相关论文: An application of the Goulden-Jackson cluster theo…

Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the…

We investigate the homology representation of the symmetric group on rank-selected subposets of subword order. We show that the homology module for words of bounded length, over an alphabet of size $n,$ decomposes into a sum of tensor…

The Goulden-Jackson cluster method is a powerful tool for obtaining generating functions for counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which…

Using combinatorial techniques, we prove that the weighted sum of the inverse number of automorphisms of all finite abelian $p$-groups $\sum_G |G|^{-u} |\text{Aut}(G)|^{-1}$ is equal to $\prod_{j=u+1}^\infty\left(1-1/p^j\right)^{-1}$, where…

For a Lie algebra L over an algebraically closed field of non-zero characteristic, every finite-dimensional L-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character.…

In this paper we prove that for any infinite word W whose set of factors is closed under reversal, the following conditions are equivalent: (I) all complete returns to palindromes are palindromes; (II) P(n) + P(n+1) = C(n+1) - C(n) + 2 for…

A word over an ordered alphabet is said to be clustering if identical letters appear adjacently in its Burrows-Wheeler transform. Such words are strictly related to (discrete) interval exchange transformations. We use an extended version of…

By $(\mathbb{Z}^+)^{\infty}$ we denote the set of all the infinite sequences $\mathcal{S}=\{s_i\}_{i=1}^{\infty}$ of positive integers (note that all the $s_i$ are not necessarily distinct and not necessarily monotonic). Let $f(x)$ be a…

In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.

We establish, for the character table of the symmetric group, the positivity of the row sums indexed by irreducible characters, when restricted to various subsets of the conjugacy classes. A notable example is that of partitions with all…

Let $G$ be a finite abeilian group. A sequence $S$ with terms from $G$ is zero-sum if the sum of terms in $S$ equals zero. It is a minimal zero-sum sequence if no proper, nontrivial subsequence is zero-sum. The maximal length of a minimal…

We give several modifications of the Goulden-Jackson Cluster method for finding generating functions for words avoiding a given set of forbidden words. Our modifications include functions which can take into account various 'weights' on…

Given a set $F$ of words, one associates to each word $w$ in $F$ an undirected graph, called its extension graph, and which describes the possible extensions of $w$ on the left and on the right. We investigate the family of sets of words…

Let G be a complex, affine algebraic group and D a meromorphic connection on the trivial G-bundle over P^1, with a pole of order 2 at zero and a pole of order 1 at infinity. We show that the map S taking the residue of D at zero to the…

Let $A$ be a subset of a finite field $\mathbb{F}$. When $\mathbb{F}$ has prime order, we show that there is an absolute constant $c > 0$ such that, if $A$ is both sum-free and equal to the set of its multiplicative inverses, then $|A| <…

Universal Cycles, or U-cycles, as originally defined by de Bruijn, are an efficient method to exhibit a large class of combinatorial objects in a compressed fashion, and with no repeats. de Bruijn's theorem states that U-cycles for $n$…

A classical tool in the study of real closed fields are the fields $K((G))$ of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian…

In this paper, we present a reciprocity on finite abelian groups involving zero-sum sequences. Let $G$ and $H$ be finite abelian groups with $(|G|,|H|)=1$. For any positive integer $m$, let $\mathsf M(G,m)$ denote the set of all zero-sum…

This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence $(S(n))_{n\ge 0}$…

In this paper, we prove the following result: {quote} Let $\A$ be an infinite set of positive integers. For all positive integer $n$, let $\tau_n$ denote the smallest element of $\A$ which does not divide $n$. Then we have $$\lim_{N \to +…