\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>>}
\citation{ET}
\citation{G}
\citation{L}
\citation{CGK}
\citation{ErSu}
\citation{GerRob}
\citation{Dixon}
\citation{JM}
\citation{LNY2014}
\citation{ES}
\citation{PS}
\citation{BBW}
\citation{CGL}
\citation{CGL2017}
\citation{CGLS2016}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{2}{section.1}}
\citation{DasNath1}
\citation{NathDash1}
\citation{Nath2011}
\citation{DasNath1}
\citation{NathDash1}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Basic definitions and notation}{4}{subsection.1.1}}
\citation{DasNath1}
\citation{DasNath2}
\citation{NathDash1}
\citation{DasNath1}
\citation{DasNath2}
\citation{NathDash1}
\citation{DasNath1}
\citation{DasNath2}
\citation{NathDash1}
\newlabel{Stab.prod.def}{{1}{5}{}{theorem.1}{}}
\citation{DasNath3}
\citation{Hall1940}
\citation{Buckley2014}
\newlabel{def.isoclinic}{{8}{6}{}{theorem.8}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Preliminaries}{6}{section.2}}
\newlabel{commute-conjug}{{10}{6}{}{theorem.10}{}}
\citation{G}
\newlabel{conjug-decomp}{{11}{7}{}{theorem.11}{}}
\newlabel{commute}{{12}{7}{Gustafson}{theorem.12}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Calculation of ${\Spec  }_{3}(G)$ and ${\Spec  }_{4}(G)$}{7}{section.3}}
\citation{DasNath2}
\newlabel{inverse.general}{{13}{8}{}{theorem.13}{}}
\newlabel{threecommute}{{14}{8}{}{theorem.14}{}}
\newlabel{Pr3}{{3}{8}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{theorem.14}{}}
\newlabel{specthree}{{15}{9}{}{theorem.15}{}}
\newlabel{Case.4.1}{{16}{9}{}{theorem.16}{}}
\citation{BP}
\citation{DL}
\newlabel{Case.4.2}{{17}{10}{}{theorem.17}{}}
\newlabel{mainfour}{{18}{10}{}{theorem.18}{}}
\newlabel{small.exchange}{{19}{13}{}{theorem.19}{}}
\newlabel{small.ex.1}{{1}{13}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.1}{}}
\newlabel{small.ex.2}{{2}{13}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.2}{}}
\newlabel{small.ex.3}{{3}{14}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.3}{}}
\newlabel{small.ex.4}{{4}{14}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.4}{}}
\newlabel{mainfour.2}{{20}{14}{}{theorem.20}{}}
\newlabel{mainfour.3}{{21}{15}{}{theorem.21}{}}
\newlabel{small.cyclic}{{22}{15}{}{theorem.22}{}}
\newlabel{small.cyclic.1}{{5}{15}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.5}{}}
\newlabel{small.cyclic.2}{{6}{16}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.6}{}}
\newlabel{small.cyclic.1.1}{{7}{16}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.7}{}}
\newlabel{small.cyclic.3}{{8}{16}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.8}{}}
\newlabel{small.cyclic.4}{{9}{16}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.9}{}}
\newlabel{small.cyclic.3.1}{{10}{16}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.10}{}}
\citation{DasNath1}
\citation{NathDash1}
\newlabel{mainfour.4}{{23}{17}{}{theorem.23}{}}
\newlabel{mainfour.last}{{24}{17}{}{theorem.24}{}}
\newlabel{spect.1}{{11}{17}{Calculation of ${\Spec }_{3}(G)$ and ${\Spec }_{4}(G)$}{equation.3.11}{}}
\citation{DL}
\citation{Sloan}
\newlabel{d4q8.four}{{25}{18}{}{theorem.25}{}}
