the skew sum and direct sum of permutations are two operations to combine shorter permutations into longer ones. Given a permutation π of length m and the...

Robinson–Schensted correspondence Sum of permutations: Direct sum of permutations Skew sum of permutations Stanley–Wilf conjecture Symmetric function Szymanski's...

separable permutation is a permutation that can be obtained from the trivial permutation 1 by direct sums and skew sums. Separable permutations may be characterized...

swaps elements 1 and 2. If π and σ are two permutations represented in this way (these variable names are standard for permutations and are unrelated to...

stack-sortable permutations, they are also a subset of the separable permutations, the permutations formed by recursive combinations of direct and skew sums. Albert...

of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of...

that can be obtained by direct sums and skew sums of the trivial permutation Separable polynomial, a polynomial whose number of distinct roots is equal...

Bidwell Wilson (1913). "Chapter II. Direct and Skew Products of Vectors". Vector Analysis. Founded upon the lectures of J. William Gibbs. New Haven: Yale...

|(1/n)\sum {((x_{i}-{\hat {\mu }})/{\hat {\sigma }})^{3}}|)} . Concern has been expressed about the inference of skew normal distributions using the direct parameterization...

chosen in K minimizing the sums of the squares of the norms of the off-diagonal terms of k(X ). Since K preserves the sums of all the squares, this is equivalent...

{\mbox{Mat}}_{n}={\mbox{Sym}}_{n}\oplus {\mbox{Skew}}_{n},} where ⊕ {\displaystyle \oplus } denotes the direct sum. Let X ∈ Mat n {\displaystyle X\in {\mbox{Mat}}_{n}}...

the rightmost axis. Among all permutations of (x,y,z), only two place that axis first; one is an even permutation and the other odd. Choosing parity...

corresponding permutation (which is + 1 {\displaystyle +1} for an even number of permutations and is − 1 {\displaystyle -1} for an odd number of permutations). Once...

sum is taken over the symmetric group of permutations on the symbols ⁠ { 1 , … , r } {\displaystyle \{1,\dots ,r\}} ⁠. This extends by linearity and homogeneity...

semisimple algebra. For the definition of the direct sum of representations please refer to the section on direct sums of representations. A representation...

whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective...

approximated using permutation tests and Monte Carlo simulations. Some books tabulate statistics equivalent to U, such as the sum of ranks in one of the samples...

{C} ^{n}} whose entries sum to zero. The roots then consist of all the n(n − 1) permutations of (1, −1, 0, ..., 0). A choice of simple roots is ( 1 , −...

into an algorithm that naively expands the formula, summing over all permutations and within the sum multiplying out each matrix entry. This requires n...

n {\displaystyle G=A_{n}} is the alternating group and S {\displaystyle S} is a set of permutations given by { ( 12 i ) ± 1 } {\displaystyle \{(12i)^{\pm...

context of address translation, as explained below. Other schemes have been suggested, such as the skewed cache, where the index for way 0 is direct, as above...

number of randomly drawn permutations (exchanging a minor loss in precision if a permutation is drawn twice—or more frequently—for the efficiency of not...

distinguishable permutations of the V-Cube 6 (6×6×6 Rubik's Cube). Chess: Shannon number, 10120, a lower bound of the game-tree complexity of chess. Physics:...

generalizations of Lie algebras in which the underlying vector space has a Z2-grading, and skew-symmetry and Jacobi identity properties of the Lie bracket...

are a permutation of the set {1,...,n}. The permutation i′ is selected randomly, with equal probabilities placed on all n! possible permutations. This...

between states of a system. In particular the entries are non-negative, and every row of the matrix sums to one, being the sum of probabilities of transitions...

key idea of the FKT algorithm is to convert the problem into a Pfaffian computation of a skew-symmetric matrix derived from a planar embedding of the graph...

between −1 and +1. The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect...

moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis...

algebra s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations...