ترغب بنشر مسار تعليمي؟ اضغط هنا

Elementary proof of congruences modulo 25 for broken $k$-diamond partitions

77   0   0.0 ( 0 )
 نشر من قبل Tang Dazhao
 تاريخ النشر 2018
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

Let $Delta_{k}(n)$ denote the number of $k$-broken diamond partitions of $n$. Quite recently, the second author proved an infinite family of congruences modulo 25 for $Delta_{k}(n)$ with the help of modular forms. In this paper, we aim to provide an elementary proof of this result.



قيم البحث

اقرأ أيضاً

67 - Dazhao Tang 2017
The notion of broken $k$-diamond partitions was introduced by Andrews and Paule. Let $Delta_{k}(n)$ denote the number of broken $k$-diamond partitions of $n$ for a fixed positive integer $k$. In this paper, we establish new infinite families of broken $k$-diamond partition congruences.
75 - Dazhao Tang 2017
Let $p_{-k}(n)$ enumerate the number of $k$-colored partitions of $n$. In this paper, we establish some infinite families of congruences modulo 25 for $k$-colored partitions. Furthermore, we prove some infinite families of Ramanujan-type congruences modulo powers of 5 for $p_{-k}(n)$ with $k=2, 6$, and $7$. For example, for all integers $ngeq0$ and $alphageq1$, we prove that begin{align*} p_{-2}left(5^{2alpha-1}n+dfrac{7times5^{2alpha-1}+1}{12}right) &equiv0pmod{5^{alpha}} end{align*} and begin{align*} p_{-2}left(5^{2alpha}n+dfrac{11times5^{2alpha}+1}{12}right) &equiv0pmod{5^{alpha+1}}. end{align*}
206 - Dazhao Tang 2018
Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for $p_{k,3}(n)$ wi th $k=1, 3$, and $9$. For example, for all integers $ngeq0$ and $alphageq1$, we prove that begin{align*} p_{3,3}left(3^{alpha}n+dfrac{3^{alpha}+1}{2}right) &equiv0pmod{3^{alpha+1}} end{align*} and begin{align*} p_{3,3}left(3^{alpha+1}n+dfrac{5times3^{alpha}+1}{2}right) &equiv0pmod{3^{alpha+4}}. end{align*}
Let $mge 2$ be a fixed positive integer. Suppose that $m^j leq n< m^{j+1}$ is a positive integer for some $jge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this paper, we show t hat $b_m(n)$ can be represented as a $j$-fold summation by constructing a one-to-one correspondence between the $m$-ary partitions and a special class of integer sequences rely only on the base $m$ representation of $n$. It directly reduces to Andrews, Fraenkel and Sellers characterization of the values $b_{m}(mn)$ modulo $m$. Moreover, denote $c_{m}(n)$ the number of $m$-ary partitions of $n$ without gaps, wherein if $m^i$ is the largest part, then $m^k$ for each $0leq k<i$ also appears as a part. We also obtain an enumeration formula for $c_m(n)$ which leads to an alternative representation for the congruences of $c_m(mn)$ due to Andrews, Fraenkel, and Sellers.
A set partition is said to be $(k,d)$-noncrossing if it avoids the pattern $12... k12... d$. We find an explicit formula for the ordinary generating function of the number of $(k,d)$-noncrossing partitions of ${1,2,...,n}$ when $d=1,2$.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا