Partitions Of 9

[Angelique Syria]

Innumber theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways:

Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general.

There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named after Norman Macleod Ferrers, and as Young diagrams, named after Alfred Young. Both have several possible conventions; here, we use English notation, with diagrams aligned in the upper-left corner.

An alternative visual representation of an integer partition is its Young diagram (often also called a Ferrers diagram). Rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. Thus, the Young diagram for the partition 5 + 4 + 1 is

While this seemingly trivial variation does not appear worthy of separate mention, Young diagrams turn out to be extremely useful in the study of symmetric functions and group representation theory: filling the boxes of Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a family of objects called Young tableaux, and these tableaux have combinatorial and representation-theoretic significance.[1] As a type of shape made by adjacent squares joined together, Young diagrams are a special kind of polyomino.[2]

No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument.,[3] as follows:

Srinivasa Ramanujan discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of n \displaystyle n ends in the digit 4 or 9, the number of partitions of n \displaystyle n will be divisible by 5.[4]

Alternatively, we could count partitions in which no number occurs more than once. Such a partition is called a partition with distinct parts. If we count the partitions of 8 with distinct parts, we also obtain 6:

This is a general property. For each positive number, the number of partitions with odd parts equals the number of partitions with distinct parts, denoted by q(n).[8][9] This result was proved by Leonhard Euler in 1748[10] and later was generalized as Glaisher's theorem.

There is a natural partial order on partitions given by inclusion of Young diagrams. This partially ordered set is known as Young's lattice. The lattice was originally defined in the context of representation theory, where it is used to describe the irreducible representations of symmetric groups Sn for all n, together with their branching properties, in characteristic zero. It also has received significant study for its purely combinatorial properties; notably, it is the motivating example of a differential poset.

Often a maximum of 4,000 partitions per broker is recommended (Kafka definitive guide, Apache Kafka + Confluent Blog). My understanding is that more partitions per Broker require more RAM and probably also more CPU due to the additional operations that are required per partition.

However I can easily scale CPU and RAM vertically in the Cloud and I wonder whether this recommendations still apply to newer Kafka versions (v2.6.0+) and if so what is the problem with more than 4k partitions? How can I tell whether my broker suffers from problems due to the number of partitions.

@weeco is referencing this blog post. But this blog post makes no mention of the underlying H/W on which 4K partitions runs seamlessly. One should also keep in mind the costs associated with unplanned downtime when loading up a broker with 4K partitions. A hard shutdown on a broker servicing a large number of partitions takes really long to recover.

@atulyab9 Thanks for the hint about longer shutdown (and probably also startup) times. That makes sense, but honestly at 4k partitions this is still fine for us and I think the major impact would be number of segments and the size of the data.

@weeco, the short answer is cost. If you are scaling to service lots of partitions, many of which you do not need, then you are simply throwing money away.

How many partitions you need should have been determined during performance testing as you attempt to hit the required throughput whilst maintaining resiliance etc.

The longer asnwer is, I suspect, it all depends. Hardware, load, size of messages, geography and so on will all play a part in what your upper limit is. And, unfortunately, that will change over time as those factors change.

Therefore, I'd like to use partitioning to split query into multiple little queries. I use a discrete partition key that has a cardinality of about 200. When I run the job for just one key everything is ok.

2; Create a scenario (that starts every hour for example) with a first step "python custom". Put inside this code below is listing all the partitions of a "main dataset" (here Clusters_summary) and then concat them in one string comma separated and assign variable

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To avoid accidentally destroying data HD Tune refuses to write to a drive unless there are no partitions on the drive. If you do attempt to write to a drive with partitions, it posts the message "Writing is disabled. To enable writing please remove all partitions."

The first thing I tried was to use the Windows 7 (64-bit) Disk Management tool (diskmgmt.msc) to delete the partition. It would not let me. The context menu choice to delete that volume was not available.

Next I opened up a command prompt window with Admin authority and ran diskpart. Diskpart deleted the volume for me. However, when I attempted to run an HD Tune write test on the drive I still got the "Writing is disabled" message. Huh???

So I fired up a utility I happen to own which allows viewing drives at the sector level and verified that the partition table in the Master Boot Record was empty. No partitions. Yet HD Tune still thought there were partitions on the drive?

After doing the above, I plugged the ADATA into my MacBook. I was then able to format it as either a GPT or MBR partitioned drive with no problems. I am not looking for suggestions on how to format this drive. I can do that.

BTW, if I plug the drive I formatted on my MacBook back into my Windows 7 64-bit system I still run into road blocks with the Disk Management tool. For example, I cannot delete all the GPT partitions on the ADATA so I can convert it into an MBR drive. I followed Microsoft's instructions, the instructions just do not work with this ADATA flash drive.

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Just format the card or USB flash drive to NTFS, and then you can delete that NTFS partition using EASEUS Partition Manager without restarting, and voil, I was able to test write speed with HDD Tune Pro.

Hello everyone,

I'm working with datasets with partitions, I found a post showing how to read a partition from a dataset but I have not found a way to write that partition down on another dataframe in the same partition name.

Hi Roy E,

Thank you very much for your explanation. I was working on something different the last days but your explanation will help me a lot in future projects as this is something I had come across a few times and now I know how to fix it.

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