exceeding k-2 equals the number of overpartitions of n with non-overlined parts not congruent to $0,\pm 1$ modulo k. This identity can be considered as a refined version of the anti-lecture hall theorem of Corteel and Savage. To prove this result, we find two Rogers-Ramanujan type identities for overpartition which are analogous to the Rogers-Ramanjan type identities due to Andrews. When k is odd, we give an alternative proof by using a generalized Rogers-Ramanujan identity due to Andrews, a bijection of Corteel and Savage and a refined version of a bijection also due to Corteel and Savage.Comment: 16 page

The Rogers-Ramanujan-Gordon Theorem for Overpartitions

Let $B_{k,i}(n)$ be the number of partitions of $n$ with certain difference condition and let $A_{k,i}(n)$ be the number of partitions of $n$ with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that $B_{k,i}(n)=A_{k,i}(n)$. Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases $i=1$ and $i=k$. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let $D_{k,i}(n)$ be the number of overpartitions of $n$ satisfying certain difference condition and $C_{k,i}(n)$ be the number of overpartitions of $n$ whose non-overlined parts satisfy certain congruences condition. We show that $C_{k,i}(n)=D_{k,i}(n)$. By using a function introduced by Andrews, we obtain a recurrence relation which implies that the generating function of $D_{k,i}(n)$ equals the generating function of $C_{k,i}(n)$. We also find a generating function formula of $D_{k,i}(n)$ by using Gordon marking representations of overpartitions, which can be considered as an overpartition analogue of an identity of Andrews for ordinary partitions.Comment: 26 page

On the Nature of X(4260)

We study the property of $X(4260)$ resonance by re-analyzing all experimental data available, especially the $e^+e^- \rightarrow J/\psi\,\pi^+\pi^-,\,\,\,\omega\chi_{c0}$ cross section data. The final state interactions of the $\pi\pi$, $K\bar K$ couple channel system are also taken into account. A sizable coupling between the $X(4260)$ and $\omega\chi_{c0}$ is found. The inclusion of the $\omega\chi_{c0}$ data indicates a small value of $\Gamma_{e^+e^-}=23.30\pm 3.55$eV.Comment: Refined analysis with new experimental data included. 13 page

A Two-Dimensional CA Traffic Model with Dynamic Route Choices Between Residence and Workplace

The Biham, Middleton and Levine (BML) model is extended to describe dynamic route choices between the residence and workplace in cities. The traffic dynamic in the city with a single workplace is studied from the velocity diagram, arrival time probability distribution, destination arrival rate and convergence time. The city with double workplaces is also investigated to compared with a single workplace within the framework of four modes of urban growth. The transitional region is found in the velocity diagrams where the system undergoes a continuous transition from a moving phase to a completely jamming phase. We perform a finite-size scaling analysis of the critical density from a statistical point of view and the order parameter of this jamming transition is estimated. It is also found that statistical properties of urban traffic are greatly influenced by the urban area, workplace area and urban layout.Comment: 18 pages, 13 figure