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...

{\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}}...

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

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...

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

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...

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...

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

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

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...

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...

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...

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

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...

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

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

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

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...

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...

{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 , −...

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...

definitions. The Lie algebra corresponding to Lie groups O(n, F ) and SO(n, F ) consists of the skew-symmetric n × n matrices, with the Lie bracket [ , ] given...

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

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...

run in Θ(1) time (amortized, if the cost of insert is) while delete-min still runs in O(log n). Applied to skew binomial heaps, it yields Brodal-Okasaki...

of set systems constructed from three permutations (Alantha Newman, Aleksandar Nikolov, 2011) Bloch–Kato conjecture (Vladimir Voevodsky, 2011) (and Quillen–Lichtenbaum...

{a}}_{i}} as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals...