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Register a new userCongruences modulo powers of 3 for 2-color partition triples
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Dazhao Tang
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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 $p_{k,3}(n)$ with $k=1, 3$, 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*}
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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*}
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.
The sequence $A(n)_{n geq 0}$ of Apery numbers can be interpolated to $mathbb{C}$ by an entire function. We give a formula for the Taylor coefficients of this function, centered at the origin, as a $mathbb{Z}$-linear combination of multiple zeta values. We then show that for integers $n$ whose base-$p$ digits belong to a certain set, $A(n)$ satisfies a Lucas congruence modulo $p^2$.
In this paper, we consider the possible types of regular maps of order $2^n$, where the order of a regular map is the order of automorphism group of the map. For $n le 11$, M. Conder classified all regular maps of order $2^n$. It is easy to classify regular maps of order $2^n$ whose valency or covalency is $2$ or $2^{n-1}$. So we assume that $n geq 12$ and $2leq s,tleq n-2$ with $sleq t$ to consider regular maps of order $2^n$ with type ${2^s, 2^t}$. We show that for $s+tleq n$ or for $s+t>n$ with $s=t$, there exists a regular map of order $2^n$ with type ${2^s, 2^t}$, and furthermore, we classify regular maps of order $2^n$ with types ${2^{n-2},2^{n-2}}$ and ${2^{n-3},2^{n-3}}$. We conjecture that, if $s+t>n$ with $s<t$, then there is no regular map of order $2^n$ with type ${2^s, 2^t}$, and we confirm the conjecture for $t=n-2$ and $n-3$.
In 1983, C. McGibbon and J. Neisendorfer have given a proof for one conjecture in J.-P. Serres famous paper (1953). In 1985, another proof was given by J. Lannes and L. Schwartz. Since then, one considers a more general conjecture: if the reduced mod 2 cohomology of any 1-connected polyGEM is of finite type and is not trivial, then it contains at least one element of infinite height, i.e., non nilpotent. This conjecture has been verified in several special situations, more precisely, by Y. Felix, S. Halperin, J.-M. Lemaire and J.-C. Thomas in 1987, by J. Lannes and L. Schwartz in 1988, and by J. Grodal in 1996. In this note, we construct an example, for which this conjecture fails.
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