\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]{}
\citation{Dedekind_1897}
\citation{Dedekind_1897}
\citation{Dedekind_1897}
\citation{Church_1940}
\citation{Ward_1946}
\citation{Church_1965}
\citation{Berman_Koehler_1976}
\citation{Markowsky_1989}
\citation{Wiedemann_1991}
\citation{Fidytek_etal_2001}
\citation{Lunnon_1971}
\citation{Kisielewicz_1988}
\citation{Sloane_DB}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\newlabel{sec_introduction}{{1}{1}{Introduction}{section.1}{}}
\citation{Berman_Koehler_1976}
\citation{Birkhoff_1967}
\citation{Lunnon_1971}
\citation{Markowsky_1989}
\citation{Riviere_1968}
\citation{Cvetkovic_1972}
\citation{Arocha_1982}
\citation{Kilibarda_Jovovic_2003}
\citation{Pippenger_1999}
\citation{Kahn_2002}
\citation{Carroll_etal_2009}
\citation{Bakoev_2012}
\citation{Gilbert_1954}
\citation{Hansel_1966}
\citation{Kleitman_1969_2}
\citation{Kleitman_Markowsky_1975}
\citation{Korshunov_1981}
\citation{Sapozhenko_1989}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces  Known Dedekind numbers.}}{2}{table.1}}
\newlabel{table_Dedekind_Zahlen}{{1}{2}{Known Dedekind numbers}{table.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Preliminaries}{3}{section.2}}
\newlabel{sec_Preliminaries}{{2}{3}{Preliminaries}{section.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Notation}{3}{subsection.2.1}}
\newlabel{subsec_notation}{{2.1}{3}{Notation}{subsection.2.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces  $\Lambda _4$ and $M_5$.}}{4}{figure.1}}
\newlabel{figure_L4_M5}{{1}{4}{$\Lambda _4$ and $M_5$}{figure.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Downsets and downset-lattices}{4}{subsection.2.2}}
\newlabel{subsec_Downsets}{{2.2}{4}{Downsets and downset-lattices}{subsection.2.2}{}}
\newlabel{isom_HPDQ_HQDP}{{1}{4}{Downsets and downset-lattices}{equation.2.1}{}}
\citation{Birkhoff_1967}
\citation{Birkhoff_1967}
\newlabel{formeln_inf_sup}{{2}{5}{Downsets and downset-lattices}{equation.2.2}{}}
\newlabel{lemma_intervallpotenzen}{{1}{5}{}{theorem.1}{}}
\newlabel{isom_DDPQ}{{3}{5}{Downsets and downset-lattices}{equation.2.3}{}}
\citation{Wiedemann_1991}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces  Free distributive lattices with up to three generators.}}{6}{figure.2}}
\newlabel{figure_FDLs}{{2}{6}{Free distributive lattices with up to three generators}{figure.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Powers and exponential sums}{6}{section.3}}
\newlabel{sec_DDQk}{{3}{6}{Powers and exponential sums}{section.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Powers of $d^2 ( Q )$ and $ d^2 ( A_{1} +Q ) $}{6}{subsection.3.1}}
\newlabel{subsec_powers}{{3.1}{6}{Powers of $\dDQ $ and $\dDQk {1}$}{subsection.3.1}{}}
\newlabel{ungl_1_2}{{4}{6}{Powers of $\dDQ $ and $\dDQk {1}$}{equation.3.4}{}}
\newlabel{ungl_3_2}{{5}{6}{Powers of $\dDQ $ and $\dDQk {1}$}{equation.3.5}{}}
\newlabel{def_connected}{{2}{6}{}{theorem.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces  Illustrations to Theorem \ref  {theorem_rho_inj} and Corollary \ref  {corr_HPB_HP}. a) The posets $B \equiv (C_2)^3$ and $S$ with their points represented by binary words. b) Assignment of the sets $ \bot , \ell , r, \ldots  $ to the points of $B$ and $S$.}}{7}{figure.3}}
\newlabel{figure_Proof_Theorem}{{3}{7}{Illustrations to Theorem \ref {theorem_rho_inj} and Corollary \ref {corr_HPB_HP}. a) The posets $B \equiv (C_2)^3$ and $S$ with their points represented by binary words. b) Assignment of the sets $ \bot , \ell , r, \ldots $ to the points of $B$ and $S$}{figure.3}{}}
\newlabel{corr_gamma_def}{{3}{7}{}{theorem.3}{}}
\newlabel{gamma_BB}{{6}{7}{}{equation.3.6}{}}
