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.

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...

In this paper we define a variant of the Ising model in which spins are replaced with permutations. The energy between two is function relative disorder one spin, permutation, to other. This motivated by complexity measure for declarative systems. For such systems state permutation and sorting measures average sequential neighbouring states. To use symmetrized version descent statistic that has...

An elusive permutation group is a transitive permutation group with no fixed point free elements of prime orders. In this paper we proved that each minimal normal subgroup of an elusive permutation group with squarefree degree is abelian. As an almost immediate consequence, it follows that there are no elusive permutation groups of degree 2pq where p, q are distinct odd primes.

We consider the relation between various permutation statistics and properties of permutation tableaux. We answer some of the open problems of Steingŕımsson and Williams [8], in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We also consider and enumerate sets of permutation tableaux related to some pattern restrictions on permu...