No Arabic abstract
We observe that the necklace polynomials $M_d(x) = frac{1}{d}sum_{emid d}mu(e)x^{d/e}$ are highly reducible over $mathbb{Q}$ with many cyclotomic factors. Furthermore, the sequence $Phi_d(x) - 1$ of shifted cyclotomic polynomials exhibits a qualitatively similar phenomenon, and it is often the case that $M_d(x)$ and $Phi_d(x) - 1$ have many common cyclotomic factors. We explain these cyclotomic factors of $M_d(x)$ and $Phi_d(x) - 1$ in terms of what we call the emph{$d$th necklace operator}. Finally, we show how these cyclotomic factors correspond to certain hyperplane arrangements in finite abelian groups.
Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.
In this note, by the umbra calculus method, the Sun and Zagiers congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.
The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also found a quadratic recursion for the alternating major index $q$-analog of the alternating descent polynomials. As an interesting application of this quadratic recursion, we show that $(1+q)^{lfloor n/2rfloor}$ divides $sum_{piinmathfrak{S}_n}q^{rm{altmaj}(pi)}$, where $mathfrak{S}_n$ is the set of all permutations of ${1,2,ldots,n}$ and $rm{altmaj}(pi)$ is the alternating major index of $pi$. This leads us to discover a $q$-analog of $n!=2^{ell}m$, $m$ odd, using the statistic of alternating major index. Moreover, we study the $gamma$-vectors of the alternating descent polynomials by using these two recursions and the ${textbf{cd}}$-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.
The aim of the article is to understand the combinatorics of snake graphs by means of linear algebra. In particular, we apply Kasteleyns and Temperley--Fishers ideas about spectral properties of weighted adjacency matrices of planar bipartite graphs to snake graphs. First we focus on snake graphs whose set of turning vertices is monochromatic. We provide recursive sequences to compute the characteristic polynomials; they are indexed by the upper or the lower boundary of the graph and are determined by a neighbour count. As an application, we compute the characteristic polynomials for L-shaped snake graphs and staircases in terms of Fibonacci product polynomials. Next, we introduce a method to compute the characteristic polynomials as convergents of continued fractions. Finally, we show how to transform a snake graph with turning vertices of two colours into a graph with the same number of perfect matchings to which we can apply the results above.
The problem of expressing an element of K_2(F) in a more explicit form gives rise to many works. To avoid a restrictive condition in a work of Tate, Browkin considered cyclotomic elements as the candidate for the element with an explicit form. In this paper, we modify and change Browkins conjecture about cyclotomic elements into more precise forms, in particular we introduce the conception of cyclotomic subgroup. In the rational function field cases, we determine completely the exact numbers of cyclotomic elements and cyclotomic subgroups contained in a subgroup generated by finitely many different cyclotomic elements, while in the number field cases, using Faltings theorem on Mordell conjecture we prove that there exist subgroups generated by an infinite number of cyclotomic elements to the power of some prime, which contain no nontrivial cyclotomic elements.