Related papers: Clustering of consecutive numbers in permutations …
Quite recently, in [8] the authoor of this paper considered the distribution of primes in the sequence $(S_n)$ whose $n$th term is defined as $S_n=\sum_{k=1}^{2n}p_k$, where $p_k$ is the $k$th prime. Some heuristic arguments and the…
A small set of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on $[0, \infty)$. Such sequences are known as Stieltjes moment sequences. This article focuses on some classical sequences…
There are several approaches to study occurrences of consecutive patterns in permutations such as the inclusion-exclusion method, the tree representations of permutations, the spectral approach and others. We propose yet another approach to…
Let $\mathbb{S}_n$ denote the symmetric group on $[n]=\{1,\ldots,n\}$ with the uniform probability measure. For a permutation $\pi \in \mathbb{S}_n$ let $X_{\pi}$ denote the simplicial complex on the vertex set $[n]$ whose simplices are all…
Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For a positive integer $k$, define $S_{n,k}$ to be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles, i.e., \[ S_{n,k} = \{\pi \in S_{n}: \pi =…
The starting point of this work is an equality between two quantities $A$ and $B$ found in the literature, which involve the {\em doubling-modulo-an-odd-integer} map, i.e., $x\in {\mathbb N} \mapsto 2x \bmod{(2n+1)}$ for some positive…
Initiated by Davis, Nelson, Petersen and Tenner (2018), the enumerative study of pinnacle sets of permutations has attracted a fair amount of attention recently. In this article, we provide a recurrence that can be used to compute…
Selecting N random points in a unit square corresponds to selecting a random permutation. By putting 5 types of symmetry restrictions on the points, we obtain subsets of permutations : involutions, signed permutations and signed…
The simple permutations in two permutation classes --- the 321-avoiding permutations and the skew-merged permutations --- are enumerated using a uniform method. In both cases, these enumerations were known implicitly, by working backwards…
Assume that a convergent series of real numbers $\sum\limits_{n=1}^\infty a_n$ has the property that there exists a set $A\subseteq \N$ such that the series $\sum\limits_{n \in A} a_n$ is conditionally convergent. We prove that for a given…
In a uniform random permutation \Pi of [n] := {1,2,...,n}, the set of elements k in [n-1] such that \Pi(k+1) = \Pi(k) + 1 has the same distribution as the set of fixed points of \Pi that lie in [n-1]. We give three different proofs of this…
Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…
In "Random complex fewnomials, I," B. Shiffman and S. Zelditch determine the limiting formula as N goes to infinity of the (normalized) expected distribution of complex zeros of a system of k random n-nomials in m variables where the…
The circular peak set of a permutation $\sigma$ is the set $\{\sigma(i)\mid \sigma(i-1)<\sigma(i)>\sigma(i+1)\}$. Let $\mathcal{P}_n$ be the set of all the subset $S\subseteq [n]$ such that there exists a permutation $\sigma$ which has the…
A Parity Alternating Permutation of the set $[n] = \{1, 2,\ldots, n\}$ is a permutation with even and odd entries alternatively. We deal with parity alternating permutations having an odd entry in the first position, PAPs. We study the…
Vincular and covincular patterns are generalizations of classical patterns allowing restrictions on the indices and values of the occurrences in a permutation. In this paper we study the integer sequences arising as the enumerations of…
The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$…
We consider the enumeration of ordered set partitions avoiding a permutation pattern, as introduced by Godbole, Goyt, Herdan and Pudwell. Let $\op_{n,k}(p)$ be the number of ordered set partitions of $\{1,2,\ldots,n\}$ into $k$ blocks that…
A collection of finite sets $\{A_1, A_2,\ldots, A_{p}\}$ is said to be a double-covering if each $a\in \cup_{k=1}^{p}A_k$ is included in exactly two sets of the collection. For fixed integers $l$ and $p$, let $\mu_{l,p}$ be the number of…
Packing density is a permutation occurrence statistic which describes the maximal number of permutations of a given type that can occur in another permutation. In this article we focus on containment of sets of permutations. Although this…