%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2021}
\articlenumber{73}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}	% もともとは thm に従属させない
\newtheorem{prop}[thm]{Propsition}
\newtheorem{Def}[thm]{Definition}
\newtheorem{Rem}[thm]{Remark}
\newtheorem{Ex}[thm]{Example}


\usepackage{graphicx}
\usepackage{overpic}
%\usepackage{tikz-cd}
%\usetikzlibrary{decorations.pathreplacing}
%\usepackage[all]{xy}

%\usepackage{lipsum}
\usepackage{comment}
\usepackage{multicol}



\newcommand{\Mapsto}{}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% macro in math styles %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\TT}{ \mathbb{T} }
\newcommand{\PP}{ \mathbb{P} }
%\newcommand{\AA}{ \mathbb{A} }

\newcommand{\NN}{ \mathbb{N} }
\newcommand{\ZZ}{ \mathbb{Z} }
\newcommand{\QQ}{ \mathbb{Q} }
\newcommand{\RR}{ \mathbb{R} }
\newcommand{\CC}{ \mathbb{C} }
\newcommand{\HH}{ \mathbb{H} }
\newcommand{\OO}{ \mathbb{O} }

\newcommand{\ep}{ \varepsilon }
\newcommand{\ph}{ \varphi }


\newcommand{\CASE}[1]{ {\left\{\begin{array}{ll} #1 \end{array}\right.} }


\newcommand{\numm}{ \,\,\, \mathrm{if \,\, and \,\, only \,\, if} \,\,\, }
\newcommand{\num}{ \,\,\, \mathrm{if} \,\,\, }

\newcommand{\unique}{ ! }

\newcommand{\et}{ \,\,\, \mathrm{and} \,\,\, }
\newcommand{\ou}{ \,\,\, \mathrm{or} \,\,\, }
\newcommand{\aut}{ \,\,\, \mathrm{or} \,\,\, }

\newcommand{\then}{ \mbox{ then } }
\newcommand{\wehave}{ \mbox{we have } }
%\newcommand{\implies}{ \mbox{ implies } }

\newcommand{\is}{ \mbox{ is } }
\newcommand{\isnot}{ \mbox{ is not } }


%\newcommand{\lq}{\mbox{`}}
%\newcommand{\rq}{\mbox{'}}
\newcommand{\lqq}{\mbox{``}}
\newcommand{\rqq}{\mbox{''}}

\newcommand{\set}[2]{
\left\{ \! \begin{array}{c|c} \! #1 \! & \! #2 \! \end{array} \! \right\}
}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\pl}{ \prec }
\newcommand{\pleq}{ \preceq }
\newcommand{\pg}{ \succ }
\newcommand{\pgeq}{ \succeq }


\newcommand{\tl}{ \triangleleft }
\newcommand{\tg}{ \triangleright }

\newcommand{\Tl}{ \lhd }
\newcommand{\Tleq}{ \unlhd }
\newcommand{\Tg}{ \rhd }
\newcommand{\Tgeq}{ \unrhd }

%arrows

\newcommand{\arrLR}{ \leftrightarrow }
\newcommand{\arrL}{ \leftarrow }
\newcommand{\arrR}{ \rightarrow }
\newcommand{\ArrLR}{ \Leftrightarrow }
\newcommand{\ArrL}{ \Leftarrow }
\newcommand{\ArrR}{ \Rightarrow }
\newcommand{\longarrLR}{ \longleftrightarrow }
\newcommand{\longarrL}{ \longleftarrow }
\newcommand{\longarrR}{ \longrightarrow }
\newcommand{\LongarrLR}{ \Longleftrightarrow }
\newcommand{\LongarrL}{ \Longleftarrow }
\newcommand{\LongarrR}{ \Longrightarrow }

\newcommand{\arrUD}{ \updownarrow }
\newcommand{\arrU}{ \uparrow }
\newcommand{\arrD}{ \downarrow }
\newcommand{\ArrUD}{ \Updownarrow }
\newcommand{\ArrU}{ \Uparrow }
\newcommand{\ArrD}{ \Downarrow }
\newcommand{\longarrUD}{ \updownarrow }
\newcommand{\longarrU}{ \uparrow }
\newcommand{\longarrD}{ \downarrow }
\newcommand{\LongarrUD}{ \Updownarrow }
\newcommand{\LongarrU}{ \Uparrow }
\newcommand{\LongarrD}{ \Downarrow }

\newcommand{\arrUR}{ \nearrow }
\newcommand{\arrDR}{ \searrow }
\newcommand{\arrDL}{ \swarrow }
\newcommand{\arrUL}{ \nwarrow }
\newcommand{\ArrUR}{ \Nearrow }
\newcommand{\ArrDR}{ \Searrow }
\newcommand{\ArrDL}{ \Swarrow }
\newcommand{\ArrUL}{ \Nwarrow }


% relation

\newcommand{\lsim}{ \lesssim }
\newcommand{\gsim}{ \gtrsim }
\newcommand{\simm}{ \approx }
\newcommand{\lsimm}{ \lessapprox }
\newcommand{\gsimm}{ \gtrapprox }

\newcommand{\simeqq}{ \cong }


% map

\newcommand{\id}[1]{ \mathrm{id}_{ #1 } }
\renewcommand{\Mapsto}{\rightarrow}            %%%% NOTE %%%%
\newcommand{\inj}{ \hookrightarrow }
\newcommand{\surj}{ \twoheadrightarrow }

\newcommand{\inU}{\setlength{\unitlength}{1.75mm}
  \begin{picture}(1,1)
  \put(0.5,0.875){\oval(1,1.5)[b]}
  \put(0.5,0.875){\line(0,-1){0.75}}
  \end{picture}
}
\newcommand{\maps}[4]{\!\!
  \mbox{
  \begin{tabular}[t]{ccc}
  \( #1 \) & \!\( \Mapsto \)\! & \( #4 \) \\
  \( \inU \) &               & \( \inU \) \\
  \( #2 \) & \!\( \mapsto \)\! & \( #3 \)
  \end{tabular}}
}







\newcommand{\plus}{\hspace{2.24mm}}%{+}
\newcommand{\nl}{\plus}




\newcommand{\st}{ \mbox{ s.t. } }


\newcommand{\resp}{ \mathrm{resp.}\, }
\newcommand{\ie}{ \,\,\, \mathrm{i.e.} \,\,\, }


\newcommand{\widecheck}[1]{ \widehat{ #1 } }
\newcommand{\widebar}[1]{ \overline{ #1 } }
\newcommand{\widevec}[1]{ \overline{ #1 } }

\newcommand{\overvec}[1]{ \stackrel{ \rightarrow }{ #1 } }
\newcommand{\overcev}[1]{ \stackrel{ \leftarrow }{ #1 } }
\newcommand{\overconj}[1]{ \bar{ #1 } }




% Linear Algebra

%%%% - vector

\newcommand{\vect}[1]{ \mathbf{#1} }

\newcommand{\VEC}[1]{ {\left[ \!\!\begin{array}{c} #1 \end{array}\!\! \right]} } % number vector as point
\newcommand{\COORD}[1]{ {\left[ \!\!\begin{array}{c} #1 \end{array}\!\! \right]} } % coordinate vector

%%%% - matrix

\newcommand{\Mat}[2]{ {\mathrm{Mat}_{#1}\left(#2\right)} }

\newcommand{\MATa}[1]{ {\left[\! \begin{array}{c} #1 \end{array} \!\right]} }
\newcommand{\MATb}[1]{ {\left[\! \begin{array}{cc} #1 \end{array} \!\right]} }
\newcommand{\MATc}[1]{ {\left[\! \begin{array}{ccc} #1 \end{array} \!\right]} }
\newcommand{\MATd}[1]{ {\left[\! \begin{array}{cccc} #1 \end{array} \!\right]} }
\newcommand{\MATe}[1]{ {\left[\! \begin{array}{ccccc} #1 \end{array} \!\right]} }
\newcommand{\MATf}[1]{ {\left[\! \begin{array}{cccccc} #1 \end{array} \!\right]} }
\newcommand{\MATg}[1]{ {\left[\! \begin{array}{ccccccc} #1 \end{array} \!\right]} }
\newcommand{\BASIS}[1]{ {\left[\! \begin{array}{c} #1 \end{array} \!\right]} } % BASIS matirx

\newcommand{\Tr}{ \mathrm{Tr}\, } % Trace
\newcommand{\tr}{ \mathrm{tr} }   % normaliezed trace

%%%% - permutation

\newcommand{\PERMa}[1]{ {\left( \begin{array}{c} #1 \end{array} \right)} }
\newcommand{\PERMb}[1]{ {\left( \begin{array}{cc} #1 \end{array} \right)} }
\newcommand{\PERMc}[1]{ {\left( \begin{array}{ccc} #1 \end{array} \right)} }
\newcommand{\PERMd}[1]{ {\left( \begin{array}{cccc} #1 \end{array} \right)} }
\newcommand{\PERMe}[1]{ {\left( \begin{array}{ccccc} #1 \end{array} \right)} }
\newcommand{\PERMf}[1]{ {\left( \begin{array}{cccccc} #1 \end{array} \right)} }
\newcommand{\PERMg}[1]{ {\left( \begin{array}{ccccccc} #1 \end{array} \right)} }


% Morphism

\newcommand{\Mor}[2]{ {\mathrm{Mor}_{#1}(#2)} }

\newcommand{\Aut}[2]{ {\mathrm{Aut}_{#1}(#2)} }
\newcommand{\End}[2]{ {\mathrm{End}_{#1}(#2)} }
\newcommand{\Iso}[2]{ {\mathrm{Iso}_{#1}(#2)} }
\newcommand{\Hom}[2]{ {\mathrm{Hom}_{#1}(#2)} }


