Surjective symmetry classes |

Then, by the Principle of Inclusion and Exclusion, we obtain for the desired number of surjective fixed points ofY^{X}_{(h,g),I}:= {f ÎY^{X}_{(h,g)}| " nÎI:f^{-1}[Y_{n}] = Æ}.

| Y^{X}_{s,(h,g)}| = | Y^{X*}_{(h,g)}| = å_{I Íc( bar (h))}(-1)^{ | I | }| Y^{X}_{(h,g),I}|

Now we recall that= å_{I Íc( bar (h))}(-1)^{c ( bar (h))- | I | }| Y^{X}_{(h,g), c( bar (h)) \I}| .

This set can be identified withY^{X}_{(h,g), c( bar (h)) \I}= {f ÎY^{X}_{(h,g)}| " n not ÎI:f^{-1}[Y_{n}]= Æ}.

We can make this more explicit by an application of lemma which yields:| Y^{X}_{(h,g), c( bar (h)) \I}| = | [~Y]^{X}_{( [~h] ,g)}| = Õ_{j}| [~Y]_{ [~h] j}|^{aj( bar (g))}.

Putting these things together we conclude| [~Y]_{ [~h] j}| =a_{1}( [~h]^{j})= å_{d | j}d ·a_{d}( [~h] ).

An application of the Cauchy-Frobenius Lemma finally yields the desired numbers of surjective symmetry classes:Corollary:The number of surjective fixed points of(h,g)iswhere the middle sum is taken over all the sequences| Y^{X}_{s,(h,g)}| = å^{c( bar (h))}_{k=1}(-1)^{c( bar ( h))-k}å_{a}Õ^{ | Y | }_{i=1}[a_{i}( bar (h)) choose a_{i}] Õ_{j=1}^{ | X | }( å_{d | j}d ·a_{d})^{aj( bar (g))},a=(aof natural numbers_{1}, ...,a_{ | Y | })asuch that_{j}åa(they correspond to all possible choices of_{j}=k[~h]out ofh, whereaof the chosen cyclic factors of_{i}[~h]arei-cycles). Hence the numbers of surjective fixed points ofgand ofhamount to:and| Y^{X}_{s,g}| = å_{k=1}^{ | Y | }(-1)^{ | Y | -k}[ | Y | choose k]k^{c( bar (g))},where the sum is taken over all the sequences| Y^{X}_{s,h}| = å_{k=1}^{c( bar (h))}(-1)^{c( bar (h))-k}å_{a}( Õ_{i}[a_{i}( bar (h)) choose a_{i}])a_{1}^{ | X | },(a,_{1}, ...,a_{ | Y | })aand_{i}ÎNåa._{i}=k

Try to compute the number of surjective symmetry classes for various group actions.Theorem:The number| (H ´G) \\Yof surjective^{X}_{s}|H ´G-classes iswhere the inner sum is taken over the sequences(1)/(| H | | G |)å_{(h,g)}å_{k=1}^{c( bar (h))}(-1)^{c( bar (h))-k}å_{a}Õ_{i=1}^{ | Y | }[a_{i}( bar (h)) choose a_{i}] Õ_{j=1}^{ | X | }( å_{d | j}d ·a_{d})^{aj( bar (g))},a=(adescribed in the corollary above. This implies, by restriction, the equations_{1}, ...a_{ | Y | })and| G \\Y^{X}_{s}| =(1)/(| G |)å_{g}å_{k=1}^{ | Y | }(-1)^{ | Y | -k}[ | Y | choose k]k^{c( bar (g))},where the last sum is to be taken over all the sequences| H \\Y^{X}_{s}| =(1)/(| H |)å_{h}å_{k=1}^{c( bar (h))}(-1)^{c( bar (h))-k}å_{a}( Õ_{i}[a_{i}( bar (h)) choose a_{i}])a_{1}^{ | X | },a=(asuch that_{1}, ...,a_{ | Y | })aand_{i}ÎNåa._{i}=k

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Surjective symmetry classes |