Several permutation strategies are often possible for tests of individual terms in analysis-of-variance (ANOVA) designs. These include restricted permutations, permutation of whole groups of units, permutation of some form of residuals or some combination of these. It is unclear, especially for complex designs involving random factors, mixed models or nested hierarchies, just which permutation ...

It is shown that there are arbitrarily long “good” (in the sense of Gilbert) binary block codes that are preserved under very large permutation groups. This result contrasts sharply with the properties of linear codes: it is conjectured that long cyclic codes are bad, and known that long al&e-invariant codes are bad. 1 T IS the object of this paper to show that there exist arbitrarily long bina...

The VMPC function is a combination of triple permutation composition and integer addition. It differs from a simple triple permutation composition with one integer addition operation performed on some of the elements of the permutation. The consequence of this addition operation is corruption of cycle structure of the transformed permutation the fundamental source of the function's resistance t...

A coloring of a graph G is an assignment of colors to its vertices so that no two adjacent vertices have the same color. We study the problem of coloring permutation graphs using certain properties of the lattice representation of a permutation and relationships between permutations, directed acyclic graphs and rooted trees having speci/c key properties. We propose an e0cient parallel algorithm...

Let In(π) denote the number of involutions in the symmetric group Sn which avoid the permutation π. We say that two permutations α, β ∈ Sj may be exchanged if for every k and ordering τ of j + 1, . . . , k, we have In(ατ) = In(βτ) for every n. Here we prove that 12 and 21 may be exchanged and that 123 and 321 may be exchanged. The first of these theorems gives a number of known results for patt...

In this paper, we study pattern-avoidance in the set of words over the alphabet [k]. We say that a word w ∈ [k] contains a pattern τ ∈ [l], if w contains a subsequence order-isomorphic to τ . This notion generalizes pattern-avoidance in permutations. We determine all the Wilf-equivalence classes of word patterns of

The linear extension diameter of a finite poset P is the diameter of the graph on all linear extensions of P as vertices, two of them being adjacent whenever they differ in a single adjacent transposition. We determine the linear extension diameter of the subposet of the Boolean lattice induced by the 1st and kth levels and describe an explicit construction of all diametral pairs. This partiall...

For about 10 years, the classification up to Wilf equivalence of permutation patterns was thought completed up to length 6. In this paper, we establish a new class of Wilf-equivalent permutation patterns, namely, (n −1, n −2, n, τ ) ∼ (n −2, n, n −1, τ ) for any τ ∈ Sn−3. In particular, at level n = 6, this result includes the only missing equivalence (546213) ∼ (465213), and for n = 7 it compl...

A permutation is an arrangement of a finite number of distinct elements of a linear order, for example, e, π, 0, √ 2 and 3412. Two permutations are order isomorphic if the have the same relative ordering. We say a permutation τ contains or involves the permutation β if deleting some of the entries of τ gives a permutation that is order isomorphic to β, and we write β ≤ τ . For example, 534162 (...

Permutation pattern classes that are defined by avoiding two permutations only and which contain only finitely many simple permutations are characterized and their growth rates are determined.