\newcommand{\Inn}[2]{ {\mathrm{Inn}_{#1}\left(#2\right)} } % inner automorphism group
\newcommand{\Out}[2]{ {\mathrm{Out}_{#1}\left(#2\right)} } % outer automorphism group


%%% image - coimage - kernel - cokernel

\newcommand{\ch}{ \mathrm{ch} }         % characteristic

%\newcommand{\Im}{ {\mathrm{Im}\,} }
\newcommand{\im}{ {\mathrm{im}\,} }
\newcommand{\Image}{ {\mathrm{Im}\,} }

\newcommand{\Coim}{ {\mathrm{Coim}\,} }
\newcommand{\coim}{ {\mathrm{coim}\,} }
\newcommand{\Coimage}{ {\mathrm{Coim}\,} }
\newcommand{\coimage}{ {\mathrm{coim}\,} }

\newcommand{\Ker}{ {\mathrm{Ker}\,} }
%\newcommand{\ker}{ {\mathrm{ker}\,} }

\newcommand{\Coker}{ {\mathrm{Coker}\,} }
%\newcommand{\coker}{ {\mathrm{coker}\,} }
\newcommand{\Cok}{ {\mathrm{Cok}\,} }
%\newcommand{\cok}{ {\mathrm{cok}\,} }

%

\newcommand{\DD}{ \mathbb{D} }
\newcommand{\FF}{ \mathbb{F} }



\newcommand{\acts}{ : }  % left action
\newcommand{\acted}{ : } % right action

\newcommand{\Lact}[1]{ {\displaystyle \mathop{\curvearrowright}^{#1} } }
\newcommand{\Ract}[1]{ {\displaystyle \mathop{\curvearrowleft}^{#1} }  }

\newcommand{\actR}[1]{ {\displaystyle \mathop{\curvearrowright}^{#1} } }
\newcommand{\actL}[1]{ {\displaystyle \mathop{\curvearrowleft}^{#1} }  }

\newcommand{\tensor}[1]{ {\displaystyle \mathop{\otimes}_{#1} } }

\newcommand{\SG}[1]{ {\mathfrak{S}_{#1}} } % symmetric group
\newcommand{\AG}[1]{ {\mathfrak{A}_{#1}} } % alternating group
\newcommand{\CG}[1]{ {\mathfrak{C}_{#1}} } % cyclic group with order #1
\newcommand{\DG}[1]{ {\mathfrak{D}_{#1}} } % dihedral group with order 2 #1
\newcommand{\BSG}[1]{ {\widehat{\mathfrak{S}}_{#1}} } % binary symmetric group
\newcommand{\BAG}[1]{ {\widehat{\mathfrak{A}}_{#1}} } % binary alternating group
\newcommand{\BCG}[1]{ {\widehat{\mathfrak{S}}_{#1}} } % binary cyclic group with order 2 #1
\newcommand{\BDG}[1]{ {\widehat{\mathfrak{D}}_{#1}} } % binary dihedral group with order 4 #1



%Wely group

\newcommand{\sgn}{\mathrm{sgn}\,}



%Lie group

\newcommand{\AGL}[2]{ {\mathrm{AGL}_{#1} \left( #2 \right)} } % subset of Mat(#1 +1)
\newcommand{\ASL}[2]{ {\mathrm{ASL}_{#1} \left( #2 \right)} } % subset of Mat(#1 +1)


\newcommand{\Isom}[2]{ {\mathrm{Isom}_{#1}(#2)} }




\newcommand{\GL}[2]{ {\mathrm{GL}_{#1}(#2)} }
\newcommand{\SL}[2]{ {\mathrm{SL}_{#1}(#2)} }
\newcommand{\GU}[2]{ {\mathrm{U}_{#1}(#2)} }
\newcommand{\SU}[2]{ {\mathrm{SU}_{#1}(#2)} }
\newcommand{\GO}[2]{ {\mathrm{O}_{#1}(#2)} }
\newcommand{\SO}[2]{ {\mathrm{SO}_{#1}(#2)} }
\newcommand{\Gpin}[2]{ {\mathrm{pin}_{#1}(#2)} }
\newcommand{\pin}[2]{{ \mathrm{pin}_{#1} (#2) }}
\newcommand{\Spin}[2]{ {\mathrm{Spin}_{#1}(#2)} }
\newcommand{\GS}[2]{ {\mathrm{Sp}_{#1}(#2)} }
%\newcommand{\SS}[2]{ {\mathrm{SU}_{#1}(#2)} }

\newcommand{\PGL}[2]{ {\mathrm{PGL}_{#1}(#2)} }
\newcommand{\PSL}[2]{ {\mathrm{PSL}_{#1}(#2)} }
\newcommand{\PGU}[2]{ {\mathrm{PU}_{#1}(#2)} }
\newcommand{\PSU}[2]{ {\mathrm{PSU}_{#1}(#2)} }
\newcommand{\PGO}[2]{ {\mathrm{PO}_{#1}(#2)} }
\newcommand{\PSO}[2]{ {\mathrm{PSO}_{#1}(#2)} }
\newcommand{\PS}[2]{ {\mathrm{PSp}_{#1}(#2)} }
\newcommand{\PSp}[2]{ {\mathrm{PSp}_{#1}(#2)} }
\newcommand{\PGSp}[2]{ {\mathrm{PSp}_{#1}(#2)} }


%Lie algebra

\newcommand{\gl}[2]{ {\mathrm{gl}_{#1}(#2)} }
\newcommand{\Lsl}[2]{ {\mathrm{sl}_{#1}(#2)} }

\newcommand{\isom}[2]{ {\mathrm{isom}_{#1}(#2)} }

\newcommand{\Lsu}[2]{ {\mathrm{su}_{#1}(#2)} }
\newcommand{\gu}[2]{ {\mathrm{u}_{#1}(#2)} }
\newcommand{\su}[2]{ {\mathrm{su}_{#1}(#2)} }
\newcommand{\go}[2]{ {\mathrm{o}_{#1}(#2)} }
\newcommand{\so}[2]{ {\mathrm{so}_{#1}(#2)} }
\newcommand{\gs}[2]{ {\mathrm{sp}_{#1}(#2)} }
%\newcommand{\ss}[2]{ {\mathrm{su}_{#1}(#2)} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{ \h }{ \mathfrak{h} }   %%%% Cartan subalgebra %%%%
\newcommand{ \g }{ \mathfrak{g} }   %%%% Lie algebra %%%%
\newcommand{ \rroot }{ \Phi }       %%%% real root system %%%%

\newcommand{ \iw }{ P }             %%%% the set of integral weights %%%%
\newcommand{ \pdiw }{ P_{\geq -1} } %%%% the set of pre-dominant integral weights %%%%
\newcommand{\sigiw}{ P_{\!\mathrm{sig}} }
\newcommand{\nulliw}{P_0}
\newcommand{ \Gpdiw }[1]{ \pdiw^{\alpha_{#1}} }
\newcommand{\diw}{ \iw_{\geq 0} }
\newcommand{\infpdiw}{ \pdiw^{\rm \,inf} }
\newcommand{\finpdiw}{ \pdiw^{\rm \,fin} }
\newcommand{ \nliw}{ P_{0} }







\newcommand{ \G }[2]{ \mathcal{G}_{#1}( #2 ) }
\newcommand{ \F }[3]{ \mathcal{F}^{(#1)}_{#2}( #3 ) }
\newcommand{ \GF }[2]{ \mathcal{G}_{\alpha_{#1}}( #2 ) }
\newcommand{ \US }[3]{ \mathcal{F}^{(#1)}_{\alpha_{#2}}( #3 ) }




\newcommand{ \diag }[1]{ \mathrm{D}( #1 ) }
\newcommand{ \twdiag }[2]{ \mathrm{D}^{ \alpha_{ #1 } }( #2 ) }
\newcommand{ \gendiag }[2]{ \mathrm{GD}^{ \alpha_{ #1 } }( #2 ) }
\newcommand{ \Gdiag }[2]{ \mathrm{GD}^{ \alpha_{ #1 } }( #2 ) }



\newcommand{\Hook}[2]{ \mathrm{H}_{#1} \left( #2 \right)}
\newcommand{\inv}[1]{ \Phi \left( #1 \right) }
\newcommand{\htt}[1]{ \mathrm{ht} \left( #1 \right)}



\newcommand{ \Path }[1]{ \mathrm{Path}( #1 ) }
\newcommand{ \basPath }[2]{ \mathrm{Path}_{ \alpha_{#1} }( #2 ) }
\newcommand{ \twPath }[2]{ s_{#1} \left( \mathrm{Path}( s_{ \alpha_{#1} }( #2 ) ) \right) }
\newcommand{ \GPath }[2]{ \mathrm{GPath}_{ \alpha_{#1} }( #2 ) }
\newcommand{\QPath}[2]{\mathrm{QPath}_{\alpha_{#1}}\!\left( #2 \right)}


\newcommand{ \shiftable }[1]{ #1 \mbox{-shiftable} }

%\newcommand{ \Path }[1]{ \mathrm{Path}( #1 ) }
\newcommand{ \MPath }[1]{ \mathrm{MPath}( #1 ) }
\newcommand{ \STab }[1]{ \mathrm{STab} ( #1 ) }
\newcommand{ \Red }[1]{ \mathrm{Red} ( #1 ) }





\newcommand{\shape}[1]{ {\diag{#1}^{\!\vee}} }

\newcommand{\covered}{ \stackrel{\,_{\bullet}}{<} }
\newcommand{\bruhat}[3]{ #1 \stackrel{ #2 }{ \triangleleft } #3 }
\newcommand{\horder}[1]{ \stackrel{ #1 }{ \triangleleft } }

