\contentsline {section}{\tocsection {}{1}{Introduction and main results}}{}
\contentsline {section}{\tocsection {}{2}{Permutation statistics}}{5}
\contentsline {section}{\tocsection {}{3}{Bijections in the literature}}{6}
\contentsline {subsection}{\tocsubsection {}{3.1}{Knuth's bijection, 1973}}{6}
\contentsline {subsection}{\tocsubsection {}{3.2}{Knuth--Rotem's bijection, 1975}}{8}
\contentsline {subsection}{\tocsubsection {}{3.3}{Simion--Schmidt's bijection, 1985}}{8}
\contentsline {subsection}{\tocsubsection {}{3.4}{Knuth--Richards' bijection, 1988}}{9}
\contentsline {subsection}{\tocsubsection {}{3.5}{West's bijection, 1995}}{9}
\contentsline {subsection}{\tocsubsection {}{3.6}{Krattenthaler's bijection, 2001}}{10}
\contentsline {subsection}{\tocsubsection {}{3.7}{Billey--Jockusch--Stanley--Reifegerste's bijection, 2002}}{10}
\contentsline {subsection}{\tocsubsection {}{3.8}{Elizalde-Deutsch's bijection, 2003}}{11}
\contentsline {subsection}{\tocsubsection {}{3.9}{Mansour--Deng--Du's bijection, 2006}}{12}
\contentsline {section}{\tocsection {}{4}{A recursive description of the Knuth--Richards bijection}}{14}
\contentsline {section}{\tocsection {}{5}{Proof of Theorem\nonbreakingspace 2\hbox {}}}{15}
\contentsline {subsection}{\tocsubsection {}{5.1}{Simion--Schmidt versus \ensuremath {\Phi }}}{15}
\contentsline {subsection}{\tocsubsection {}{5.2}{Simion--Schmidt versus Krattenthaler}}{17}
\contentsline {subsection}{\tocsubsection {}{5.3}{Knuth--Richards versus \ensuremath {\Phi }}}{19}
\contentsline {subsection}{\tocsubsection {}{5.4}{Simion--Schmidt versus Mansour--Deng--Du}}{20}
\contentsline {subsection}{\tocsubsection {}{5.5}{Billey--Jockusch--Stanley--Reifegerste versus Knuth--Rotem}}{21}
\contentsline {section}{\tocsection {}{6}{Proof of Theorem\nonbreakingspace 1\hbox {}}}{23}
\contentsline {subsection}{\tocsubsection {}{6.1}{Simion--Schmidt's bijection}}{23}
\contentsline {subsection}{\tocsubsection {}{6.2}{Billey--Jockusch--Stanley--Reifegerste's bijection}}{24}
\contentsline {subsection}{\tocsubsection {}{6.3}{West's bijection}}{26}
\contentsline {subsection}{\tocsubsection {}{6.4}{Knuth's bijection}}{27}
\contentsline {subsection}{\tocsubsection {}{6.5}{Elizalde-Deutsch's bijection}}{28}
\contentsline {section}{\tocsection {}{7}{Appendix}}{28}
\contentsline {section}{\tocsection {}{8}{Acknowledgments}}{29}
\contentsline {section}{\tocsection {}{}{References}}{29}