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We use cosmography to present constraints on the kinematics of the Universe,without postulating any underlying theoretical model. To this end, we use aMonte Carlo Markov Chain analysis to perform comparisons to the supernova IaUnion 2 compilation, combined with the Hubble Space Telescope measurements ofthe Hubble constant, and the Hubble parameter datasets. We introduce a sixthorder cosmographic parameter and show that it does not enlarge considerably theposterior distribution when comparing to the fifth order results. We alsopropose a way to construct viable parameter variables to be used asalternatives of the redshift $z$. These can overcome both the problems ofdivergence and lack of accuracy associated with the use of $z$. Moreover, weshow that it is possible to improve the numerical fits by re-parameterizing thecosmological distances. In addition, we constrain the equation of state of theUniverse as a whole by the use of cosmography. Thus, we derive expressionswhich can be directly used to fit the equation of state and the pressurederivatives up to fourth order. To this end, it is necessary to depart from apure cosmographic analysis and to assume the Friedmann equations as valid. Allour results are consistent with the $\Lambda$CDM model, although alternativefluid models, with nearly constant pressure and no cosmological constant, matchthe results accurately as well.

Cosmography and constraints on the equation of state of the Universe in various parametrizations