\newcommand{\PSTab}[1]{ \mathrm{PSTab}( #1 ) }
\newcommand{\Numb} [1]{\mathrm{Numb}( #1 )}
\newcommand{\MNumb}[1]{\mathrm{MNumb}( #1 )}

\newcommand{\SLab} [1]{\mathrm{SLab}( #1 )}




\newcommand{\reach}[2]{ #2 \mbox{ is reachable from } #1 }


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%  q-hook formula   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\newcommand{\qq}{ \mathbf{q} }
%\renewcommand{\erat}[1]{ \frac{ \qq^{#1} }{ 1 - \qq^{#1} } }
\newcommand{\erat}[1]{ \frac{ e^{#1} }{ 1 - e^{#1} } }
\newcommand{\qrat}[1]{ \frac{ q^{\htt{#1}} }{ 1 - q^{\htt{#1}} } }

\newcommand{\qqrat}[1]{ \frac{ \qq^{#1} }{ 1 \!-\! \qq^{#1} }     }

\newcommand{\terat}[1]{ \textstyle{ \erat{#1} } }
\newcommand{\txqqrat}[1]{ \textstyle{ \qqrat{#1} } }



\newcommand{\Skew}{ \mathrm{\Sigma} }
\newcommand{\SSkew}{ \mathrm{\Sigma\!\!\!\;\Sigma} }
\newcommand{\strict}{ {}^{\times} }

%%%% W-ordered set %%%%

\newcommand{\Jf}[1]{ \mathrm{J_f}\left( #1 \right) }
\newcommand{\Part}[1]{ \mathsf{A}\left( #1 \right) }
\newcommand{\Chain}[1]{ \mathrm{Chain}\left( #1 \right) }

\newcommand{\xx}{\mathbf{x}}
\newcommand{\xrat}[1]{ \frac{ \xx^{#1} }{ 1-\xx^{#1} } }
\newcommand{\xxrat}[1]{ \frac{ \xx^{#1} }{ 1-\xx^{#1} } }
\newcommand{\LE}[1]{ \mathcal{L}\left( #1 \right) }



%%%%%%%%%%%%%%%%%



%\newcommand{\Ref}[1]{ \mathrm{Ref}\left( #1 \right) }


\newcommand{\ACmove}[2]{ \langle #1 ,\, #2 ^\vee \rangle \!=\! \langle #2 ,\, #1 ^\vee \rangle \!=\! 2 }
\newcommand{\Shift}[1]{ #1 \mbox{-shiftable} }
\newcommand{\shift}[2]{ #1 \mbox{ is } #2 \mbox{-shiftable} }
\newcommand{\notshift}[2]{ #1 \mbox{ is not } #2 \mbox{-shiftable} }
\newcommand{\comm}[3]{ #1 \mbox{ is } #2 \mbox{-commutative at } #3}
\newcommand{\notcomm}[3]{ #1 \mbox{ is not } #2 \mbox{-commutative at } #3}

\newcommand{\Rpp}[1]{ \mathrm{Rpp} \left( #1 \right) }


\newcommand{\chose}[2]{ \left( \!\! \begin{array}{c} {\small #1} \\ {\small #2} \end{array} \!\! \right) }
\newcommand{\sigset}{ \{ 1, 0 ,-1 \} }
\newcommand{\parity}{\epsilon}

\newcommand{\position}[2]{ \stackrel{#1}{\stackrel{\smile}{#2}} }
\newcommand{\oo}[1]{ \stackrel{ {}_{{}_{{}_{{}_{\circ}}}} }{#1} }

%\newcommand{\return}{ \!\!\! \\ \!\!\! }













\newcommand{\MSSkew}{ \mathrm{M}\SSkew }
\newcommand{\Comm}[1]{ \mbox{\( #1 \)-commutative} }
\newcommand{\CQmove}[3]{
 #1 \isnot \Shift{\alpha_{#3}}, #1 \neq #2 \alpha_{#3} \et \langle #1 ,\, \alpha_{#3}^\vee \rangle = #2 2
}
\newcommand{\Stair}[3]{ \mathcal{S}^{( #1 )}_{\alpha_{#2}}\left( #3 \right) }
\newcommand{\RSTab}[1]{ \mathrm{RSTab}\left( #1 \right)}

\newcommand{\SPath}[1]{ \mathrm{SPath}{ \left( #1 \right) } }
\newcommand{\SShape}[1]{ \mathrm{SShape}{ \left( #1 \right) } }
\newcommand{\Init}[1]{ \mathrm{Init} \left( #1 \right) }

\newcommand{\corder}{\prec}
\newcommand{\aal}{ {\boldsymbol \alpha} }
\newcommand{\shook}[3]{ \mathrm{M}^{#1}_{#2}\! \left( #3 \right) }
\newcommand{\dd}[1]{ {}_{        {}\atop{ #1 }       } }

\newcommand{\Q}[1]{ \mathrm{Q}^{\alpha_{#1}}   }
\newcommand{\bas}[1]{ \mathrm{bas}^{\alpha_{#1}}   }
\newcommand{\tw}[1]{ \mathrm{tw}^{\alpha_{#1}}   }

\newcommand{\parset}{ \{ 1 , -1 \} }


\newcommand{\heap}{\preceq}
\newcommand{\norder}{\stackrel{0}{<}}
\newcommand{\incomp}{\,||\,}


\newcommand{\FA}[2]{{\mathrm{F}{#1}\mathrm{A}{#2}}}
\newcommand{\D}[1]{\underline{#1}}
\newcommand{\VT}{\mathrm{VT}}
\renewcommand{\L}{\mathrm{L}}
\newcommand{\Code}[3]{{\mathrm{C}^{#1}_{#2}(#3)}}
\newcommand{\Lev}{\mathrm{L}}
\newcommand{\Helb}{\mathrm{H}}

\newcommand{\NP}{}%\newpage}
\newenvironment{Sb}{
  \begin{subarray}{c}
}{
  \end{subarray}
}

\setlength{\unitlength}{5mm}
\newcommand{\BOX}[3]{
  \put(#1.05,#2.05){\framebox(0.9,0.9){#3}}
}
\newcommand{\MAYAh}[8]{%
  \begin{picture}(9.5,.8)
  \put(.6,0.2){
  \put(-.5, .5){\line(1,0){9}}
  \put(-.5,-.5){\line(0,1){1}}
  \put(-.5,-.5){\BOX{0}{0}{#1} \BOX{1}{0}{#2} \BOX{2}{0}{#3} \BOX{3}{0}{#4} \BOX{4}{0}{#5} \BOX{5}{0}{#6} \BOX{6}{0}{#7} \BOX{7}{0}{#8}}
  \put(8,0){\makebox(0,0){$\cdots$}}
  \put(-.5,-.5){\line(1,0){9}}
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  \end{picture}
}

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  \begin{picture}(9.3,.8)
  \put(.6,0.2){
  \put(-.5, .5){\line(1,0){9}}
  \put(-.5,-.5){\line(0,1){1}}
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  \put(#1,0){\makebox(0,0){$\bigcirc$}}
  \put(8,0){\makebox(0,0){$\cdots$}}
  \put(-.5,-.5){\line(1,0){9}}
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}

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  \begin{picture}(9.3,.8)
  \put(.6,0.2){
  \put(-.5, .5){\line(1,0){9}}
  \put(-.5,-.5){\line(0,1){1}}
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  \put(#2,0){\makebox(0,0){$\bigcirc$}}
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  \put(-.5, .5){\line(1,0){9}}
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  \begin{picture}(9.3,.8)
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  \put(-.5, .5){\line(1,0){9}}
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\renewcommand{\d}{\mathrm{d}}
\newcommand{\osum}{\sideset{}{^\oplus }\sum}

\newcommand{\SGF}[1][]{{\mathrm{SG}_{#1}}}
\newcommand{\M}[2]{{\left(\begin{array}{@{\hspace{0cm}}c|c@{\hspace{0cm}}} \displaystyle #1 \!&\! \displaystyle #2 \end{array}\right)}}
\newcommand{\N}[1]{{\mathrm{N}(#1)}}
\newcommand{\Anim}[1]{{\mathrm{Anim}(#1)}}





%% define your title in the usual way
\title[Inset games]
{ Inset games and strategies }

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{
  Kento NAKADA\thanks{\href{mailto:nakada@okayama-u.ac.jp}{nakada@okayama-u.ac.jp}.
  This paper is partially supported by KAKENHI 18H01435.}\addressmark{1}
}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}
	Okayama University,
	Graduate School of Education, Master Program,	\\
	1-1-1 Tsushima-naka, Kita-ku, Okayama-shi,   700-8530, JAPAN.
}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
  This paper proposes a new class of combinatorial $2$-player games with calculatable winning strategy.
  Our games are obtained as patchwork of the Sato-Welter games.
  This paper gives the explicit formula of the Sprague-Grundy functions of the games.
}

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\keywords{d-complete poset, inset, combinatorial games.}

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\maketitle
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\renewcommand{\NP}{\newpage}

\section{ Introduction }

Since the beginning of combinatorial game theory, i.e. the 1930's \cite{sprague}\cite{grundy},
winning strategies have been studied.
In this paper, we focus on impartial games \cite{conway},
 which are a sort of $2$-player games with perfect information.
Winning strategies of impartial games are analyzed by the Sprague-Grundy value.
Since the Sprague-Grundy value is recursively defined,
 it is very difficult to calculate the Sprague-Grundy value for most games.
Therefore, finding large classes for which it is possible to calculate the Sprague-Grundy value
 is a very important issue.
Typical examples of games possible to calculate the Sprague-Grundy value by ``good'' algorithm are:
\begin{itemize}
  \item
  Nim.
  See \cite{berlekamp}\cite{conway} for details.
  \item
  Sato-Welter game.
  This game is introduced by M.~Sato \cite{sato} and C.~P.~Welter \cite{welter} independently.
  See section \ref{sec:typeA} for definition and properties.
  \item
  Turning turtles.
  See section \ref{sec:typeD} for definition and properties.
  See \cite{berlekamp} for further details.
\end{itemize}

In \cite{Kawanaka}, N.~Kawanaka generalize these examples to \( \mathrm{d} \)-complete posets,
 which are introduced by R.~A.~Proctor \cite{ProctorI}.
In his setting,
\begin{itemize}
  \item 
  Sato-Welter game is a game on a shape (a Young diagram).
  \item
  Turning turtles is a game on a shifted shape (a shifted Young diagram).
\end{itemize}
He has given a formula which calculates the Sprague-Grudy function over his games.
Unfortunately, his formula does not give a formula for individual d-complete posets.

In this paper,
 we focus on the class of insets, which is one of the classes of $\mathrm{d}$-complete posets.
We define a game over insets --- which we call the \emph{inset games} ---
 and a closed formula for their Sprague-Grundy function.
Our games are costructed as patchworks of the Sato-Welter games.

This paper is organized as follows:
Section \ref{sec:prelim} explains fundamental notions and definitions of games and the Sprague-Grundy values.
Section \ref{sec:typeA} explains precise definition of the Sato-Welter games.
Section \ref{sec:typeD} explains precise definition of the turning turtles.
Section \ref{sec:Inset} explains our main results.


\begin{comment}
\[\]

For $x,y\in \ZZ $, we put
\begin{equation*}
 (x \mid y) = (x\oplus y)\oplus (x\oplus y-1),
\end{equation*}
where the symbol $\oplus $ denotes Nim-addition (see Section 2.2).
Following Conway~\cite{conway},
 we call a function $f:\ZZ \Mapsto \ZZ $ of the form
\begin{equation*}
  f(x) = x \oplus \osum_{i=1}^r (x \mid a_i) \oplus c
\end{equation*}
with some $ a_i \in \ZZ $ an animating function.
We shall see, in Section 4, that the theory of animating functions given in \cite{conway} has a natural counterpart 
in this situation.

In Section 5, we introduce the notion of a $k$-animating function \textit{in several variables}, and show that 
the set of $k$-animating functions $f(x_1,x_2,...,x_n)$ in several variables is closed under natural operations such as 
``the outer sum", ``the $x_i$-inversion", and ``the $x_i$-composition".\\
A most natural example of animating functions in several variables would be the function $\ph_n$ defined by
\begin{equation*}
  \ph_n(x_1,x_2,...,x_n)
  = \osum_{1 \leq i \leq n} x_i \oplus \osum_{1 \leq i<j \leq n} (x_i \mid x_j) ,
\end{equation*}
where $x_1,x_2,...,x_n$ are integer-valued variables.

In Section 3, we introduce the notion of the energy of a game.
This is a function on the set of positions satisfying the ``energy equation" and the ``inequality condition".
If the energy of a game exists, its value at each position coincides with the corresponding Sprague-Grundy number.
A game is called an e-game if its energy exists. Nim, 
Hackenbush unrestrained, and the Sato-Welter game are e-games, and these facts partly explain their complete solvability.
Particulary interesting is the Sato-Welter game; it was shown, in \cite{sato}\cite{welter}\cite{conway},
that the Sprague-Grundy number (i.e. the energy) of this game is given by the animating function $\varphi_n^{(0)}$. 
Now we ask: Can we reverse the process?
More precisely, we ask: Starting with a ``good" animating function $f=f(x_1,x_2,...,x_n)$, 
can we construct an e-game whose energy is given by $f$?
Our first attempt in this direction is described in Section 6.
There, starting with the animating function $\varphi _n^{(k)}$, we construct a game,
 the $k$-th Sato-Welter game, whose energy is given by $\varphi _n^{(k)}$.

Motivated by this example, we introduce, in Section 7,
 the notion of a pre-energy, which is an animating function in several
variables satisfying the ``pre-energy equation".  A pre-energy is ``almost" the energy of a game in a sense that
can be made precise.  We show that the set of pre-energies is closed under the natural operations (the outer sum, 
the $x_i$-inversion, the $x_i$-composition) alluded above.  
Thus we can ``generate" a class of completely solvable games from a generating set such as \{$\varphi_n^{(k)}\}_{n,k}$.\\
Apart from some side remarks, this paper is essentially self-contained.


\end{comment}

%\NP
\section{ Preliminaries }\label{sec:prelim}


\subsection{Games}
We begin by defining the games we concern with.

\begin{Def}
  Let $P$ be a set, and $\arrR$ a binary relation over $P$.
  For an element $p \in P$,
  we put $\alpha(p) := \set{ q \in P }{ p \arrR q }$.
  The pair $(P; \arrR)$ is called a \textit{game} if it satisfies:
  \begin{enumerate}
    \item
    For any $p \in P$, the set \( \alpha(p) \) is finite;
    \item
    There exists no infinite sequence
    \begin{equation*}\label{path}
      p_0,p_1,p_2,p_3,\ldots	\quad p_i \in P  
    \end{equation*}
    with 
    \begin{equation*}\label{option}
	p_i \arrR p_{i+1} , \quad i=0,1,2,3,\ldots 
    \end{equation*}
  \end{enumerate}
\end{Def}

We call an element $p$ of $P$ a \emph{position}, $\alpha(p)$ the 
\textit{option set} at the position $p$.
If $\alpha(p)=\varnothing$, then we say $p$ is an \emph{ending position}.
%A sequence \eqref{path} satisfying \eqref{option} is called a \textit{path}.
Any position $p=p_0\in P$ can be interpreted as an opening position of a $2$-player game (in the usual sense of the word);
 two players alternatively choose positions:
%terms of a path:
\begin{equation*}
  \begin{array}{cl}
	p_0 \arrR p_1	& \text{(the first player's move)},	\\
	p_1 \arrR p_2	& \text{(the second player's move)},	\\
	p_2 \arrR p_3	& \text{(the first player's move)},	\\
			& \cdots\cdots
  \end{array}
\end{equation*}
until one of them reaches an ending position $p_n$.
If $n$ is odd (resp. even), we say the first (resp. second) 
player \emph{wins}.
If \( (P;\arrR) \) and \( (Q;\arrR) \) are isomorphic to each other as digraphs,
then we say \( (P;\arrR) \) is game-isomorphic to \( (Q;\arrR) \).

\begin{Ex}[$1$-heap nim]
  Denote by \( \NN \) the set of nonnegative integers.
  Then, the pair \( (\NN;>) \) is a game,
  where \( > \) denotes the ordinary order relation `greater than'.
  This game is called the \emph{$1$-heap nim}.
\end{Ex}

According to \cite{sprague}\cite{grundy}, we define:
\begin{Def}
  For a game \( (P;\arrR) \), let \( \SGF = \SGF[P] : P \Mapsto \NN \) be the map defined by
  \begin{equation*}
    \SGF(p) = \min \left( \NN \setminus \set{ \SGF(q) \in \NN }{ p \arrR q } \right), \qquad (p \in P).
  \end{equation*}
  The map \( \SGF \) is called the \emph{Sprague-Grundy function} of \( P \).
  The value \( \SGF(p) \) is called the \emph{Sprague-Grundy number} (or \emph{Sprague-Grundy value}) of \( p \).
\end{Def}
\begin{Ex}[$1$-heap nim]
  The Sprague-Grundy function \( \SGF \) of the $1$-heap nim \( (\NN;>) \) is the identity map:
  \begin{equation*}
    \SGF(x) = x \,\,\,(x \in \NN) .
  \end{equation*}
\end{Ex}

\begin{prop}[\cite{sprague, grundy}]
  Let \( (P;\arrR) \) be a game and \( p \in P \).
  Then we have:
  \begin{enumerate}
    \item
    If \( \SGF[P](p) = 0 \), then, for any \( q \in \alpha(p) \), we have \( \SGF[P](q) > 0 \).
    \item
    If \( \SGF[P](p) > 0 \), then, for some \( q \in \alpha(p) \), we have \( \SGF[P](q) = 0 \).
    \item
    If \( p \) is an ending position, then we have \( \SGF[P](p) = 0 \).
  \end{enumerate}
\end{prop}

\begin{Rem}
  For a position \( p \in P \),
  the following two conditions are equivalent:
  \begin{enumerate}
    \item
    the position \( p \) has a winning strategy.
    \item
    \( \SGF[P](p) > 0 \).
  \end{enumerate}
  Indeed, if the first player is at the position \( p \) with \( \SGF[P](p)>0 \).
  Then there exists a next position \( q \in \alpha(p) \) with \( \SGF[P](q)=0 \).
  The strategic move \( p \arrR q \) leads the first player to win,
  because any next position \( r \in \alpha(q) \) chosen by the second player must be satisfying \( \SGF[P](r)>0 \).
\end{Rem}



%\newpage
\subsection{ Nim-Addition }
Denote by $\ZZ$ the set of integers.
We shall write the binary expression of an integer 
$ a \in \ZZ $ as
\begin{equation*}%\label{2adic}
   a=[a_i]=[a_i]_{i \in \NN}=[\ldots,a_i,\ldots,a_3,a_2,a_1,a_0].
\end{equation*}
For example, we have:
\begin{equation*}
  \begin{array}{r@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c}
          11&=&1&+&2&+&0  &+&2^3&+&0  &+&0  &+&\cdots&=&[\cdots 001011],\\
         - 1&=&1&+&2&+&2^2&+&2^3&+&2^4&+&2^5&+&\cdots&=&[\cdots 111111],\\
         -12&=&0&+&0&+&2^2&+&0  &+&2^4&+&2^5&+&\cdots&=&[\cdots 110100].
  \end{array}
\end{equation*}
We recall the definition of Nim-addition $\oplus$ in $\ZZ$.
For $a=[a_i]$, $b=[b_i]$, and $c=[c_i]$ in $\ZZ$,
we write
\begin{equation*}
  a\oplus b=c
\end{equation*}
if
\begin{equation*}
  a_i+b_i\equiv c_i \pmod{2}, \qquad  i \in \NN.
\end{equation*}
For example, we have
\begin{equation*}
  3 \oplus 5
  =
  [\cdots00011] \oplus [\cdots00101]
  =
  [\cdots00110]
  =
  6 .
\end{equation*}
The system \( (\ZZ;\oplus) \) forms an abelian group with
\begin{equation*}
  a\oplus a=0, \quad \text{for any}\,\,\, a \in \ZZ.
\end{equation*}
Note that $\NN$ is an index $2$ subgroup of $(\ZZ;\oplus )$.
We have
\begin{equation*}
  (-1)\oplus a=-a-1.
\end{equation*}
Here, the symbol \( - \) denotes the inverse on the usual addition (the binary operation \( + \)). 
For $a \in \ZZ$, we put
\begin{equation*}%\label{zero}
        \N{a} = a\oplus (a-1).
\end{equation*}
%In particular, we have
%\begin{equation*}%\label{zerozero}
%  \N{0}=-1.
%\end{equation*}
We also put, for $a,b\in \ZZ$,
\begin{equation*}%\label{mating}
  \M{a}{b} = \N{a\oplus b}.
\end{equation*}
We have the following (\cite[ch.13]{conway}, \cite{sato}, \cite{welter}).
\begin{lem}\label{lem:mate}
  Let $a,b,c \in \ZZ$.
  We have:
  \begin{enumerate}
    \item
    If $a$ is a multiple of $2^t \,\,\,(t \in \NN)$, and is not a multiple of $2^{t+1}$, then
    \begin{equation*}
      \N{a} = [\cdots 0\overbrace{11\cdots1}^{t+1}] = 2^{t+1}-1.
    \end{equation*}
    \item
    $\N{a}$ is negative if and only if $ a = 0 $. 
%    \item
%    If $ \M{a}{x} = \M{b}{x} $ for any $x\in \ZZ$, then $a=b$.
    \item
    $ \M{a}{b} = \N{a-b} $.
    \item\label{item:isom}
    $ \M{a}{b} = \M{a+c}{b+c} = \M{a\oplus c}{b\oplus c} $.
    \item
    If \( c > 0 \) (resp. \( c < 0 \)), then we have
    \begin{equation*}\hspace*{-5mm}
      a\oplus \osum_{h=0}^{c-1}\!\! \M{a}{h} = a-c
      \,\,\,
      \left( \text{resp.}\,\,\,
        a\oplus \osum_{h=c}^{-1} \!\!\M{a}{h} = a-c
      \right)\!\!,
    \end{equation*}
    where the symbol $\osum$ denotes Nim-summation. 
  \end{enumerate}
\end{lem}

\NP
\subsection{ Animating Functions }
Following Conway \cite{conway},
 we call a function $f:\ZZ \Mapsto \ZZ$ of the form
\begin{equation*}
  f(x)= \cdots ((((x\oplus a)+b)\oplus c)+d)\oplus \cdots
\end{equation*}
an \emph{animating function}.
Clearly an animating function is a bijective map from $ \ZZ $ to $ \ZZ $,
 and its inverse is animating again.
As is shown in \cite{conway}, a function $f$ is animating if and only if it can be written as
\begin{equation}\label{eq:animating}
  f(x)=x\oplus \osum_{i=1}^r \M{x}{\alpha_i} \oplus \beta
\end{equation}
with some $ \alpha_i , \beta \in \ZZ$.
Moreover, the expression \eqref{eq:animating} is unique as long as $\alpha_1,\alpha_2,\ldots,\alpha_r$ are distinct.
We denote by \( \Anim{\ZZ} \) the set of animating functions.

Some of the fundamental properties of animating functions are listed in the following:
\begin{lem}[Sato \cite{sato} and Conway \cite{conway}]\label{lem:anim}
  Let $x, y\in \ZZ$.
  \begin{enumerate}
    \item\label{item:Anim_group}
    If $f$ and $g$ are elements of $\Anim{\ZZ}$,
    then the composition $f\circ g$ and the inverse $f^{-1}$ are elements of $\Anim{\ZZ}$.
    (Hence $(\Anim{\ZZ},\circ )$ forms a group.)
    \item\label{item:Anim_isom}
    If $f$ is an element of $\Anim{\ZZ}$, then we have
    \begin{equation*}
      \M{f(x)}{f(y)} = \M{x}{y}, \quad (x,y \in \ZZ).
    \end{equation*}
    \item\label{item:Anim_inverse}
    If $y=f(x)$ with $f(x)$ given by \eqref{eq:animating}, then the inverse \( x = f^{-1}(y) \) is given by
    \begin{equation*}
      f^{-1}(y) = y \oplus \osum_{i=1}^r \M{y}{f(\alpha_i)} \oplus \beta.
    \end{equation*}
  \end{enumerate}
\end{lem}
\begin{comment}
\begin{proof}
  \eqref{item:Anim_group}\quad
  The statement follows from the definition of animating functions.

  \eqref{item:Anim_isom}\quad
  The statement follows from Lemma \ref{lem:mate} \eqref{item:isom} and the definition of animating functions.

  \eqref{item:Anim_inverse}\quad
  By Part \eqref{item:Anim_group} we can write
  \begin{equation*}
   f^{-1}(y) = y \oplus \osum_{i=1}^s \M{y}{\ep_i} \oplus \eta .
  \end{equation*}
  Substituting $y=f(x)$, we have
  \begin{equation*}
    x = f(x)\oplus \osum_{i=1}^s \M{f(x)}{\ep_i} \oplus \eta ,
  \end{equation*}
  which, by Parts \eqref{item:Anim_group} and \eqref{item:Anim_isom}, can be rewritten as
  \begin{equation*}
    x = f(x)\oplus \osum_{i=1}^s \M{x}{ f^{-1}(\ep_i) } \oplus \eta ,
  \end{equation*}
  i.e. as
  \begin{equation*}
    f(x) = x\oplus \osum_{i=1}^s \M{x}{ f^{-1}(\ep_i) } \oplus \eta .
  \end{equation*}
  Hence, by the uniqueness of expression in \eqref{eq:animating}, we get
  \begin{equation*}
    \begin{split}
    & \eta = \beta,  r=s \et
    \\& \{\alpha_1,\alpha_2,\ldots,\alpha_r\} = \{ f^{-1}(\ep_1),f^{-1}(\ep_2),\ldots,f^{-1}(\ep_s)\} ,
    \end{split}
  \end{equation*}
  i.e.
  \begin{equation*}
    \begin{split}
    & \eta = \beta, \et
    \{f(\alpha_1),f(\alpha_2),\ldots,f(\alpha_r)\} = \{ \ep_1,\ep_2,\ldots,\ep_s\} .
    \end{split}
  \end{equation*}
  Hence Part \eqref{item:Anim_inverse} follows.
\end{proof}
\end{comment}



%\subsection{ Multivariate Animating Functions }

\begin{Def}
  A multivariate function \( E : \ZZ^n \Mapsto \ZZ \) is said to be \emph{animating} if
  \begin{enumerate}
    \item
    \( E_i(x_i) := E(x_1,\cdots,x_i,\cdots,x_n) \) is animating for each \( x_i \),
    \item
    \( E \) is symmetric in \( x_1, \cdots, x_n \).
  \end{enumerate}
  The set of animating functions over \( \ZZ^n \) is denoted by \( \Anim{\ZZ^n} \).
\end{Def}

\newcommand{\Psato}[1]{{P_{#1}}}
\newcommand{\PtypeA}{{P_n}}
\newcommand{\arrRA}{{\arrR_{\mathrm{A}}}}
%\newcommand{\Bin}[1]{\binom{\NN}{#1}}
\newcommand{\Bin}[1]{{P_{#1}}}
\NP
\section{The Sato-Welter Game}\label{sec:typeA}

\begin{Def}\label{Def:typeA}
  Fix an integer \( n \geq 1 \).
  Put
  \begin{equation*}
    \Bin{n} := \set{ \mathbf{x} \subseteq \NN }{ |\mathbf{x}| = n }.
  \end{equation*}
%  For \( \mathbf{x} = \{x_1,x_2,\cdots,x_n\} \) and \( \mathbf{y} = \{y_1,y_2,\cdots,y_n\} \) in \( \Bin{n} \),
  For \( \mathbf{x}, \mathbf{y} \in \Bin{n} \),
   we denote \( \mathbf{x} \arrRA \mathbf{y} \) if
   the following two coditions hold:
  \begin{multicols}{2}
  \begin{itemize}
    \item
    \( | \mathbf{x} \cap \mathbf{y} | = n-1 \);
    \item
    if \( x \in \mathbf{x} \setminus \mathbf{y} \) and \( y \in \mathbf{y} \setminus \mathbf{x} \),
    then \( y < x \).
  \end{itemize}
  \end{multicols}
  We call the game \( (\Bin{n};\arrRA) \) the \emph{Sato-Welter game with $n$ balls}.
\end{Def}
  The Sato-Welter game with $n$ balls can be visually interpreted as follows:
  $n$ balls are lined up.
  At each move, a player moves one ball ``$\bigcirc$'' leftwards to any empty box.
  The player to make the last move wins.
%
%  This is another game whose analysis involves the interplay of nim-addition and ordinary addition.
%  It is a form of Nim with unequal heaps, but in keep track of empty heaps, only one of which is allowed,
%  it' s better to play it with coins on a strip of squares, numbered \( 0, 1, 2, \ldots \),
%  with at most one coin on a square.
%  A move is to shift a coin leftwards to any unoccupied square, possibly passing over other coins.
%  The game ends when the $k$ coins are on the leftmost squares \( 0, 1, \ldots , k - 1\).
%  Figure 8 shows a position with \( k = 7 \).
\begin{Ex}\label{Ex:typeA}
  For a position
%  \begin{equation*}
  \( \mathbf{x} = \{3,5\} = \MAYAb35 \in P_2 \)
%  \end{equation*}
  of the Sato-Welter game of $2$ balls,
  the elements of the option set \( \alpha(\mathbf{x}) \) are:
  \begin{equation*}
    \begin{split}
    \{3,4\} &= \MAYAb34 ,
    \\
    \{3,2\} &= \MAYAb32 ,
    \\
    \{3,1\} &= \MAYAb31 ,
    \\
    \{3,0\} &= \MAYAb30 ,
    \end{split}
    \quad
    \begin{split}
    \{2,5\} &= \MAYAb25 ,
    \\
    \{1,5\} &= \MAYAb15 ,
    \\
    \{0,5\} &= \MAYAb05 .
    \end{split}
  \end{equation*}
\end{Ex}
\begin{Rem}
  By regarding a position \( \mathbf{x} \in P_n \) of the Sato-Welter game
  as a beta number,
  we can also regard \( \mathbf{x} \) as Young diagrams.
  For exmaple, a position \( \mathbf{x} = \{3,5\} \in P_2 \)
  is regarded as a Young diagram with partition \( (4,3) \).
\end{Rem}

\begin{Rem}
  The Sato-Welter game \( P_1 \) of $1$ ball is game-isomorphic to the $1$-heap nim.
\end{Rem}

For \( \mathbf{x} = \{x_1,x_2,\cdots,x_n\} \in \Bin{n} \), we put
\begin{equation*}
  \ph_n(\mathbf{x})
  :=
  \osum_{i} x_i \oplus \osum_{i<j} \M{x_i}{x_j} .
\end{equation*}
For ${\bf x} \in \Psato{n}$, we have \( \ph_n ({\bf x})\geq 0 \).
Since \( \ph \) is a symmetric function in \( x_1,\cdots,x_n \),
we denote \( \ph_n(\mathbf{x}) = \ph_n(x_1,x_2,\cdots,x_n) \).

\begin{thm}[Sato \cite{sato}, and Welter \cite{welter}]\label{thm:typeA}
  The Sprague-Grundy function of
  the Sato-Welter game $(\Bin{n};\arrRA)$ is given by
  \begin{equation*}
    \ph_n({\bf x}), \quad {\bf x} \in \Psato{n}.
  \end{equation*}
\end{thm}

\begin{Ex}
  For \( \mathbf{x} = \{3,5\} \in \Bin{2} \),
  the Sprague-Grundy value \( \ph_2(\mathbf{x}) \) is
  \begin{equation*}
    \begin{split}
    \ph_2(\mathbf{x})
    &= 3 \oplus 5 \oplus \M35
    = 5 \neq 0.
    \end{split}
  \end{equation*}
  Hence, the position \( \mathbf{x} \) has a winning strategy.
  The (unique) winning move is:
  \begin{equation*}
    \{3,5\} \arrRA \{3,2\} .
  \end{equation*}
\end{Ex}

\begin{comment}
\begin{Ex}
  For a position
  \begin{equation*}
    \mathbf{x} = \{1,2,4,6\} = \MAYAd1246 \in P_4
  \end{equation*}
  of the Sato-Welter game of $4$ balls,
  the elements of option set \( \alpha(\mathbf{x}) \) are:
  \begin{equation*}
    \begin{split}
    \{1,2,4,5\} &= \MAYAd1245 ,
    \\
    \{1,2,4,3\} &= \MAYAd1243 ,
    \\
    \{1,2,4,0\} &= \MAYAd1240 ,
    \\
    \{1,2,3,6\} &= \MAYAd1236 ,
    \\
    \{1,2,0,6\} &= \MAYAd1206 ,
    \\
    \{1,0,4,6\} &= \MAYAd1046 ,
    \\
    \{0,2,4,6\} &= \MAYAd0246 .
    \end{split}
  \end{equation*}
\end{Ex}

\begin{Ex}
  For \( \mathbf{x} = \{1,2,4,6\} \in \Bin{4} \),
  the Sprague-Grundy value \( \ph_4(\mathbf{x}) \) is
  \begin{equation*}
    \begin{split}
    & \ph_4(\mathbf{x})
    \\&= 1 \oplus 2 \oplus 4 \oplus 6
    \\&\quad\oplus \M12 \oplus \M14 \oplus \M16 \oplus \M24 \oplus \M26 \oplus \M46
    \\&= 7 \neq 0.
    \end{split}
  \end{equation*}
  Hence, the position \( \mathbf{x} \) has a winning strategy.
  The (unique) winnig move is:
  \begin{equation*}
    \{1,2,4,6\} \arrRA \{0,2,4,6\} .
  \end{equation*}
\end{Ex}
\end{comment}


%\input{Game_typeA_proof}



\newcommand{\PtypeD}{{P_{\text{odd}}}}
\newcommand{\odd}{\text{odd}}
\newcommand{\arrRD}{{\arrR_{\mathrm{D}}}}
\NP
\section{The Turning turtles}\label{sec:typeD}




\begin{Def}\label{Def:typeD}
  Put
  \begin{equation}
    \label{Def:PtypeD}
    \begin{split}
    \PtypeD
    &:=
    \bigcup_{n :\odd} \Bin{n}
    =
    \bigcup_{n :\odd} \set{ \mathbf{x} \subseteq \NN }{ |\mathbf{x}|=n }.
    \end{split}
  \end{equation}
  For \( \mathbf{x} \in \Bin{n} \) and \( \mathbf{y} \in \Bin{m} \),
   we denote \( \mathbf{x} \arrRD \mathbf{y} \) if
  \begin{multicols}{2}
  \begin{itemize}
    \item
    \( n = m	\et \mathbf{x} \arrRA \mathbf{y} \); \quad or
    \item
    \( n = m+2	\et \mathbf{x} \supset \mathbf{y} \).
  \end{itemize}
  \end{multicols}
  We call the game \( (\PtypeD;\arrRD) \) the \emph{turning turtles}.
\end{Def}
  The turning turtles can be visually interpreted as follows:
  Several turtles are lined up.
  An odd number of them are awake and the others sleeping.
  At each move, a player chooses two turtles with both hands and turns them over.
  The player is allowed to turn ``$\bigcirc$''(awake turtle) into ``empty''(sleeping turtle) and also ``empty'' into ``$\bigcirc$''.
  However, the turtle that he grabs with his right hand must be an awake turtle.
  The player to make the last move wins.
\begin{Ex}
  For a position
%  \begin{equation*}\,\,\,\,
  \( \mathbf{x} = \{2,3,5\} = \MAYAc235 \in \PtypeD \)
%  \end{equation*}
  of the turning turtles,
  the elements of the option set \( \alpha(\mathbf{x}) \) are:
  \begin{equation*}
    \begin{split}
    \{2,3,4\} &= \MAYAc234 ,
    \\
    \{2,3,1\} &= \MAYAc231 ,
    \\
    \{2,3,0\} &= \MAYAc230 ,
    \\
    \{2,1,5\} &= \MAYAc215 ,
    \\
    \{2,0,5\} &= \MAYAc205 ,
    \end{split}
    \quad
    \begin{split}
    \{1,3,5\} &= \MAYAc135 ,
    \\
    \{0,3,5\} &= \MAYAc035 ,
    \\
    \{5\} &= \MAYAa5 ,
    \\
    \{3\} &= \MAYAa3 ,
    \\
    \{2\} &= \MAYAa2 .
    \end{split}
  \end{equation*}
\end{Ex}
\begin{Rem}
  By regarding a position \( \mathbf{x} \in \PtypeD \) of the turning tutles
  as a strict partition,
  we can also regard \( \mathbf{x} \) as shifted Young diagrams.
  For exmaple, a position \( \mathbf{x} = \{2,3,5\} \in \PtypeD \)
  is regarded as a shifted Young diagram with strict partition \( (5,3,2) \).
\end{Rem}

For \( \mathbf{x} = \{x_1,x_2,\cdots,x_n\} \in \PtypeD \), we put
\begin{equation*}
  \psi(\mathbf{x})
  :=
  x_1\oplus x_2\oplus \cdots \oplus x_n .
\end{equation*}
For ${\bf x} \in \PtypeD$, we have \( \psi ({\bf x})\geq 0 \).
See \cite{berlekamp} for further details.
\begin{thm}\label{thm:typeD}
  The Sprague-Grundy function of
  the turning turltes $(\PtypeD;\arrRD)$ is given by
  \begin{equation*}
    \psi({\bf x}), \quad {\bf x} \in \PtypeD.
  \end{equation*}
\end{thm}
%\begin{Rem}
%  The theorem is (essentially) due to Sato \cite{sato} and Welter \cite{welter}.
%  Our proof of Theorem 5.1 is quite similar, in spirit,  to the one in  
%  Sato \cite{sato}.
%\end{Rem}
\begin{Ex}
  For \( \mathbf{x} = \{2,3,5\} \in \PtypeD \),
  the Sprague-Grundy value \( \psi(\mathbf{x}) \) is
  \begin{equation*}
    \psi(\mathbf{x}) = 2 \oplus 3 \oplus 5 = 4 \neq 0.
  \end{equation*}
  Hence, the position \( \mathbf{x} \) has a winning strategy.
  The (unique) winning move is:
  \begin{equation*}
    \{2,3,5\} \arrRD \{2,3,1\} .
  \end{equation*}
\end{Ex}

