Related papers: The peak and descent statistics over ballot permut…
Permutation statistics constitute a classical subject of enumerative combinatorics. In her study of the genus zeta function, Denert discovered a new Mahonian statistic for permutations, which is called the Denert's statistic ({\bf $\den$})…
Drawing on a problem posed by Hertzsprung in 1887, we say that a given permutation $\pi\in\mathcal{S}_n$ contains the Hertzsprung pattern $\sigma\in\mathcal{S}_k$ if there is factor $\pi(d+1)\pi(d+2)\cdots\pi(d+k)$ of $\pi$ such that…
The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice…
We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random…
Recently, Kitaev and Remmel [Classifying descents according to parity, Annals of Combinatorics, to appear 2007] refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers. Results in that paper…
A four-variable distribution on permutations is derived, with two dual combinatorial interpretations. The first one includes the number of fixed points "fix", the second the so-called "pix" statistic. This shows that the duality between…
A consecutive pattern in a permutation $\pi$ is another permutation $\sigma$ determined by the relative order of a subsequence of contiguous entries of $\pi$. Traditional notions such as descents, runs and peaks can be viewed as particular…
A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary…
We consider a multinomial distribution, where the number of cells increases and the cell-probabilities decreases as the number of observations grows. The probabilities of large deviations of statistics, which has form of a sum of Borel…
In 1916, MacMahon showed that permutations in $S_n$ with a fixed descent set $I$ are enumerated by a polynomial $d_I(n)$. Diaz-Lopez, Harris, Insko, Omar, and Sagan recently revived interest in this descent polynomial, and suggested the…
For a finite subset $I$ of positive integers, the descent polynomial $\mathcal{D}(I;n)$ counts the number of permutations in $S_n$ that have descent set $I$. We generalize descent polynomials by considering permutations with a specific…
By using the matrix formulation of the two-step approach to distributions of patterns in random sequences, recurrence and explicit formulas for the generating functions of successions in random permutations of arbitrary multisets are…
This paper develops techniques to study the number of descents in random permutations via martingales. We relax an assumption in the Berry-Esseen theorem of Bolthausen (1982) to extend the theorem's scope to martingale differences of…
We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.
A conjecture by Deutsch, Kitaev, and Remmel states that the triples of permutation statistics $(S_{10}, S_{12}, S_{17})$ and $(S_{12}, S_{10} ,S_{17})$ are equidistributed over the symmetric group $\mathfrak{S}_n$. Here, $S_{10}$ enumerates…
Using earlier results we prove a formula for the number $W_{(n,k)}$ of 2-stack sortable permutations of length $n$ with $k$ runs, or in other words, $k-1$ descents. This formula will yield the suprising fact that there are as many 2-stack…
In this paper we look at polynomials arising from statistics on the classes of involutions, $I_n$, and involutions with no fixed points, $J_n$, in the symmetric group. Our results are motivated by F. Brenti's conjecture which states that…
We define a map between the set of permutations that avoid either the four patterns $3214,3241,4213,4231$ or $3124,3142,4123,4132$, and the set of Dyck prefixes. This map, when restricted to either of the two classes, turns out to be a…
We introduce Lipschitz functions on a finite partially ordered set $P$ and study the associated Lipschitz polytope $L(P)$. The geometry of $L(P)$ can be described in terms of descent-compatible permutations and permutation statistics that…
Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain…