Value-Peaks of Permutations

Value-Peaks of Permutations

In this paper, we focus on a “local property” of permutations: value-peak. A permutation σ has a value-peak σ(i) if σ(i − 1) < σ(i) > σ(i + 1) for some i ∈ [2, n − 1]. Define V P (σ) as the set of value-peaks of the permutation σ. For any S ⊆ [3, n], define V Pn(S) such that V P (σ) = S. Let Pn = {S | V Pn(S) 6= ∅}. we make the set Pn into a poset Pn by defining S T if S ⊆ T as sets. We prove t...

Derivative polynomials and enumeration of permutations by number of interior and left peaks

Realization of meander permutations by boundary value problems

Sharply $(n-2)$-transitive Sets of Permutations

Let $S_n$ be the symmetric group on the set $[n]={1, 2, ldots, n}$. For $gin S_n$ let $fix(g)$ denote the number of fixed points of $g$. A subset $S$ of $S_n$ is called $t$-emph{transitive} if for any two $t$-tuples $(x_1,x_2,ldots,x_t)$ and $(y_1,y_2,ldots ,y_t)$ of distinct elements of $[n]$, there exists $gin S$ such that $x_{i}^g=y_{i}$ for any $1leq ileq t$ and additionally $S$ is called e...

Intersecting families of permutations and partial permutations

The above result motivated the study of intersecting families of permutations which was initiated by Deza and Frankl, see [2]. Let Sn be the symmetric group on [n], that is the group of all permutations of [n]. For a positive integer t, a subset A of Sn is said to be t-intersecting if, for any g, h ∈ A with g 6= h, we have |{x : g(x) = h(x)}| ≥ t. By an intersecting family, we mean an 1-interse...