\newlabel{SPEC4}{{26}{18}{}{theorem.26}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Probabilities of permutation equations, number of alternating cycles, and Hultman decomposition}{18}{section.4}}
\newlabel{def.cycle.graph}{{27}{19}{}{theorem.27}{}}
\newlabel{def.hult.num}{{28}{19}{}{theorem.28}{}}
\newlabel{Theorem.8}{{31}{19}{Doignon and Labarre's theorem 8}{theorem.31}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Exchange operation}{20}{subsection.4.1}}
\newlabel{def.exchange}{{33}{20}{}{theorem.33}{}}
\newlabel{exchange-product}{{38}{21}{}{theorem.38}{}}
\citation{BP}
\newlabel{exchange-circ}{{39}{22}{}{theorem.39}{}}
\citation{BP}
\newlabel{rho.def}{{12}{23}{Exchange operation}{equation.4.12}{}}
\newlabel{exchange-block-trans}{{40}{23}{}{theorem.40}{}}
\newlabel{exchange.same.num.cycles}{{41}{23}{}{theorem.41}{}}
\citation{BP}
\newlabel{exchange.orbit.def}{{42}{24}{}{theorem.42}{}}
\newlabel{exchange.operation.keeps.prob}{{43}{24}{}{theorem.43}{}}
\newlabel{Case1a.Eq1}{{13}{24}{Exchange operation}{equation.4.13}{}}
\newlabel{Case1a.Eq2}{{14}{25}{Exchange operation}{equation.4.14}{}}
\newlabel{special.case.eq.1}{{15}{26}{Exchange operation}{equation.4.15}{}}
\newlabel{special.case.eq.2}{{16}{26}{Exchange operation}{equation.4.16}{}}
\newlabel{Case2a.Eq1}{{17}{26}{Exchange operation}{equation.4.17}{}}
\newlabel{Case2a.Eq2}{{18}{26}{Exchange operation}{equation.4.18}{}}
\newlabel{Case2.Eq1}{{19}{27}{Exchange operation}{equation.4.19}{}}
\newlabel{Case2.Eq2}{{20}{27}{Exchange operation}{equation.4.20}{}}
\newlabel{Case3.Eq1}{{21}{27}{Exchange operation}{equation.4.21}{}}
\newlabel{Case3.Eq2}{{22}{27}{Exchange operation}{equation.4.22}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Cyclic operation}{29}{subsection.4.2}}
\newlabel{cyclic-cdot}{{44}{29}{}{theorem.44}{}}
\newlabel{howcyclicworks}{{50}{30}{}{theorem.50}{}}
\newlabel{HCW.case1}{{23}{32}{Cyclic operation}{equation.4.23}{}}
\newlabel{HCW.case1.new}{{24}{32}{Cyclic operation}{equation.4.24}{}}
\newlabel{HCW.case1.1}{{25}{33}{Cyclic operation}{equation.4.25}{}}
\newlabel{HCW.case1.2}{{26}{33}{Cyclic operation}{equation.4.26}{}}
\newlabel{HCW.case1.1.new}{{27}{33}{Cyclic operation}{equation.4.27}{}}
\newlabel{HCW.case1.2.new}{{28}{33}{Cyclic operation}{equation.4.28}{}}
\newlabel{HCW.case2}{{29}{33}{Cyclic operation}{equation.4.29}{}}
\newlabel{HCW.case2.new}{{30}{34}{Cyclic operation}{equation.4.30}{}}
\newlabel{HCW.case3}{{31}{34}{Cyclic operation}{equation.4.31}{}}
\newlabel{HCW.case3.new}{{32}{34}{Cyclic operation}{equation.4.32}{}}
\newlabel{HCW.case.last}{{33}{35}{Cyclic operation}{equation.4.33}{}}
\newlabel{HCW.case.last.new}{{34}{35}{Cyclic operation}{equation.4.34}{}}
\newlabel{cyclic.orbit.def}{{51}{35}{}{theorem.51}{}}
\newlabel{cyclic.operation.keeps.prob}{{52}{35}{}{theorem.52}{}}
\newlabel{cyclic.first}{{35}{35}{Cyclic operation}{equation.4.35}{}}
\newlabel{cyclic.first.case1}{{36}{36}{Cyclic operation}{equation.4.36}{}}
\newlabel{cyclic.first.case1.2}{{37}{36}{Cyclic operation}{equation.4.37}{}}
\newlabel{cyclic.first.case2}{{38}{36}{Cyclic operation}{equation.4.38}{}}
\newlabel{cyclic.first.case2.2}{{39}{37}{Cyclic operation}{equation.4.39}{}}
\newlabel{cyclic.second}{{40}{37}{Cyclic operation}{equation.4.40}{}}
\newlabel{cyclic.second1}{{41}{37}{Cyclic operation}{equation.4.41}{}}
\newlabel{cyclic.second2}{{42}{38}{Cyclic operation}{equation.4.42}{}}
\newlabel{cyclic.third}{{43}{38}{Cyclic operation}{equation.4.43}{}}
\newlabel{cyclic.third.inv1}{{44}{38}{Cyclic operation}{equation.4.44}{}}
\newlabel{cyclic.third.inv2}{{45}{38}{Cyclic operation}{equation.4.45}{}}
\newlabel{cyclic.third.inv3}{{46}{38}{Cyclic operation}{equation.4.46}{}}
\newlabel{cyclic.third.inv4}{{47}{39}{Cyclic operation}{equation.4.47}{}}
\newlabel{cyclic.third.final}{{48}{39}{Cyclic operation}{equation.4.48}{}}
\newlabel{cyclic.fourth}{{49}{39}{Cyclic operation}{equation.4.49}{}}
\newlabel{cyclic.fourth1}{{50}{39}{Cyclic operation}{equation.4.50}{}}
\newlabel{cyclic.fourth2}{{51}{39}{Cyclic operation}{equation.4.51}{}}
\citation{DasNath3}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3}Main results}{40}{subsection.4.3}}
\newlabel{xy.equiv.def}{{53}{40}{}{theorem.53}{}}
\newlabel{equiv-alternating}{{54}{40}{}{theorem.54}{}}
\newlabel{equivalent.keeps.prob}{{55}{40}{}{theorem.55}{}}
\newlabel{DasNash}{{56}{40}{Theorem 6.8 of Das and Nath}{theorem.56}{}}
\newlabel{generic.group}{{57}{40}{}{theorem.57}{}}
\newlabel{d4q8.general}{{58}{41}{}{theorem.58}{}}
\newlabel{d4q8.formula}{{52}{41}{}{equation.4.52}{}}
\newlabel{permutation.augment}{{59}{41}{}{theorem.59}{}}
\newlabel{tech.lem.0}{{60}{41}{}{theorem.60}{}}
\newlabel{tech.lem.0a}{{61}{42}{}{theorem.61}{}}
\newlabel{tech.lem.1}{{62}{42}{}{theorem.62}{}}
\newlabel{tech.lem.2}{{63}{43}{}{theorem.63}{}}
\newlabel{tech.lem.3}{{64}{44}{}{theorem.64}{}}
\newlabel{tech.lem.4}{{65}{45}{}{theorem.65}{}}
\newlabel{last.lemma.eq.1}{{53}{45}{Main results}{equation.4.53}{}}
\newlabel{hultman-exchange-cyclic}{{66}{45}{}{theorem.66}{}}
\citation{DL}
\newlabel{main.theorem.1}{{67}{46}{}{theorem.67}{}}
\newlabel{main.theorem.2}{{68}{46}{}{theorem.68}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Two explicit formulae for $\qopname  \relax m{Pr}^{2t}(G)$}{47}{section.5}}
\newlabel{main.result.3}{{69}{47}{}{theorem.69}{}}
\newlabel{Eq.Stab.Prod}{{54}{47}{}{equation.5.54}{}}
\newlabel{Eq.weak.ring}{{55}{47}{}{equation.5.55}{}}
\newlabel{final.equation1}{{56}{49}{Two explicit formulae for $\Pr ^{2t}(G)$}{equation.5.56}{}}
\citation{DasNath2}
\citation{DasNath2}
\@writefile{toc}{\contentsline {section}{\numberline {6}Conclusion}{50}{section.6}}
\newlabel{Commutator}{{70}{50}{}{theorem.70}{}}
\newlabel{Thm_Isoclinic}{{71}{50}{}{theorem.71}{}}
\citation{BP}
\bibcite{BP}{1}
\bibcite{BBW}{2}
\bibcite{Buckley2014}{3}
\bibcite{CGL}{4}
\bibcite{CGLS2016}{5}
\bibcite{CGL2017}{6}
\bibcite{CGK}{7}
\bibcite{DasNath1}{8}
\bibcite{DasNath2}{9}
\bibcite{DasNath3}{10}
\newlabel{isoclinic}{{72}{51}{}{theorem.72}{}}
\@writefile{toc}{\contentsline {section}{\numberline {7}Acknowledgments}{51}{section.7}}
\bibcite{Dixon}{11}
\bibcite{DL}{12}
\bibcite{ES}{13}
\bibcite{ET}{14}
\bibcite{ErSu}{15}
\bibcite{G}{16}
\bibcite{GerRob}{17}
\bibcite{Hall1940}{18}
\bibcite{JM}{19}
\bibcite{L}{20}
\bibcite{LNY2014}{21}
\bibcite{NathDash1}{22}
\bibcite{Nath2011}{23}
\bibcite{PS}{24}
\bibcite{Sloan}{25}