\newcommand{\Pmain}{P_{N,1}}
\NP
\section{ Main result }\label{sec:Inset}
The definition \ref{Def:typeD} means
the position set \( \PtypeD \) of turning turtles is a ``patchwork'' of position sets \( \PtypeA \) of Sato-Welter games.
This is our motivation of this study.

\begin{Def}\label{Def:Inset}
  Fix an integer \( N \geq 2 \).
  Put
  \begin{equation*}
    \begin{split}
    \Pmain
    &:=
    \Bin{N} \cup \Bin{1} .
    \end{split}
  \end{equation*}
  Let \( n,m \in \{ 1, N \} \).
  For \( \mathbf{x} \in \Bin{n} \) and \( \mathbf{y} \in \Bin{m} \),
   we denote \( \mathbf{x} \arrR \mathbf{y} \) if
  \begin{itemize}
    \item
    \( n = m		\et \mathbf{x} \arrRA \mathbf{y} \); \quad or
    \item
    \( n = N,	m=1	\et \mathbf{x} \supset \mathbf{y} \).
  \end{itemize}
  We call the game \( (\Pmain;\arrR) \) the \emph{inset game}.
\end{Def}
\begin{Rem}
  If \( N = 2,3 \), then this game is not new (see subsection \ref{subsec:N=2}, \ref{subsec:N=3}).
  For \( N \geq 4 \), this game is new.
\end{Rem}
\begin{Ex}
  For a position
  \begin{equation*}\,\,\,\,
    \mathbf{x} = \{1,2,4,5\} = \MAYAd1245 \in P_{4,1}
  \end{equation*}
  of our game with \( N=4 \),
  the elements of option set \( \alpha(\mathbf{x}) \) are:
  \begin{equation*}
    \begin{split}
    \{1,2,4,3\} &= \MAYAd1243,
    \\
    \{1,2,4,0\} &= \MAYAd1240,
    \\
    \{1,2,3,5\} &= \MAYAd1235,
    \\
    \{1,2,0,5\} &= \MAYAd1205,
    \\
    \{1,0,4,5\} &= \MAYAd1045,
    \end{split}
    \quad
    \begin{split}
    \{0,2,4,5\} &= \MAYAd0245,
    \\
    \{5\} &= \MAYAa5,
    \\
    \{4\} &= \MAYAa4,
    \\
    \{2\} &= \MAYAa2,
    \\
    \{1\} &= \MAYAa1.
    \end{split}
  \end{equation*}
\end{Ex}

\begin{Rem}
  With a position \( \mathbf{x} = \{ x_1,x_2,\cdots,x_n \} \in P_{N,1} \),
  the inset
  \begin{center}
    \begin{overpic}[width=10cm]{inset2.eps}
	\put(14, 38)	{\makebox(0,0){\scriptsize$ n-1 $}}
	\put(64, 38)	{\makebox(0,0){\scriptsize$ x_{n}-(n-1) $}}
	\put(50, 29)	{\makebox(0,0){\scriptsize$ x_{n-1}-(n-2) $}}
	\put(41,  1)	{\makebox(0,0){\scriptsize$x_1$}}
    \end{overpic}
  \end{center}
\begin{comment}
  \begin{equation*}
    \xymatrix{
	\circ \ar@{-}[r]	& \cdots \ar@{-}[r]	& \circ \ar@{-}[d] \ar@{-}[r]	& \circ \ar@{-}[d] \ar@{-}[r]	& \circ \ar@{-}[d] \ar@{-}[r]	& \cdots \ar@{-}[r]	& \circ \ar@{-}[d] \ar@{-}[r]	& \cdots \ar@{-}[r]	& \circ	\ar@{-}[d] \ar@{-}[r]	& \cdots \ar@{-}[r]	& \circ	\\
				&			& \circ	\ar@{-}[r]		& \circ \ar@{-}[d] \ar@{-}[r]	& \circ \ar@{-}[d] \ar@{-}[r]	& \cdots \ar@{-}[r]	& \circ \ar@{-}[d] \ar@{-}[r]	& \cdots \ar@{-}[r]	& \circ	\\
				&			&				& \vdots \ar@{-}[d]		& \vdots			& \vdots \ar@{-}[d]	& \vdots			&	\\
				&			&				& \circ	\ar@{-}[r]		& \cdots \ar@{-}[r]		& \circ			&	\\
    }
  \end{equation*}

\begin{tikzpicture}[commutative diagrams/every diagram]
\matrix
[column sep=8mm, row sep=8mm]
%[matrix of math nodes, name=m,inner xsep=0pt] 
{
 \node (x) {$A_1$}; & \node (y) {$B_2$}; & \node (z){$C_3$}; & \node (w){$D_4$}; \\
 {$E_5$}; & \node (s) {$F_6$}; & \node (t){$G_7$}; & {$H_8$}; \\
};
\draw[decorate,decoration={brace}]
   (x) -- (y) node[midway,auto] {$A$ to $B$};
\draw[decorate,decoration={brace}]
   (z) -- (w) node[midway,auto] {$C$ to $D$};
\draw[decorate,decoration={brace,mirror}]
   (s) -- (t) node[midway,auto,swap] {$F$ to $G$};
\end{tikzpicture}

\vspace*{1cm}

\begin{tikzpicture}
\matrix [column sep=2cm, row sep=7mm] {
                  & \node (x) {$x$}; &                  \\
 \node (t) {$t$}; &                  & \node (z) {$z$}; \\
                  & \node (y) {$y$}; &                  \\
                & \node (below y) {}; &                 \\
};
\draw[->] (t) -- (x);
\draw[->] (x) -- (z);
\draw[->] (t) -- (y);
\draw[->] (y) -- (z);
\draw[->, double, rounded corners]
 (t) -- (t |- below y) -- (below y -| z) -- (z);
\end{tikzpicture}
\end{comment}
\noindent
  can be associated,
  if \( n = N \et x_1 < x_2 < \cdots < x_N \).
  If, on the other hand, \( n=1 \),
  then we attach the \( x_1 \)-chain.
\end{Rem}

For \( \mathbf{x} = \{x_1,x_2,\cdots,x_n\} \in \Pmain \), we put
\begin{equation*}
  \ph_{N,1}(\mathbf{x})
  :=
  \CASE{
	\displaystyle
	\osum_i \M{ -1 }{ \ph_{N-1}(\mathbf{x}^{(i)}) } \oplus \ph_N(\mathbf{x})	& n=N	\\
	x_1 & n=1	\\
  }
\end{equation*}
where \( \mathbf{x}^{(i)} = \mathbf{x} \setminus \{ x_{i} \} \).
For ${\bf x} \in \Pmain$, we have
\begin{equation}\label{eq:nonnegative}
  \ph_{N,1}({\bf x}) \geq 0 .
\end{equation}
Now we can state the main result:
\begin{thm}\label{thm:Inset}
  The Sprague-Grundy function of
  the inset game $(\Pmain;\arrR)$ is given by
  \begin{equation*}
    \ph_{N,1}({\bf x}), \quad {\bf x} \in \Pmain.
  \end{equation*}
\end{thm}
A sketch of proof is given in the subsection \ref{subsec:proof}.
%\begin{Rem}
%  The theorem is (essentially) due to Sato \cite{sato} and Welter \cite{welter}.
%  Our proof of Theorem 5.1 is quite similar, in spirit,  to the one in  
%  Sato \cite{sato}.
%\end{Rem}
\begin{Ex}
  For \( \mathbf{x} = \{1,2,4,5\} \in P_{4,1} \),
  the Sprague-Grundy value \( \ph_{4,1}(\mathbf{x}) \) is
  \begin{equation*}
    \begin{split}
    \ph_{4,1}(\mathbf{x})
    &=
    \M{-1}{\ph_3(2,4,5)}
    \oplus \M{-1}{\ph_3(1,4,5)}
    \\&\qquad
    \oplus \M{-1}{\ph_3(1,2,5)}
    \oplus \M{-1}{\ph_3(1,2,4)}
    \oplus \ph_4(1,2,4,5)
    \\&=
    \M{-1}{0}
    \oplus \M{-1}{7}
    \oplus \M{-1}{1}
    \oplus \M{-1}{4}
    \oplus 6
    =
    10.
    \end{split}
  \end{equation*}
  Hence, the position \( \mathbf{x} \) has a winning strategy.
  The (unique) winning move is:
  \begin{equation*}
    \{1,2,4,5\} \arrR \{0,1,4,5\} .
  \end{equation*}
\end{Ex}


\subsection{ Case $ N=2 $ }\label{subsec:N=2}
Define an injection \( f : P_{2,1} \Mapsto P_2 \) by
\begin{equation*}
    f(\{x_1,x_2\}) = \{x_1+1,x_2+1\} ,\qquad
    f(\{x_1\}) = \{0,x_1+1\} .
\end{equation*}
It is straightforward to see the case \( N=2 \):
\begin{prop}\label{prop:N=2}
  The map \( f \) is a game isomorphism from \( P_{2,1} \) to the image \( f(P_{2,1}) \).
  In particular, for \( \mathbf{x} \in P_{2,1} \), we have
  \( \ph_{2,1}(\mathbf{x}) = \ph_2(f(\mathbf{x})) \).
\end{prop}
\begin{Ex}
  For a position
  \begin{equation*}\,\,\,\,
    \mathbf{x} = \{2,4\} = \MAYAb24 \in P_{2,1}
  \end{equation*}
  of our game with \( N=2 \),
  the elements of option set \( \alpha(\mathbf{x}) \) are:
  \begin{equation*}
    \begin{split}
    \{2,3\} &= \MAYAb23 ,
    \\
    \{2,1\} &= \MAYAb21 ,
    \\
    \{2,0\} &= \MAYAb20 ,
    \\
    \{2\}   &= \MAYAa2 ,
    \end{split}
    \quad
    \begin{split}
    \{1,4\} &= \MAYAb14 ,
    \\
    \{0,4\} &= \MAYAb04 ,
    \\
    \{4\}   &= \MAYAa4 .
    \end{split}
  \end{equation*}
  These positions correspond to the positions in Example \ref{Ex:typeA}.
\end{Ex}

