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Dedekind's Problem Generating Function For Eulerian Numbers Permutation Loops Polyomino Enumerations Powers of Primes Dividing Factorials The Dartboard Sequence Binary Games 414298141056 Quarto Draws Suffice!, etc.
   Cumulative Permutation Sequences
Let p1, p2, ...,pk (where k = n!) denote the permutations of n elements, and let "*" denote composition. (For example, pi*pj signifies the permutation given by first applying pj, and then applying pi.) Of course, the n! permutations can be ordered in (n!)! different ways.
   Permutation Loops
Suppose a,b,c,d,e are elements of a group. We'll call a circular arrangement of such elements a "loop". For example, we might choose to arrange those elements in alphabetical order, giving the loop {abcde}, where the last element is understood to wrap around to the first. Note that {abcde} and {eabcd} are both valid representations of the same loop, since they are just rotations of each other.
   double cosets
Thus when n is odd, the formula is 1/(2n)^2 { n^2 ((n-1)/2)! 2^((n-1)/2) + sum{d|n} sigma(d)^2 (n/d)! d^(n/d)}. So for even n, the formula is 1/(2n)^2 {2(n/2)(n/2)!2^(n/2)+(n/2)^2(n/2)!2^(n/2) +(n/2)^2(n/2-1)!2^(n/2) +sum{d|n} sigma(d)^2 (n/d)! d^(n/d)} =1/(2n)^2 {2^(n/2)(n/2)![n+(n/2)^2+n/2]+sum{d|n}...} =1/(2n)^2 {2^(n/2)(n/2)!(n/2)(n/2+3) +sum{d|n} sigma(d)^2 (n/d)! d^(n/d)}.

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