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A permutation, also called an "arrangement number" or "order," is a rearrangement of the elements of an ordered list S into a one-to-one correspondence with S itself. The ...

The number of different triangles which have integer side lengths and perimeter n is T(n) = P(n,3)-sum_(1<=j<=|_n/2_|)P(j,2) (1) = [(n^2)/(12)]-|_n/4_||_(n+2)/4_| (2) = ...

A binary bracketing is a bracketing built up entirely of binary operations. The number of binary bracketings of n letters (Catalan's problem) are given by the Catalan numbers ...

A rook polynomial is a polynomial R_(m,n)(x)=sum_(k=0)^(min(m,n))r_kx^k (1) whose number of ways k nonattacking rooks can be arranged on an m×n chessboard. The rook ...

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 graph G having chromatic number gamma(G)=k is called a k-chromatic graph (Harary 1994, p. 127). In contrast, a graph having gamma(G)<=k is said to be a k-colorable graph. A ...

Given a point set P={x_n}_(n=0)^(N-1) in the s-dimensional unit cube [0,1)^s, the star discrepancy is defined as D_N^*(P)=sup_(J in Upsilon^*)D(J,P), (1) where the local ...

There are a number of functions in mathematics denoted with upper or lower case Qs. 1. The nome q. 2. A prefix denoting q-analogs and q-series. 3. Q_n or q_n with n=0, 1, 2, ...

Iff p is a prime, then (p-1)!+1 is a multiple of p, that is (p-1)!=-1 (mod p). (1) This theorem was proposed by John Wilson and published by Waring (1770), although it was ...

Given a real number x, find the powers of a base b that will shift the digits of x a number of places n to the left. This is equivalent to solving b^x=b^nx (1) or x=n+log_bx. ...