Related papers: Generating functions of permutations with respect …

For a prime $p$ and nonnegative integers $j$ and $n$ let $\vartheta_p(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are exactly divisible by $p^j$. Moreover, for a finite sequence $w=(w_{r-1}\cdots w_0)\neq…

We address the problem of the number of permutations that can be sorted by two stacks in series. We do this by first counting all such permutations of length less than 20 exactly, then using a numerical technique to obtain nineteen further…

This paper is about the positive part $U_q^+$ of the $q$-deformed enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$. The literature contains at least three PBW bases for $U_q^+$, called the Damiani, the Beck, and the alternating PBW…

Given a barrier $0 \leq b_0 \leq b_1 \leq ...$, let $f(n)$ be the number of nondecreasing integer sequences $0 \leq a_0 \leq a_1 \leq ... \leq a_n$ for which $a_j \leq b_j$ for all $0 \leq j \leq n$. Known formul\ae for $f(n)$ include an $n…

Permutation statistics $\wnm$ and $\rlm$ are both arising from permutation tableaux. $\wnm$ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While $\rlm$ is…

The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we…

We characterise and enumerate permutations that are sortable by n-4 passes through a stack. We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a…

This paper is concerned with the joint distribution of the number of exterior peaks and the number of proper double descents over permutations on $[n] =\{1,2,\ldots,n\}$. The notion of exterior peaks of a permutation was introduced by…

We reconsider the problem of regularizing the divergent series $\sum_{n=1}^{\infty}n^{\alpha}$ for $\operatorname{Re}\alpha>-1$, and offer a regularization prescription that yields the Riemann zeta regularization as a special case. The…

Let $F_{n}$ and $L_n$ be the $n$th Fibonacci and Lucas number, respectively. For each positive integer $m$, the order of appearance of $m$ in the Fibonacci sequence, denoted by $z(m)$, is the smallest positive integer $k$ such that $m$…

We find the generating function for the permutation class $\mathcal{A}'=\text{Av}(52341,53241,52431,35142,42513,351624)$ whose permutations index local complete intersection Schubert varieties. The method we apply is the extension of how…

A Catalan word $w$ is said to be flattened if the subsequence of $w$ obtained by taking the first letter of each weakly increasing run is nondecreasing. Let $\mathcal{F}_n$ denote the set of flattened Catalan words of length $n$, which has…

We present a prime-generating polynomial $(1+2n)(p -2n) + 2$ where $p>2$ is a lower member of a pair of twin primes less than $41$ and the integer $n$ is such that $\: \frac {1-p}{2} < n < p-1$.

For a fixed integer $m\ge 1$, let $\mathcal{A}_n^{(m)}$ be the set of permutations $\pi\in S_n$ that avoid the pattern $132$ and satisfy the adjacency bound $|\pi_{i+1}-\pi_i|\le m$ for all $i$. Here, a pattern $132$ means three indices…

In this paper, by making use of properties of elliptic functions, we describe meromorphic solutions of Fermat-type functional equations $f(z)^{n}+f(L(z))^{m}=1$ over the complex plane $\mathbb{C}$, where $L(z)$ is a nonconstant entire…

We use the generating function of the characters of $C_2$ to obtain a generating function for the multiplicities of the weights entering in the irreducible representations of that simple Lie algebra. From this generating function we derive…

We introduce a basis of rational polynomial-like functions $P_0,\ldots,P_{n-1}$ for the free module of functions $Z/nZ\to Z/mZ$. We then characterize the subfamily of congruence preserving functions as the set of linear combinations of the…

Let $m,n\in \Bbb{N}$. We represent the additive subgroups of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of…

In this paper, we introduce a context-free grammar $G\colon x \rightarrow xy,\, y \rightarrow zu,\, z \rightarrow zw,\, w \rightarrow xv,\, u \rightarrow xyz^{-1}v,\, v \rightarrow x^{-1}zwu$ over the variable set $V=\{x,y,z,w,u,v\}$. We…

Let V be an infinite matrix with rows and columns indexed by the positive integers, and entries in a field F. Suppose that v_{i,j} only depends on i-j and is 0 for |i-j| large. Then V^n is defined for all n, and one has a "generating…