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A matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, every linear transformation can be represented by a ...

Any symmetric polynomial (respectively, symmetric rational function) can be expressed as a polynomial (respectively, rational function) in the elementary symmetric ...

The circumsphere of given set of points, commonly the vertices of a solid, is a sphere that passes through all the points. A circumsphere does not always exist, but when it ...

7 7 6 6 3 1; 6 5 4 2 ; 3 3 ; 2 A descending plane partition of order n is a two-dimensional array (possibly empty) of positive integers less than or equal to n such that the ...

In general, an arrangement of objects is simply a grouping of them. The number of "arrangements" of n items is given either by a combination (order is ignored) or permutation ...

Every finite group of order n can be represented as a permutation group on n letters, as first proved by Cayley in 1878 (Rotman 1995).

A subset tau in S_n of a permutation {1,...,n} is said to contain alpha in S_k if there exist 1<=i_1<...<i_k<=n such that tau=(tau_i,...,tau_k) is order isomorphic to ...

Let A, B, and C be three polar vectors, and define V_(ijk) = |A_i B_i C_i; A_j B_j C_j; A_k B_k C_k| (1) = det[A B C], (2) where det is the determinant. The V_(ijk) is a ...

The number of elements greater than i to the left of i in a permutation gives the ith element of the inversion vector (Skiena 1990, p. 27).

A primitive subgroup of the symmetric group S_n is equal to either the alternating group A_n or S_n whenever it contains at least one permutation which is a q-cycle for some ...