Separate section number in headers

Question

I'm writing a book and want my separate odd and even headers, with centered chapter/section titles on even/odd pages respectively. On both pages I want to print the section number on the outside of the page.

I thought this would be easy to do with, for example,

\fancyhead[LE]{\thesection}
\fancyhead[RO]{\thesection}

Unfortunately, this fails when a new section heading starts at the top of a page, in which case the previous page gets labelled with the new section number. I know this happens because \thesection is expanded too late, but I want an easy fix. Also, I can't use the \rightmark to store the section number because I'm using that to store the section name.

Here's a minimum working example that illustrates the problem:

\documentclass{article}
\usepackage{lipsum}

\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead{}
\fancyhead[L]{\leftmark}
\fancyhead[R]{\thesection}

\renewcommand{\sectionmark}[1]{\markboth{\thesection. #1}{}} 

\begin{document}
\section{Introduction}
\lipsum[1-4]
\lipsum[2]

\section{Blah}
\lipsum[1]
\end{document}

Answer 1

The following extracts the section number from \leftmark that you are using for the full section title. The section number is identified as anything at the beginning until .␣ is seen. Otherwise nothing is set.

\documentclass{article}
\usepackage{lipsum}

\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead{}
\fancyhead[L]{\leftmark}
\fancyhead[R]{\leftmarksection}

\renewcommand{\sectionmark}[1]{\markboth{\thesection. #1}{}}

\makeatletter
\newcommand*{\leftmarksection}{%
  \begingroup
    \protected@edef\temp{\leftmark}%
    \expandafter\@leftmarksection\temp. \@nil
  \endgroup
}
\def\@leftmarksection#1. #2\@nil{%
  \def\temp{#2}%
  \ifx\temp\@empty
  \else
    #1%
  \fi  
}
\makeatother

\begin{document}
Title page
\newpage  
\section{Introduction}
\lipsum[1-4]
\lipsum[2]  

\section{Blah}
\lipsum[1]
\end{document}

Answer 2

Perhaps you want this, I'm not sure which mark you need in your real case.

\documentclass{article}
\usepackage{lipsum}

\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead{}
\fancyhead[L]{\leftmark}
\fancyhead[R]{\expandafter\numberonly\romannumeral`\^^@\leftmark!!}
\def\numberonly#1. #2!!{#1}

\renewcommand{\sectionmark}[1]{\markboth{\thesection. #1}{}} 

\begin{document}
\section{Introduction}
\lipsum[1-4]
\lipsum[2]

\section{Blah}
\lipsum[1]
\end{document}

Answer 3

Thanks to both David and Heiko for your answers. I wanted a solution that I could understand, so I used the xstring package to implement unpacking the section number and title from the \leftmark:

\documentclass{article}
\usepackage{lipsum}
\usepackage{xstring}

\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead{}
\fancyhead[L]{\StrBehind{\leftmark}{+}}
\fancyhead[R]{\StrBefore{\leftmark}{+}}

\renewcommand{\sectionmark}[1]{\markboth{\thesection+#1}{}} 

\begin{document}
\section{Introduction}
\lipsum[1-4]
\lipsum[2]

\section{Blah}
\lipsum[1]
\end{document}