Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Stirling numbers of the second kind S(n, k) count the number of ways to partition a set of n elements into k nonempty subsets.What if there were duplicate elements in the set?That is,the set is a multiset?

share|cite|improve this question

migrated from Jan 9 '11 at 15:40

This question came from our site for professional and enthusiast programmers.

It would be good to add the link to the definition of Stirling numbers: – David Jan 9 '11 at 3:37
The answer is likely to be somewhat complicated to express. For example, if the set consists of a single type of element, you get the partition numbers p_{n,k} : – Qiaochu Yuan Jan 9 '11 at 15:54
I think [math] is a pretty useless tag in this site; I've deleted it and added [counting] – Arturo Magidin Jan 10 '11 at 21:54

There is no known formulation for a general multiset. However, a paper at JIS tackles the case where the element 1 occurs multiple times.

share|cite|improve this answer

Here are two links to get you started:

  1. Eulerian numbers of the second kind may be helpful (for counting ascents, descents, etc., though i think)

  2. Additionally some more useful information may be found in Stanley's book

share|cite|improve this answer

You divide the Stirling number by the possible permutations of identical elements as this is just changing order within the stars and bars example.

share|cite|improve this answer

By definition, a set only has distinct elements. If elements are duplicated, then yes, the set in question would be a multiset.

share|cite|improve this answer
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. – homegrown Dec 31 '14 at 18:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.