Related papers: Circular Peaks and Hilbert Series

We consider a tuple $\Phi = (\phi_1,\ldots,\phi_m)$ of commuting maps on a finitary matroid $X$. We show that if $\Phi$ satisfies certain conditions, then for any finite set $A\subseteq X$, the rank of $\{\phi_1^{r_1}\cdots\phi_m^{r_m}(a):a…

We apply the homomorphism complex construction to partially ordered sets, introducing a new topological construction based on the set of maximal chains in a graded poset. Our primary objects of study are distributive lattices, with special…

We consider the cycle structure of a random permutation $\sigma$ chosen uniformly from the symmetric group, subject to the constraint that $\sigma$ does not contain cycles of length exceeding $r.$ We prove that under suitable conditions the…

There are several remarks on Hilbert series of finitely presented (f. p.) associative algebras over a field and their modules. First, given an integer $D$, the set of Hilbert series of right-sided ideals with generators and relations of…

Let $\mathfrak{S}_n$ and $\mathfrak{B}_n$ denote the respective sets of ordinary and bigrassmannian (BG) permutations of order $n$, and let $(\mathfrak{S}_n,\leq)$ denote the Bruhat ordering permutation poset. We study the restricted poset…

A finite poset $P$ is called "simplicial", if it has the smallest element $\hat{0}$, and every interval $[\hat{0}, x]$ is a boolean algebra. The face poset of a simplicial complex is a typical example. Generalizing the Stanley-Reisner ring…

A family of permutations A \subset S_n is said to be intersecting if any two permutations in A agree at some point, i.e. for any \sigma, \pi \in A, there is some i such that \sigma(i)=\pi(i). Deza and Frankl showed that for such a family,…

We study algebraic dynamical systems (and, more generally, $\sigma$-varieties) $\Phi:{\mathbb A}^n_{\mathbb C} \to {\mathbb A}^n_{\mathbb C}$ given by coordinatewise univariate polynomials by refining a theorem of Ritt. More precisely, we…

We study a sorting procedure (run-sorting) on permutations, where runs are rearranged in lexicographic order. We describe a rather surprising bijection on permutations on length $n$, with the property that it sends the set of peak-values to…

The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real…

Let $C(n)$ denote the number of permutations $\sigma$ of $[n]=\{1,2,\dots,n\}$ such that $\gcd(j,\sigma(j))=1$ for each $j\in[n]$. We prove that for $n$ sufficiently large, $n!/3.73^n < C(n) < n!/2.5^n$.

For a permutation $\pi$ in the symmetric group $S_n$ let the {\it total degree} be its valency in the Hasse diagram of the strong Bruhat order on $S_n$, and let the {\it down degree} be the number of permutations which are covered by $\pi$…

Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\ldots,a_{n-1}\in G$. We show that there is a permutation $\sigma\in S_{n-1}$ such that all the elements $sa_{\sigma(s)}\ (s=1,\ldots,n-1)$ are nonzero if and only…

Given a Stirling permutation w, we introduce the mesa set of w as the natural generalization of the pinnacle set of a permutation. Our main results characterize admissible mesa sets and give closed enumerative formulas in terms of rational…

A set $S$ of natural numbers is multiplicative Sidon if the products of all pairs in $S$ are distinct. Erd\H{o}s in 1938 studied the maximum size of a multiplicative Sidon subset of $\{1,\ldots, n\}$, which was later determined up to the…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal…

Let $n\ge 2$ be a positive integer. To each irreducible representation $\sigma$ of $\mathrm{Sp}(1)$, an $\mathrm{Sp}(1)$-Kepler problem in dimension $(4n-3)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is…

Each group G of nxn permutation matrices has a corresponding permutation polytope, P(G):=conv(G) in R^{nxn}. We relate the structure of P(G) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then…

We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{\sigma(1)} \cdots x_{\sigma(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $\sigma$…

Let \(H(n)\) denote the Hilbert number, i.e.\ the maximal number of limit cycles of planar polynomial vector fields of degree \(\le n\). A classical lower-bound mechanism for \(H(n)\) is \emph{replication}: one pulls back a vector field by…