Questions tagged [generating-functions]

Generating functions are formed by making a series $\sum_{n\geq 0} a_n x^n$ out of a sequence $a_n$. They are used to count objects in enumerative combinatorics.

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Recurrence relation of partioning $n$ into exactly 3 parts:

I wanted to find a recurrence relation for partitioning an integer $n$ into exactly $3$ parts To be clear, I know the formula $P(n,k)=P(n-1,k-1)+P(n-k,k)$, but I want to derive a relation involving ...
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Result on partitions with distinct odd parts

Let $pdo(n)$ be the number of partitions of n into distinct odd parts. Then $p(n)$ is odd if and only if $pdo(n)$ is odd. I am well aware that a proof of this is available here but I want to do it ...
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Find the generating function to determine the number of ways to choose k objects from n objects when the ith object appears at least n + i times

Find the generating function to determine the number of ways to choose k objects from n objects when the ith object appears at least n + i times for 1 ≤ i ≤ n. the generating function for picking k ...