Related papers: Schur-Concavity for Avoidance of Increasing Subseq…

We study the number of values taken by the sums $\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\dots,a_n$ is a permutation of $1,2,\dots,n$ and $1 \leq u < v \leq n+1$. In particular, we show that for a random choice of a permutation, with high…

We show that for any log-concave measure $\mu$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $\lambda\in [0,1],$ $$\mu((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu(K)^{c_n}+\lambda\mu(L)^{c_n},$$ where…

Recently, Archer et al.\ studied cyclic permutations that avoid the decreasing pattern $\delta_k=k(k-1)\cdots21$ in one-line notation and avoid another pattern $\tau$ of length $4$ in all their cycle forms. There are three cases in total to…

Ascent sequences were introduced by Bousquet-M\'{e}lou, Claesson, Dukes and Kitaev in their study of $(\bf{2+2})$-free posets. An ascent sequence of length $n$ is a nonnegative integer sequence $x=x_{1}x_{2}... x_{n}$ such that $x_{1}=0$…

We define a class L_{n, k} of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give two bijections between…

The sequence a_1,...,a_m is a common subsequence in the set of permutations S = {p_1,...,p_k} on [n] if it is a subsequence of p_i(1),...,p_i(n) and p_j(1),...,p_j(n) for some distinct p_i, p_j in S. Recently, Beame and Huynh-Ngoc (2008)…

We define a number of related combinatorial objects, each of which possesses a surprising symmetry. We include several applications such as a combinatorial explanation for certain fixed points of the involution $\omega$ on the ring of…

Starting from some considerations we make about the relations between certain difference statistics and the classical permutation statistics we study permutations whose inversion number and excedance difference coincide. It turns out that…

We prove an existing conjecture that the sequence defined recursively by $a_1=1, a_2=2, a_n=4a_{n-1}-2a_{n-2}$ counts the number of length-$n$ permutations avoiding the four generalized permutation patterns 1-32-4, 1-42-3, 2-31-4, and…

Let $A^{(n)}_{l;k}\subset S_n$ denote the set of permutations of $[n]$ for which the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions. Under the uniformly probability measure $P_n$ on…

A set $A$ of integers is said to be Schur if any two-colouring of $A$ results in monochromatic $x,y$ and $z$ with $x+y=z$. We study the following problem: how many random integers from $[n]$ need to be added to some $A\subseteq [n]$ to…

Let us define, for a compact set $A \subset \mathbb{R}^n$, the Minkowski averages of $A$: $$ A(k) = \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots +…

For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >=…

Sums of $M$ consecutive squared integers $\left(a+i\right)^{2}$ equaling squared integers (for $a\geq1$, $0\leq i\leq M-1$) yield certain linear groupings of pairs $\left(a_{1},a_{2}\right)$ of $a$ values for successive same values of $M$…

For a set of permutations (patterns) $\Pi$ in $S_k$, consider the set of all permutations in $S_n$ that avoid all patterns in $\Pi$. An important problem in current algebraic combinatorics is to find pattern sets $\Pi$ such that the…

We prove that the permutations of $\{1,\dots, n\}$ having an increasing (resp., decreasing) subsequence of length $n-r$ index a subset of the set of all $r$th Kronecker powers of $n \times n$ permutation matrices which is a basis for the…

Let $\pi$ be a cyclic permutation that can be expressed in its one-line form as $\pi = \pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and in its standard cycle form as $\pi = (c_1,c_2, ..., c_n)$ where $c_1=1$. Archer et al. introduced the notion of…

We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the…

Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to (2+2)-free posets and…

Ascent sequences are those consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it and have been shown to be equinumerous with the (2+2)-free posets of the same size.…