Calculation of moments of combinatorial statistics of symmetrically dependent random variables

We consider statistics defined on sequences of symmetrically dependent random variables such that the numbers of ascents, descents, equalities, double ascents, double descents, the number of peaks, and the number of cavities. We obtain explicit expressions for the first moments of such statistics in terms of simple characteristics of the joint distributions.
The research of the first author was supported by the program of the President of Russian Federation for support of leading scientific schools, grant 1758.2003.1, and by the program of the Russian Academy of Sciences ‘Modern Problems of Theoretical Mathematics.’