\newlabel{gamma_BB2}{{7}{7}{}{equation.3.7}{}}
\newlabel{gamma_ABAB}{{8}{7}{}{equation.3.8}{}}
\newlabel{gamma_ggc}{{9}{7}{}{equation.3.9}{}}
\newlabel{lemma_gamma}{{4}{7}{}{theorem.4}{}}
\newlabel{theorem_rho_inj}{{5}{8}{}{theorem.5}{}}
\newlabel{eq_Gamma_1}{{10}{8}{Powers of $\dDQ $ and $\dDQk {1}$}{equation.3.10}{}}
\newlabel{eq_Gamma_2}{{11}{8}{Powers of $\dDQ $ and $\dDQk {1}$}{equation.3.11}{}}
\newlabel{corr_HPB_HP}{{6}{9}{}{theorem.6}{}}
\newlabel{corr_injektion}{{7}{9}{}{theorem.7}{}}
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces  Dedekind numbers and their squares and cubes.}}{10}{table.2}}
\newlabel{table_Squares_Cubes}{{2}{10}{Dedekind numbers and their squares and cubes}{table.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Exponential sums}{10}{subsection.3.2}}
\newlabel{subsec_sums}{{3.2}{10}{Exponential sums}{subsection.3.2}{}}
\citation{Berman_Koehler_1976}
\citation{Fidytek_etal_2001}
\citation{Lunnon_1971}
\citation{Markowsky_1989}
\citation{Riviere_1968}
\citation{Yusun_2011}
\citation{Fidytek_etal_2001}
\newlabel{powersum_d}{{12}{11}{Exponential sums}{equation.3.12}{}}
\newlabel{powersum_I}{{13}{11}{Exponential sums}{equation.3.13}{}}
\newlabel{isom_DDQm}{{14}{11}{Exponential sums}{equation.3.14}{}}
\newlabel{formel_dDQe}{{15}{11}{Exponential sums}{equation.3.15}{}}
\newlabel{formel_dDQz}{{16}{11}{Exponential sums}{equation.3.16}{}}
\newlabel{lemma_bijHom_Emb}{{8}{11}{}{theorem.8}{}}
\citation{Davey_Priestley_2012}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces  Illustrations for the proof of Corollary \ref  {corr_sumsDS}.}}{13}{figure.4}}
\newlabel{figure_Proof_C1}{{4}{13}{Illustrations for the proof of Corollary \ref {corr_sumsDS}}{figure.4}{}}
\newlabel{corr_sumsDS}{{9}{13}{}{theorem.9}{}}
\newlabel{sum_A2Q_leq_sum3}{{17}{13}{}{equation.3.17}{}}
\newlabel{sumkL_leq_sumLk}{{18}{13}{}{equation.3.18}{}}
\newlabel{sum_1k_2k1}{{19}{13}{}{equation.3.19}{}}
\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces  Values of $\DOTSB \sum@ \slimits@ _{ \mathfrak  { A } \in  {\cal  D}^2 ( A_{n} )  } \delta (\mathfrak  { A })^k$.}}{14}{table.3}}
\newlabel{table_sum_Ldk}{{3}{14}{Values of $\sum _{ \mf {A} \in \DDk {n} } \delta (\mf {A})^k$}{table.3}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces  Illustrations for the proof of Corollary \ref  {corr_sumsInt}.}}{14}{figure.5}}
\newlabel{figure_Proof_C2}{{5}{14}{Illustrations for the proof of Corollary \ref {corr_sumsInt}}{figure.5}{}}
\newlabel{corr_sumsInt}{{10}{14}{}{theorem.10}{}}
\newlabel{dDQ2_SumI3_dDQ3}{{20}{15}{}{equation.3.20}{}}
\newlabel{dDQ3_SumI6_dDQ4}{{21}{15}{}{equation.3.21}{}}
\newlabel{dDQk_SumIk_dDQk1}{{22}{15}{}{equation.3.22}{}}
\newlabel{SumIk_SumI2k}{{23}{15}{}{equation.3.23}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Averages}{15}{subsection.3.3}}
\newlabel{subsec_Averages}{{3.3}{15}{Averages}{subsection.3.3}{}}
\citation{Berman_Koehler_1976}
\citation{Fidytek_etal_2001}
\citation{Lunnon_1971}
\citation{Markowsky_1989}
\citation{Shmulevich_etal_1995}
\citation{Wiedemann_1991}
\citation{Yusun_2011}
\citation{Markowsky_1989}
\citation{Shmulevich_etal_1995}
\citation{Lunnon_1971}
\citation{Markowsky_1989}
\citation{Shmulevich_etal_1995}
\@writefile{lot}{\contentsline {table}{\numberline {4}{\ignorespaces  Averages of downset sizes and products of downset sizes.}}{16}{table.4}}
\newlabel{table_averages}{{4}{16}{Averages of downset sizes and products of downset sizes}{table.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Computational time of algorithms for the calculation of Dedekind numbers}{16}{section.4}}
