Partitions

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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.

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]

By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14. Such partitions are said to be conjugate of one another.[6] In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. Of particular interest is the partition 2 + 2, which has itself as conjugate. Such a partition is said to be self-conjugate.[7]

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.

For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction. An important example is q(n) (partitions into distinct parts). The first few values of q(n) are (starting with q(0)=1):

By taking conjugates, the number pk(n) of partitions of n into exactly k parts is equal to the number of partitions of n in which the largest part has size k. The function pk(n) satisfies the recurrence

This can be used to solve change-making problems (where the set T specifies the available coins). As two particular cases, one has that the number of partitions of n in which all parts are 1 or 2 (or, equivalently, the number of partitions of n into 1 or 2 parts) is

and the number of partitions of n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of n into at most three parts) is the nearest integer to (n + 3)2 / 12.[14]

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.

Additive partitions of integers. Enumerates the partitions, unequal partitions, and restricted partitions of an integer; the three corresponding partition functions are also given. Set partitions and now compositions and riffle shuffles are included.

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When you create a table, the initial status of the table is CREATING. During this phase, DynamoDB allocates sufficient partitions to the table so that it can handle your provisioned throughput requirements. You can begin writing and reading table data after the table status changes to ACTIVE.

Global secondary indexes in DynamoDB are also composed of partitions. The data in a global secondary index is stored separately from the data in its base table, but index partitions behave in much the same way as table partitions.

The following diagram shows a table named Pets, which spans multiple partitions. The table's primary key is AnimalType (only this key attribute is shown). DynamoDB uses its hash function to determine where to store a new item, in this case based on the hash value of the string Dog. Note that the items are not stored in sorted order. Each item's location is determined by the hash value of its partition key.

DynamoDB is optimized for uniform distribution of items across a table's partitions, no matter how many partitions there may be. We recommend that you choose a partition key that can have a large number of distinct values relative to the number of items in the table.

Event Hubs throughput is scaled by using partitions and throughput-unit allocations (see below). It's a best practice for publishers to remain unaware of the specific partitioning model chosen for an event hub and to only specify a partition key that is used to consistently assign related events to the same partition.

Event Hubs retains events for a configured retention time that applies acrossall partitions. Events are automatically removed when the retention period hasbeen reached. If you specify a retention period of one day (24 hours), the event willbecome unavailable exactly 24 hours after it has been accepted. You can'texplicitly delete events.

The number of partitions is specified at the time of creating an event hub. It must be between 1 and the maximum partition count allowed for each pricing tier. For the partition count limit for each tier, see this article.

We recommend that you choose at least as many partitions as you expect that are required during the peak load of your application for that particular event hub. You can't change the partition count for an event hub after its creation except for the event hub in a dedicated cluster and premium tier. The partition count for an event hub in a dedicated Event Hubs cluster can be increased after the event hub has been created, but the distribution of streams across partitions will change when it's done as the mapping of partition keys to partitions changes, so you should try hard to avoid such changes if the relative order of events matters in your application.

Setting the number of partitions to the maximum permitted value is tempting, but always keep in mind that your event streams need to be structured such that you can indeed take advantage of multiple partitions. If you need absolute order preservation across all events or only a handful of substreams, you may not be able to take advantage of many partitions. Also, many partitions make the processing side more complex.

It doesn't matter how many partitions are in an event hub when it comes to pricing. It depends on the number of pricing units (throughput units(TUs) for the standard tier, processing units (PUs) for the premium tier, and capacity units (CUs) for the dedicated tier) for the namespace or the dedicated cluster. For example, an event hub of the standard tier with 32 partitions or with 1 partition incur the exact same cost when the namespace is set to 1 TU capacity. Also, you can scale TUs or PUs on your namespace or CUs of your dedicated cluster independent of the partition count.

As partition is a data organization mechanism that allows you to publish and consume data in a parallel manner, we recommend that you balance scaling units (TUs, PUs or CUs) and partitions to achieve optimal scale. In general, we recommend a maximum throughput of 1 MB/s per partition. Therefore, a rule of thumb for calculating the number of partitions would be to divide the maximum expected throughput by 1 MB/s. For example, if your use case requires 20 MB/s, it is recommended to choose at least 20 partitions to achieve the optimal throughput.

However, if you have a model in which your application has an affinity to a particular partition, increasing the number of partitions may not be beneficial. For more information, see availability and consistency.

You can use a partition key to map incoming event data into specific partitions for the purpose of data organization. The partition key is a sender-supplied value passed into an event hub. It is processed through a static hashing function, which creates the partition assignment. If you don't specify a partition key when publishing an event, a round-robin assignment is used.

While you can send events directly to partitions, we don't recommend it, especially when high availability is important to you. It downgrades the availability of an event hub to partition-level. For more information, see Availability and Consistency.

In a stream processing architecture, each downstream application equates to a consumer group. If you want to write event data to long-term storage, then that storage writer application is a consumer group. Complex event processing can then be performed by another, separate consumer group. You can only access partitions through a consumer group. There's always a default consumer group in an event hub, and you can create up to the maximum number of consumer groups for the corresponding pricing tier.

Some clients offered by the Azure SDKs are intelligent consumer agents that automatically manage the details of ensuring that each partition has a single reader and that all partitions for an event hub are being read from. This allows your code to focus on processing the events being read from the event hub so it can ignore many of the details of the partitions. For more information, see Connect to a partition. 2b1af7f3a8