Bijective enumeration of some colored permutations given by the product of two long cycles

Let $\gamma_n$ be the permutation on $n$ symbols defined by $\gamma_n = (1\ 2\...\ n)$. We are interested in an enumerative problem on colored permutations, that is permutations$\beta$of$n$in which the numbers from 1 to$n$are colored with$p$colors such that two elements in a same cycle have the same color. We show that the proportion of colored permutations such that$\gamma_n \beta^{-1}$is a long cycle is given by the very simple ratio$\frac{1}{n- p+1}$. Our proof is bijective and uses combinatorial objects such as partitioned hypermaps and thorn trees. This formula is actually equivalent to the proportionality of the number of long cycles$\alpha$such that$\gamma_n\alpha$has$m$cycles and Stirling numbers of size$n+1$, an unexpected connection previously found by several authors by means of algebraic methods. Moreover, our bijection allows us to refine the latter result with the cycle type of the permutations.Comment: 22 pages. Version 1 is a short version of 12 pages, entitled "Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result", published in DMTCS proceedings of FPSAC 2010, AN, 713-72

The second critical point for the Perfect Bose gas in quasi-one-dimensional traps

We present a new model of quasi-one-dimensional trap with some unknown physical predictions about a second transition, including about a change in fractions of condensed coherence lengths due to the existence of a second critical temperature Tm < Tc. If this physical model is acceptable, we want to challenge experimental physicists in this regard

Dynamics of an Open System for Repeated Harmonic Perturbation

We use the Kossakowski-Lindblad-Davies formalism to consider an open system defined as the Markovian extension of one-mode quantum oscillator S, perturbed by a piecewise stationary harmonic interaction with a chain of oscillators C. The long-time asymptotic behaviour of various subsystems of S+C are obtained in the framework of the dual W-dynamical system approach

Random point field approach to analysis of anisotropic Bose-Einstein condensations

Position distributions of constituent particles of the perfect Bose-gas trapped in exponentially and polynomially anisotropic boxes are investigated by means of the boson random point fields (processes) and by the spatial random distribution of particle density. Our results include the case of \textit{generalised} Bose-Einstein Condensation. For exponentially anisotropic quasi two-dimensional system (SLAB), we obtain \textit{three} qualitatively different particle density distributions. They correspond to the \textit{normal} phase, the quasi-condensate phase (type III generalised condensation) and to the phase when the type III and the type I Bose condensations co-exist. An interesting feature is manifested by the type II generalised condensation in one-directional polynomially anisotropic system (BEAM). In this case the particle density distribution rests truly random even in the \textit{macroscopic} scaling limit