[User]
Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.

Present the answer in LaTex format: \boxed{Your answer}

dataset: FINEGRAINED


[User]
Consider a regular 10-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 10, determine how many ways we can color the vertices such that there exists at least one rotation of the 10-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'red' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'red' vertex ends up at a position originally occupied by a 'blue' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider a regular 10-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 10, determine how many ways we can color the vertices such that there exists at least one rotation of the 10-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider a regular 10-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'red' vertices. For each k from 0 to 10, determine how many ways we can color the vertices such that there exists at least one rotation of the 10-gon where every 'red' vertex ends up at a position originally occupied by a 'blue' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'red' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'red' vertex ends up at a position originally occupied by a 'blue' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'red' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'red' vertex ends up at a position originally occupied by a 'blue' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Consider a regular 10-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 10, determine how many ways we can color the vertices such that there exists at least one rotation of the 10-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=10.

Present the answer in LaTex format: \boxed{Your answer}