0
votes
0answers
42 views

Number of set partitions of n elements into k sets with subsets of size r not allowed

This is a generalization of the question Number of ways to partition a set with $n$ elements to $k$ subsets where at least one subset has $r$ elements . At the end of answer for this question, there ...
0
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0answers
47 views

Number of Singleton Blocks in Set Partition

I'm interested in some general information on the following question: Consider the collection of partitions of an $n$-set into $m$ blocks as a uniform probability space. Let $X$ be the random ...
1
vote
1answer
85 views

Determining Stirling number

In the first part of the question I was asked to find the exponential generating function for $s_{n,r}$, the number of ways to distribute $r$ distinct objects into $n$ (a fixed constant) distinct ...
5
votes
2answers
290 views

Why does $S(n,k)=\sum 1^{a_1-1}2^{a_2-1}\cdots k^{a_k-1}$?

I've been trying to get back into some combinatorics, and in my reading, I find that $$ S(n,k)=\sum 1^{a_1-1}2^{a_2-1}\cdots k^{a_k-1} $$ where the sum is taken over all compositions ...