\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand*\HyPL@Entry[1]{}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction: drunken walks}{1}{section.1}}
\citation{EIS}
\citation{Guy}
\citation{Stanley}
\citation{Touchard}
\citation{Koshy}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Center of Nice, France, with the promenade at its southern end.}}{2}{figure.1}}
\newlabel{fig:Nice}{{1}{2}{Center of Nice, France, with the promenade at its southern end}{figure.1}{}}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces All Touchard walks of length 4 with equal quantities of \textsf  {N}\xspace  -steps and \textsf  {S}\xspace  -steps, 26 valid and 28 not, besides 16 valid \textsf  {E}\xspace  /\textsf  {W}\xspace  walks with no \textsf  {N}\xspace  /\textsf  {S}\xspace  -steps at all. The illegal steps in the Mediterranean are \relax $\@@underline {\hbox {underlined}}\mathsurround \z@ $\relax . Walks with unequal numbers of \textsf  {N}\xspace  - and \textsf  {S}\xspace  -steps are always invalid.}}{3}{table.1}}
\newlabel{tab}{{1}{3}{All Touchard walks of length 4 with equal quantities of \N -steps and \S -steps, 26 valid and 28 not, besides 16 valid \E /\W walks with no \N /\S -steps at all. The illegal steps in the Mediterranean are \underline {underlined}. Walks with unequal numbers of \N - and \S -steps are always invalid}{table.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A valid walk, \textsf  {N}\xspace  \textsf  {E}\xspace  \textsf  {W}\xspace  \textsf  {W}\xspace  \textsf  {N}\xspace  \textsf  {N}\xspace  \textsf  {E}\xspace  \textsf  {E}\xspace  \textsf  {S}\xspace  \textsf  {E}\xspace  \textsf  {N}\xspace  \textsf  {N}\xspace  \textsf  {S}\xspace  \textsf  {S}\xspace  \textsf  {S}\xspace  \textsf  {S}\xspace  \textsf  {E}\xspace  \textsf  {E}\xspace  , consisting of 18 steps, 5 \textsf  {N}\xspace  , 5 \textsf  {S}\xspace  , 6 \textsf  {E}\xspace  , and 2 \textsf  {W}\xspace  .}}{3}{figure.2}}
\newlabel{fig:walk}{{2}{3}{A valid walk, \N \E \W \W \N \N \E \E \S \E \N \N \S \S \S \S \E \E , consisting of 18 steps, 5 \N , 5 \S , 6 \E , and 2 \W }{figure.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The ten \textsf  {N}\xspace  /\textsf  {S}\xspace  -steps of Figure\nobreakspace  {}\ref  {fig:walk}, stretched out on a timeline: \textsf  {N}\xspace  \textsf  {N}\xspace  \textsf  {N}\xspace  \textsf  {S}\xspace  \textsf  {N}\xspace  \textsf  {N}\xspace  \textsf  {S}\xspace  \textsf  {S}\xspace  \textsf  {S}\xspace  \textsf  {S}\xspace  . Connecting the tails of the steps yields a Dyck path of \textsf  {NE}\xspace  /\textsf  {SE}\xspace  steps.}}{4}{figure.3}}
\newlabel{fig:NS}{{3}{4}{The ten \N /\S -steps of Figure~\ref {fig:walk}, stretched out on a timeline: \N \N \N \S \N \N \S \S \S \S . Connecting the tails of the steps yields a Dyck path of \NE /\SE steps}{figure.3}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The eight \textsf  {E}\xspace  /\textsf  {W}\xspace  -steps of Figure\nobreakspace  {}\ref  {fig:walk}: \textsf  {E}\xspace  \textsf  {W}\xspace  \textsf  {W}\xspace  \textsf  {E}\xspace  \textsf  {E}\xspace  \textsf  {E}\xspace  \textsf  {E}\xspace  \textsf  {E}\xspace  .}}{4}{figure.4}}
\newlabel{fig:EW}{{4}{4}{The eight \E /\W -steps of Figure~\ref {fig:walk}: \E \W \W \E \E \E \E \E }{figure.4}{}}
\citation{Shapiro}
\citation{Whitworth}
\citation{Dyck}
\citation{Koshy}
\citation{lattice}
\@writefile{toc}{\contentsline {section}{\numberline {2}Enumeration: Touchard's identity}{5}{section.2}}
\newlabel{s1}{{1}{5}{Enumeration: Touchard's identity}{Item.1}{}}
\newlabel{s2}{{2}{5}{Enumeration: Touchard's identity}{Item.2}{}}
\newlabel{s3}{{3}{5}{Enumeration: Touchard's identity}{Item.3}{}}
\newlabel{s4}{{4}{5}{Enumeration: Touchard's identity}{Item.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Bijection: Dyck paths}{5}{section.3}}
