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Let Pi be a permutation of n elements, and let alpha_i be the number of permutation cycles of length i in this permutation. Picking Pi at random, it turns out that ...

Given a Poisson process, the probability of obtaining exactly n successes in N trials is given by the limit of a binomial distribution P_p(n|N)=(N!)/(n!(N-n)!)p^n(1-p)^(N-n). ...

To pick a random point on the surface of a unit sphere, it is incorrect to select spherical coordinates theta and phi from uniform distributions theta in [0,2pi) and phi in ...

The game of tic-tac-toe, also spelled ticktacktoe and also known as 3-in-a-row or "naughts and crosses," is a game in which players alternate placing pieces (typically Xs for ...

A harmonic number is a number of the form H_n=sum_(k=1)^n1/k (1) arising from truncation of the harmonic series. A harmonic number can be expressed analytically as ...

The regular dodecahedron, often simply called "the" dodecahedron, is the Platonic solid composed of 20 polyhedron vertices, 30 polyhedron edges, and 12 pentagonal faces, ...

Zeros of the Riemann zeta function zeta(s) come in two different types. So-called "trivial zeros" occur at all negative even integers s=-2, -4, -6, ..., and "nontrivial ...

Consecutive number sequences are sequences constructed by concatenating numbers of a given type. Many of these sequences were considered by Smarandache and so are sometimes ...

A product involving an infinite number of terms. Such products can converge. In fact, for positive a_n, the product product_(n=1)^(infty)a_n converges to a nonzero number iff ...

The rectilinear crossing number of a graph G is the minimum number of crossings in a straight line embedding of G in a plane. It is variously denoted rcr(G), cr^_(G) ...