\subsection{ Case $ N=3 $ }\label{subsec:N=3}
Define an injection \( f : P_{3,1} \Mapsto \PtypeD \) by
\begin{equation*}
    f(\{x_1,x_2,x_3\}) = \{x_1,x_2,x_3\} ,\qquad
    f(\{x_1\}) = \{x_1\} .
\end{equation*}
It is straightforward to see the case \( N=3 \):
\begin{prop}\label{prop:N=3}
  The map \( f \) is a game isomorphism from \( P_{3,1} \) to the image \( f(P_{3,1}) \).
  In particular, for \( \mathbf{x} \in P_{3,1} \), we have
  \( \ph_{3,1}(\mathbf{x}) = \psi(f(\mathbf{x})) \).
\end{prop}



\subsection{ Sketch of proof of main theorem }\label{subsec:proof}

We need the following two lemmata:
\begin{lem}\label{lem:Inset_parity}
  Let \( \mathbf{x} \in P_{N,1} \) and \( h \in \NN \).
  Then we have
  \begin{enumerate}
    \item
    if \( 0 \leq h < \ph_{N,1}(\mathbf{x}) \),
    then the number of \( \mathbf{y} \in \alpha(\mathbf{x}) \) with \( \ph_{N,1}(\mathbf{y}) = h \) is odd.
    In particular, there exists such a next position \( \mathbf{y} \) of \( \mathbf{x} \) with \( \ph_{N,1}(\mathbf{y}) = h \).
    \item
    if \( h \geq \ph_{N,1}(\mathbf{x}) \),
    then the number of \( \mathbf{y} \in \alpha(\mathbf{x}) \) with \( \ph_{N,1}(\mathbf{y}) = h \) is even (it may be zero).
  \end{enumerate}
\end{lem}
We omit the proof of Lemma \ref{lem:Inset_parity}.

\begin{lem}\label{lem:Inset_null}
  Let \( \mathbf{x} \in P_{N,1} \) and \( \mathbf{y} \in \alpha(\mathbf{x}) \).
  Then we have
  \( \ph_{N,1}(\mathbf{x}) \neq \ph_{N,1}(\mathbf{y}) \).
\end{lem}
\begin{proof}
  Let \( \mathbf{x} = \{x_1,\cdots,x_n\} \) and \( \mathbf{y} = \{y_1,\cdots,y_m\} \).

  Case \( \mathbf{x} \in P_N \) and \( \mathbf{y} \in P_N \):\quad
  We may assume that \( x_1 > y_1 \) and \( x_i = y_i \,\,\,(2 \leq i \leq N) \).
  We have
  \begin{equation*}
    \begin{split}
    &
    \ph_{N,1}(\mathbf{x}) \oplus \ph_{N,1}(\mathbf{y})
%    &=
%    \osum_{i=2}^N \left( \M{-1}{\ph_{N-1}(\mathbf{x}^{(i)})} \oplus \M{-1}{\ph_{N-1}(\mathbf{y}^{(i)})} \right)
%    \\&\quad\oplus
%    x_1 \oplus y_1 \oplus \osum_{i=2}^N \left( \M{x_1}{x_i} \oplus \M{y_1}{y_i} \right)
   \\&=
    \osum_{i=2}^N \left(
	\begin{split}
	& \M{-1}{\ph_{N-1}(\{x_1,x_2,\cdots,x_N\}\setminus\{x_i\})}
	\\& \oplus \M{-1}{\ph_{N-1}(\{y_1,x_2,\cdots,x_N\}\setminus\{x_i\})}
	\end{split}
    \right)
%    \\&\quad
    \oplus
    x_1 \oplus y_1 \oplus \osum_{i=2}^N \left( \M{x_1}{x_i} \oplus \M{y_1}{x_i} \right) .
    \end{split}
  \end{equation*}
  Suppose \( \ph_{N,1}(\mathbf{x}) = \ph_{N,1}(\mathbf{y}) \).
  Then we have
  \begin{equation*}
    \begin{split}
    &
    \osum_{i=2}^N \M{-1}{\ph_{N-1}(\{x_1,x_2,\cdots,x_N\}\setminus\{x_i\})}
    \oplus x_1 \oplus \osum_{i=2}^N \M{x_1}{x_i}
    \\&=
    \osum_{i=2}^N \M{-1}{\ph_{N-1}(\{y_1,x_2,\cdots,x_N\}\setminus\{x_i\})}
    \oplus y_1 \oplus \osum_{i=2}^N \M{y_1}{x_i} .
    \end{split}
  \end{equation*}
  Put
  \begin{equation*}
    \begin{split}
    g_i(x) &:= \ph_{N-1}(\{x,x_2,\cdots,x_N\}\setminus\{x_i\}) ,\quad(2\leq i \leq N).
    \end{split}
  \end{equation*}
  Then \( g_i \) is an animating function.
  Put
  \begin{equation*}
    \begin{split}
    f(x) &:=
    \osum_{i=2}^N \M{-1}{g_i(x)}
    \oplus x \oplus \osum_{i=2}^N \M{x}{x_i}.
%    \\&=
%    \osum_{i=2}^N \M{g_i^{-1}(-1)}{x}
%    \oplus x \oplus \osum_{i=2}^N \M{x}{x_i} .
    \end{split}
  \end{equation*}
  Then \( f \) is an animating function and \( f(x_1) = f(y_1) \).
  Since \( f \) is bijective, we have \( x_1 = y_1 \).
  This contradicts our assumption.
  Hence, we have
  \( \ph_{N,1}(\mathbf{x}) \neq \ph_{N,1}(\mathbf{y}) \).

  Case \( \mathbf{x} \in P_N \) and \( \mathbf{y} \in P_1 \):\quad
  Put
  \begin{equation*}
    f(x) := x \oplus \osum_{j=1}^N \M{x}{x_j} \oplus \ph_N(\mathbf{x}),\quad(x \in \ZZ).
  \end{equation*}
  Then \( f \) is animating and we have
  \begin{equation}\label{eq:inverse1}
    f^{-1}(y) = y \oplus \osum_{i=1}^N \M{y}{f(x_i)} \oplus \ph_N(\mathbf{x}),\quad(y \in \ZZ)
  \end{equation}
  by Lemma \ref{lem:anim} \eqref{item:Anim_inverse}.
  Since
  \begin{equation*}
    \begin{split}
    \M{-1}{\ph_{N-1}(\mathbf{x}^{(i)})}
    &=
    \M{\M{x_i}{x_i}}{x_i \oplus \osum_{j \neq i} \M{x_i}{x_j} \oplus \ph_N(\mathbf{x})}
    \\&=
    \M{0}{x_i \oplus \osum_{j=1}^N \M{x_i}{x_j} \oplus \ph_N(\mathbf{x})}
    \\&=
    \M{0}{f(x_i)} ,
    \end{split}
  \end{equation*}
  we have
  \begin{equation}\label{eq:inverse2}
    \begin{split}
    \ph_{N,1}(\mathbf{x})
    &=
    0 \oplus \osum_{i=1}^N \M{0}{f(x_i)} \oplus \ph_N(\mathbf{x}).
    \end{split}
  \end{equation}
  Hence, by \eqref{eq:inverse1} and \eqref{eq:inverse2}, we have \(  \ph_{N,1}(\mathbf{x}) = f^{-1}(0) \).
  Therefore,
  \begin{equation*}
    \begin{split}
    0
    &=
    f(\ph_{N,1}(\mathbf{x}))
    =
    \ph_{N,1}(\mathbf{x}) \oplus \osum_{i=1}^N \M{\ph_{N,1}(\mathbf{x})}{x_i} \oplus \ph_N(\mathbf{x}).
    \end{split}
  \end{equation*}
  Since the first term (by \eqref{eq:nonnegative}) and the last terms are nonnegative in the right hand side
  and \( x_i \)'s are distinct,
  there exists no \( i \) such that \( \ph_{N,1}(\mathbf{x}) = x_i \).

  Case \( \mathbf{x} \in P_1 \) and \( \mathbf{y} \in P_1 \):\quad
  Then we have
  \begin{equation*}
    \ph_{N,1}(\mathbf{x}) = x_1 > y_1 = \ph_{N,1}(\mathbf{y}) .
  \end{equation*}
  Hence, we have
  \( \ph_{N,1}(\mathbf{x}) \neq \ph_{N,1}(\mathbf{y}) \).
\end{proof}

By Lemma \ref{lem:Inset_parity}, Lemma \ref{lem:Inset_null} and induction,
we have
\begin{equation*}
  \SGF[P_{N,1}](\mathbf{x}) = \ph_{N,1}(\mathbf{x}) , \quad (\mathbf{x} \in P_{N,1}).
\end{equation*}
This proves Theorem \ref{thm:Inset}.





\begin{comment}
\section{Conclusion}
This paper proposes a new class of combinatorial $2$-player games with calculatable winning strategy,
which we call the \emph{inset games}.
Our games are obtained as patchwork of the Sato-Welter games.
This paper gives the explicit formula (Theorem \ref{thm:Inset}) of the Sprague-Grundy functions of the inset games.
Weakness of this study is that
 it can only consider the patchwork games of type \( P_{N,1} \).
Patchwork games of types other than $ P_{N,1} $ are still open.
The author wishes this study proposes a new theme to theory of combinatorial $2$-player games with perfect information and attractive discussions to mathematics.
\end{comment}

\acknowledgements{
  The author would like to thank Dr.\ Hagiwara for his comments and suggestions.
}

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