\citation{Breck}
\citation{GKS}
\citation{two}
\citation{Fell}
\citation{Dyck}
\citation{Stanley}
\citation{Polya}
\citation{GD}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The Dyck path corresponding to the Touchard walk of Figure\nobreakspace  {}\ref  {fig:walk}.}}{6}{figure.5}}
\newlabel{fig:Dyck}{{5}{6}{The Dyck path corresponding to the Touchard walk of Figure~\ref {fig:walk}}{figure.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Extension: more dimensions}{6}{section.4}}
\newlabel{a}{{\textit  {a}}{6}{Extension: more dimensions}{Item.5}{}}
\newlabel{b}{{\textit  {b}}{6}{Extension: more dimensions}{Item.6}{}}
\citation{BF}
\citation{course}
\newlabel{c}{{\textit  {c}}{7}{Extension: more dimensions}{Item.7}{}}
\newlabel{d}{{\textit  {d}}{7}{Extension: more dimensions}{Item.8}{}}
\citation{ab}
\citation{Nagy}
\citation{Knuth}
\citation{Gould}
\citation{Guy}
\citation{Guy}
\citation{Guy}
\citation{Sands}
\citation{DR}
\citation{CMS}
\citation{GKS}
\citation{walks}
\citation{Kreweras}
\citation{Bousquet-Melou}
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Two-dimensional walks. The types \textit  {a}--\textit  {e} are as explained in the text. Each square gives the enumeration of walks with one dimension according to the row and the other according to column. (*The three starred sequences enumerate walks of even length only, returning to the point of origin.)}}{9}{table.2}}
\newlabel{tab:sum}{{2}{9}{Two-dimensional walks. The types \textit {a}--\textit {e} are as explained in the text. Each square gives the enumeration of walks with one dimension according to the row and the other according to column. (*The three starred sequences enumerate walks of even length only, returning to the point of origin.)}{table.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Restriction: three dimensions}{9}{section.5}}
\bibcite{course}{1}
\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces Three-dimensional walks, required to return to the origin (three dimensions of type \textit  {a} or \textit  {b}), axis of origin (two), or plane of origin (one). They may be constrained to a fraction of the space---octant (three of \textit  {a}, \textit  {c}, or \textit  {d}), quadrant (two), or half-space (one), or else allowed the full space (zero). The types are as explained in the text. (*The four starred sequences enumerate walks of even length only. In one case, \textit  {bbc}, the cited sequence, \href  {http://oeis.org/A138547}{\relax $\@@underline {\hbox {A138547}}\mathsurround \z@ $\relax }, has alternating signs.)}}{10}{table.3}}
\newlabel{tab:3D}{{3}{10}{Three-dimensional walks, required to return to the origin (three dimensions of type \textit {a} or \textit {b}), axis of origin (two), or plane of origin (one). They may be constrained to a fraction of the space---octant (three of \textit {a}, \textit {c}, or \textit {d}), quadrant (two), or half-space (one), or else allowed the full space (zero). The types are as explained in the text. (*The four starred sequences enumerate walks of even length only. In one case, \textit {bbc}, the cited sequence, \seqnum {A138547}, has alternating signs.)}{table.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Acknowledgments}{10}{subsection.5.1}}
\bibcite{BF}{2}
\bibcite{two}{3}
\bibcite{walks}{4}
\bibcite{Bousquet-Melou}{5}
\bibcite{Breck}{6}
\bibcite{CMS}{7}
\bibcite{DR}{8}
\bibcite{Dyck}{9}
\bibcite{Gould}{10}
\bibcite{Guy}{11}
\bibcite{GKS}{12}
\bibcite{Knuth}{13}
\bibcite{Koshy}{14}
\bibcite{Kreweras}{15}
\bibcite{lattice}{16}
\bibcite{Nagy}{17}
\bibcite{GD}{18}
\bibcite{Polya}{19}
\bibcite{Sands}{20}
\bibcite{Shapiro}{21}
\bibcite{EIS}{22}
\bibcite{Stanley}{23}
\bibcite{ab}{24}
\citation{EIS}
\bibcite{Fell}{25}
\bibcite{Touchard}{26}
\bibcite{Whitworth}{27}