arXiv 1304.65632013-04-24math.CO

相交整数分区

Intersecting integer partitions

Peter Borg

备注：: 8 pages, submitted

摘要：

如果$a_1, a_2, ..., a_k$和$n$是正整数，使得$n = a_1 + a_2 + ... + a_k$，则总和$a_1 + a_2 + ... + a_k$被称为是\emph{length $k$}的\emph{partition of $n$}，并且$a_1, a_2, ..., a_k$被称为是分区的\emph{parts}。两个分区如果只是在其部分的顺序上不同，则被认为是相同的。我们说两个分区\emph{intersect}，如果它们至少有一个公共部分。如果$A$中的任何两个分区相交，则我们将分区的集合称为\emph{intersecting} $A$。令$P_{n,k}$是长度为$k$的$n$的所有分区的集合。我们推测，如果$2 \leq k \leq n$，那么$P_{n,k}$的任何相交子集的大小最多是$P_{n-1,k-1}$的大小，这是$P_{n,k}$的相交子集的大小，该子集由具有1作为部分的那些分区组成。这个猜想对于$n \leq 2k$是平凡的，我们证明了它对于$n \geq 5k^5$。我们还概括了这一点的子集$P_{n,k}$的财产，任何两个成员至少有$t$共同的部分。

英文摘要：

If $a_1, a_2, ..., a_k$ and $n$ are positive integers such that $n = a_1 + a_2 + ... + a_k$, then the sum $a_1 + a_2 + ... + a_k$ is said to be a \emph{partition of $n$} of \emph{length $k$}, and $a_1, a_2, ..., a_k$ are said to be the \emph{parts} of the partition. Two partitions that differ only in the order of their parts are considered to be the same. We say that two partitions \emph{intersect} if they have at least one common part. We call a set $A$ of partitions \emph{intersecting} if any two partitions in $A$ intersect. Let $P_{n,k}$ be the set of all partitions of $n$ of length $k$. We conjecture that if $2 \leq k \leq n$, then the size of any intersecting subset of $P_{n,k}$ is at most the size of $P_{n-1,k-1}$, which is the size of the intersecting subset of $P_{n,k}$ consisting of those partitions which have 1 as a part. The conjecture is trivially true for $n \leq 2k$, and we prove it for $n \geq 5k^5$. We also generalise this for subsets of $P_{n,k}$ with the property that any two of their members have at least $t$ common parts.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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