# range of r parameter of second kind r-Stirling numbers

My question is about second kind r-Stirling numbers. Here are two important papers about it. https://www.sciencedirect.com/science/article/pii/0012365X84901614#:~:text=The%20r%2DStirling%20numbers%E2%98%86&text=The%20r%2DStirling%20numbers%20of,cycles%20and%20respectively%20distinct%20subsets. Broder/ r-Stirling numbers. https://www.sciencedirect.com/science/article/pii/S0012365X14001241#:~:text=3.&text=%2DLah%20numbers%20The%20%2DLah%20numbers,be%20in%20distinct%20ordered%20blocks. Ryul and Nacz / r-Lah numbers.

In Broder / r-Stirling numbers article, r-Stirling numbers of second kind are defined; $$\left\{\begin{array}{l}n \\ m\end{array}\right\}_{r}=$$ The number of partitions of the set $$\{1, \ldots, n\}$$ into $$m$$ non-empty disjoint subsets ,such that the numbers $$1,2,3...,r$$ are in distinct subsets.

I try to understand the definition of second kind r-Stirling numbers. For r-Stirling numbers r is a natural number. Can the range of r be extended to rational numbers or complex numbers? Thanks.

• When you ask a question, you should make sure it is useful for the future readers, too. You start a sentence by from what I understand ... You should write your interpretation and deduction of why $r$ is a natural number, give some context of your knowledge so that people know how to answer your question. You gave a link to two important papers , which sounds a bit like giving an assignment. Also, if you're looking for a reference, make that clear in the body, not just by a tag. Commented Feb 1, 2022 at 22:10
• thanks. i will edit it:) Do you have any opinions about my question? @invisible Commented Feb 1, 2022 at 22:12
• I'm going to read it. (: I hope I didn't sound rude and that your question gets due attention. (: Commented Feb 1, 2022 at 22:14
• I'm not sure about rational or complex numbers, but I think it can extended to negative integers. See this paper for a different version of a restriction that preserves the reciprocity between the restricted Stirling numbers of the first and second kinds. Hope that helps. Commented Feb 2, 2022 at 0:18

Not sure what are your intentions, but you can use the known expression $${n\brace k}_r=\sum _{i=0}^n\binom{n}{i}{i\brace k}r^{n-i}$$ to extend it. That comes from the fact that you can choose the numbers that are going to be in the first $$r$$ subsets(say $$n-i$$ of them) in $$\binom{n}{n-i}$$ ways, and the rest elements have to be in $$k$$ blocks in $${i\brace k}$$ ways. To distribute the $$n-i$$ elements in the $$r$$ first blocks, you can use any function in $$r^{n-i}$$ ways.
For example, for $$r<0$$ and $$n,k$$ even, one can express $${n\brace k}_r$$ using Stirling numbers of higher level (see remark 3.1 here).
• Thanks for your answer. I have two questions. The first one what about $$\left\{\begin{array}{c}n+r \\ k+r\end{array}\right\}_{r}$$ when r is negative or other conditions ? My second question the expression you wrote above are Noncentral stirling numbers S(n,k;r) to me, aren't it? So i was a little surprised. You may also look for it to Charalambides Enumerative Combinatorics, page 316. Commented Feb 4, 2022 at 20:18
• @user1062 1) That is what I mean in your notation, sorry there are two standard ways to see $r-$stirling numbers. So, what I mean by $\{n\brace k\}_r$ in your notation is actually $\{n\brace k+r\}_r$ but in your notation you want $n\geq r$ so take $n'=n-r$ and it is the same expression. For the second one, those are two different generalizations. The non-central stirling numbers are defined, in your notation, as connecting coefficients of $(x)_n$ and $(x-r)^n$ and the Stirling numbers of higher level are connecting coefficients of $x(x-1^r)(x-2^r)\cdots(x-(n-1)^r)$ and $x^n$. Commented Feb 4, 2022 at 20:38