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Combinatorics
- 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.
URL: http://www.seanet.com/~ksbrown/icombina.htm
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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.
URL: http://www.seanet.com/~ksbrown/kmath098.htm
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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.
URL: http://www.seanet.com/~ksbrown/kmath031.htm
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| 4. |
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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)}.
URL: http://members.nbci.com/Polytope/double-cosets
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