\newlabel{sec_calculation}{{4}{16}{Computational time of algorithms for the calculation of Dedekind numbers}{section.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Algorithms}{16}{subsection.4.1}}
\newlabel{subsec_RecForm}{{4.1}{16}{Algorithms}{subsection.4.1}{}}
\citation{Church_1940}
\citation{Lunnon_1971}
\citation{Berman_Koehler_1976}
\citation{Fidytek_etal_2001}
\citation{Lunnon_1971}
\citation{Markowsky_1989}
\citation{Riviere_1968}
\citation{Yusun_2011}
\citation{Fidytek_etal_2001}
\citation{Berman_Koehler_1976}
\citation{Lunnon_1971}
\citation{Markowsky_1989}
\citation{Wiedemann_1991}
\citation{Yusun_2011}
\citation{Fidytek_etal_2001}
\newlabel{alg_L0}{{24}{17}{Algorithms}{equation.4.24}{}}
\newlabel{alg_L1}{{25}{17}{Algorithms}{equation.4.25}{}}
\newlabel{alg_L1_Fid}{{26}{17}{Algorithms}{equation.4.26}{}}
\newlabel{alg_L2_BK}{{27}{17}{Algorithms}{equation.4.27}{}}
\newlabel{alg_L2_Fid1}{{28}{17}{Algorithms}{equation.4.28}{}}
\newlabel{alg_L2_Fid2}{{29}{17}{Algorithms}{equation.4.29}{}}
\citation{Fidytek_etal_2001}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces  Illustration of algorithm \textup  {\hbox {\mathsurround \z@ \normalfont  (\ignorespaces \ref  {alg_L3}\unskip \@@italiccorr )}}.}}{18}{figure.6}}
\newlabel{figure_Alg3}{{6}{18}{Illustration of algorithm \eqref {alg_L3}}{figure.6}{}}
\newlabel{alg_L3}{{30}{18}{Algorithms}{equation.4.30}{}}
\newlabel{alg_L4}{{31}{18}{Algorithms}{equation.4.31}{}}
\citation{Markowsky_1989}
\citation{Lunnon_1971}
\@writefile{lot}{\contentsline {table}{\numberline {5}{\ignorespaces  Number of calls of the innermost calculation formula.}}{19}{table.5}}
\newlabel{table_noofcalls}{{5}{19}{Number of calls of the innermost calculation formula}{table.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Computational time}{19}{subsection.4.2}}
\newlabel{subsec_effort}{{4.2}{19}{Computational time}{subsection.4.2}{}}
\citation{Fidytek_etal_2001}
\citation{Church_1940}
\citation{Church_1940}
\citation{Lunnon_1971}
\citation{Markowsky_1989}
\citation{Wiedemann_1991}
\citation{Lunnon_1971}
\citation{Markowsky_1989}
\citation{Wiedemann_1991}
\citation{Markowsky_1989}
\citation{Markowsky_1989}
\citation{Wiedemann_1991}
\citation{Fidytek_etal_2001}
\citation{Church_1965}
\citation{Fidytek_etal_2001}
\citation{Markowsky_1989}
\citation{Shmulevich_etal_1995}
\@writefile{lot}{\contentsline {table}{\numberline {6}{\ignorespaces  Number of interval enumerations for the calculation of $d^2(A_{4+n})$.}}{20}{table.6}}
\newlabel{table_noofIntEn_DAk}{{6}{20}{Number of interval enumerations for the calculation of $d^2(A_{4+n})$}{table.6}{}}
\citation{Shmulevich_etal_1995}
\citation{Church_1965}
\citation{Fidytek_etal_2001}
\bibcite{Arocha_1982}{1}
\@writefile{toc}{\contentsline {section}{\numberline {5}Appendix}{22}{section.5}}
\newlabel{sec_appendix}{{5}{22}{Appendix}{section.5}{}}
\bibcite{Bakoev_2012}{2}
\bibcite{Berman_Koehler_1976}{3}
\bibcite{Birkhoff_1967}{4}
\bibcite{Carroll_etal_2009}{5}
\bibcite{Church_1940}{6}
\bibcite{Church_1965}{7}
\bibcite{Cvetkovic_1972}{8}
\bibcite{Davey_Priestley_2012}{9}
\bibcite{Dedekind_1897}{10}
\bibcite{Fidytek_etal_2001}{11}
\bibcite{Gilbert_1954}{12}
\bibcite{Hansel_1966}{13}
\bibcite{Kahn_2002}{14}
\bibcite{Kilibarda_Jovovic_2003}{15}
\bibcite{Kisielewicz_1988}{16}
\bibcite{Kleitman_1969_2}{17}
\bibcite{Kleitman_Markowsky_1975}{18}
\bibcite{Korshunov_1981}{19}
\bibcite{Lunnon_1971}{20}
\bibcite{Markowsky_1989}{21}
\bibcite{Riviere_1968}{22}
\bibcite{Pippenger_1999}{23}
\bibcite{Sapozhenko_1989}{24}
\bibcite{Shmulevich_etal_1995}{25}
\bibcite{Sloane_DB}{26}
\bibcite{Ward_1946}{27}
\bibcite{Wiedemann_1991}{28}
\bibcite{Yusun_2011}{29}