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You can define $r$-associated Stirling number of the second kind with the following recurrence relation (wiki): $$ S_r(n+1, k)=k\ S_r(n, k)+\binom{n}{r-1}S_r(n-r+1, k-1) $$ The corresponding definition (with necessary special cases) in Mathematica is ClearAll[S]; S[_, 0, 0] = 1; S[r_, n_, 1] /; n >= r = 1; S[r_, n_, k_] /; r > 0 && n > 0 ...


7

You can use the built-in functionality PolyhedronData["Icosahedron", "VertexCoordinates"] {{0, 0, -(5/Sqrt[50 - 10 Sqrt[5]])}, ... } or this short generator {0,##2}~RotateLeft~#&@@@Tuples@{{1,2,3},s={-1,1},s(1+√5)/2} ConvexHullMesh[%]


4

You can try this: V = Table[RotateRight[{0, (-1)^j, (-1)^Floor[j/2] GoldenRatio}, Floor[(j - 1)/4]], {j, 12}]; You can plot it using ConvexHullMesh: ConvexHullMesh[V]


5

Since Mathematica 8 it is possible generate the elements of any group one by one with GroupElements. Here's for example a randomly chosen element of the permutation group on 20 elements: GroupElements[SymmetricGroup[20], {10^6 + 1}] {Cycles[{{11, 13, 19}, {12, 18, 17, 16}, {15, 20}}]} The result is immediately; there's no need to build up the full ...



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