$$\definecolor{c1}{RGB}{255, 0, 0} \definecolor{c2}{RGB}{0, 0, 255} \definecolor{c3}{RGB}{255, 165, 0} \definecolor{c4}{RGB}{75, 0, 130} \definecolor{c5}{RGB}{220, 220, 0} \definecolor{c6}{RGB}{238, 130, 238} \definecolor{c7}{RGB}{0, 128, 0} \definecolor{c8}{RGB}{100,100,100} \definecolor{c9}{RGB}{45,177,93} \definecolor{default}{RGB}{10,20,20} \color{default}$$ # cauchy_frobenius_orbit_counting_lemma AKA burnside’s lemma ## introduction ## intuition $$ \color{c1} \# \color{c2}\text{ orbits } \color{default} \color{c3}\frac{1}{\left| G \right| } \sum _{g \in G} \color{c4}fix_x(g) $$
$$ \color{c2} \text{If you shuffle a deck of $n$ cards, } \color{c3} \text{on average } \color{c1} 1 \color{c4} \text{ card will remain in it's original position.} $$
If you shuffle a deck of n cards, on average 1 card will remain in it’s original position. %% ==rigour %% ==exam clinic %% ==examples and non-examples %% ==related tags math