The 2-adic valuation of plane partitions and totally symmetric partitions

通过 $p$-adic 赋值计数 Fibonomial 和 $C$-nomial 系数的通用矩阵

Universal Matrices for Counting Fibonomial and $C$-nomial Coefficients by their $p$-adic Valuations

针对由序列 $C$ 定义的二项式系数推广形式 $C$-nomial 系数，提出当 $C$ 为强整除序列时，其按 $p$-adic 赋值计数的生成函数可通过通用矩阵乘积公式表示，并将该公式推广至 $C$-多项式系数。

Legendre's formula and $p$-adic analysis

In number theory, we know Legendre's formula $ v_p(n!) = \sum_{k \ge 1} \lfloor \frac{n}{p^k} \rfloor $, which calculates the $p$-adic valuation of the factorial, i.e. the exponent of the greatest power of a prime $p$ that divides $n!$. There is also the second (or alternative) equality $ v_p (n!) = \frac{n-s_p(n)}{p-1} $ where $s_p(n)$ is the $p$-adic weight of $n$ or the sum of digits of $n$ in base $p$. Both kinds of Legendre's formula allow us to determine valuations of the natural number, the odd factorial, binomial coefficients, Catalan numbers, and other combinatorial objects. The artic…

On abelian $\ell$-towers of multigraphs

We study how the $\ell$-adic valuation of the number of spanning trees varies in regular abelian $\ell$-towers of multigraphs. We show that for an infinite family of regular abelian $\ell$-towers of bouquets, the behavior of the $\ell$-adic valuation of the number of spanning trees behave similarly to the $\ell$-adic valuation of the class numbers in $\mathbb{Z}_{\ell}$-extensions of number fields.

On the k-regularity of the k-adic valuation of Lucas sequences

For integers $k \geq 2$ and $n \neq 0$, let $v_k(n)$ denotes the greatest nonnegative integer $e$ such that $k^e$ divides $n$. Moreover, let $u_n$ be a nondegenerate Lucas sequence satisfying $u_0 = 0$, $u_1 = 1$, and $u_{n + 2} = a u_{n + 1} + b u_n$, for some integers $a$ and $b$. Shu and Yao showed that for any prime number $p$ the sequence $v_p(u_{n + 1})$ is $p$-regular, while Medina and Rowland found the rank of $v_p(F_{n + 1})$, where $F_n$ is the $n$-th Fibonacci number. We prove that if $k$ and $b$ are relatively prime then $v_k(u_{n + 1})$ is a $k$-regular sequence, and for $k$ a pri…

On the p-adic valuation of Stirling numbers of the first kind

For all integers $n \geq k \geq 1$, define $H(n,k) := \sum 1 / (i_1 \cdots i_k)$, where the sum is extended over all positive integers $i_1 < \cdots < i_k \leq n$. These quantities are closely related to the Stirling numbers of the first kind by the identity $H(n,k) = s(n + 1, k + 1) / n!$. Motivated by the works of Erdős-Niven and Chen-Tang, we study the $p$-adic valuation of $H(n,k)$. In particular, for any prime number $p$, integer $k \geq 2$, and $x \geq (k-1)p$, we prove that $ν_p(H(n,k)) < -(k - 1)(\log_p(n/(k - 1)) - 1)$ for all positive integers $n \in [(k-1)p, x]$ whose base $p$ repre…

On $p$-adic valuations of certain $m$ colored $p$-ary partition functions

Let $k\in\N_{\geq 2}$ and for given $m\in\Z\setminus\{0\}$ consider the sequence $(S_{k,m}(n))_{n\in\N}$ defined by the power series expansion $$ \frac{1}{(1-x)^{m}}\prod_{i=0}^{\infty}\frac{1}{(1-x^{k^{i}})^{m}}=\sum_{n=0}^{\infty}S_{k,m}(n)x^{n}. $$ The number $S_{k,m}(n)$ for $m\in\N_{+}$ has a natural combinatorial interpretation: it counts the number of representations of $n$ as sums of powers of $k$, where the part equal to $1$ takes one among $mk$ colors and each part $>1$ takes $m(k-1)$ colors. We concentrate on the case when $k=p\in\mathbb{P}$. Our main result is the computation of th…

Thue-Morse生成函数的五次方和九次方系数的2-adic赋值

2-adic Valuations of Coefficients of the Fifth and Ninth Powers of the Thue--Morse Generating Function

本文证明了Thue-Morse生成函数的五次方和九次方系数的2-adic赋值的精确公式，验证了Gawron-Miska-Ulas的猜想，并证明了所有系数非零。

p-adic valuations of some sums of multinomial coefficients

Let $m$ and $n>0$ be integers. Suppose that $p$ is a prime dividing $m-4$ but not dividing $m$. We show that $ν_p(\sum_{k=0}^{n-1}\frac{\binom{2k}k}{m^k})$ and $ν_p(\sum_{k=0}^{n-1}\binom{n-1}{k}(-1)^k\frac{\binom{2k}k}{m^k})$ are at least $ν_p(n)$, where $ν_p(x)$ denotes the $p$-adic valuation of $x$. Furthermore, if $p>3$ then $$n^{-1}\sum_{k=0}^{n-1}\frac{\bi{2k}k}{m^k}=\frac{\binom{2n-1}{n-1}}{4^{n-1}} (mod p^{ν_p(m-4)})$$ and $$n^{-1}\sum_{k=0}^{n-1}\binom{n-1}{k}(-1)^k\frac{\binom{2k}k}{m^k}=\frac{C_{n-1}}{4^{n-1}} (mod p^{ν_p(m-4)}),$$ where $C_k$ denotes the Catalan number $\binom{2k}{…

2-adic valuations of certain ratios of products of factorials and applications

We prove the conjecture of Falikman--Friedland--Loewy on the parity of the degrees of projective varieties of $n\times n$ complex symmetric matrices of rank at most $k$. We also characterize the parity of the degrees of projective varieties of $n\times n$ complex skew symmetric matrices of rank at most $2p$. We give recursive relations which determine the parity of the degrees of projective varieties of $m\times n$ complex matrices of rank at most $k$. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of $n\times n$ skew symmetric real matr…

方格沙堆群中全一类的阶的2-adic赋值

The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square

研究方格沙堆群中全一类的阶的2-adic赋值，通过循环基和多项式理想方法，给出了精确的公式。

迹码的可除性

Divisibility of Trace Codes

本文研究了迹码的可除性问题，提出了一种系统的方法来确定相关迹码的p-adic估值，扩展了Ward的经典可除性准则，并应用于证明阿贝尔码的可除性结果以及确定Artin-Schreier型方程解数的p-adic估值下界。