Sensitivity of symmetric Boolean functions

In quantum computing theory, the well-known Deutsch's problem and Deutsch-Jozsa problem can be equivalent to symmetric Boolean functions. Meanwhile, sensitivity of Boolean functions is a quite important complexity measure in the query model. So far, whether symmetry means high-sensitivity problems is still considered as a challenge. In symmetric setting, based on whether all inputs in {0,1}n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{0,1\}<^>{n}$$\end{document} are defined, this paper investigates sensitivity of total and partial Boolean functions, respectively. Firstly, we point out that the computation of sensitivity requires at most n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} classical queries or n quantum queries. Secondly, we show that the lower bound of sensitivity is not less than n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{n}{2}$$\end{document} except for the sensitivity 0. Finally, we discover and prove some non-trivial bounds on the number of symmetric (total and partial) Boolean functions with each possible sensitivity.