# Tagged Questions

There are two kinds of Stirling numbers. Stirling numbers of the first kind $[{n \atop k}]$ count the number of ways to arrange $n$ objects into $k$ cycles. Stirling numbers of the second kind $\{ {n \atop k} \}$ count the number of ways to partition a set of $n$ objects into $k$ subsets.

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### Show $\sum_{k=0}^n b_r(n,k) = (r-1)!\frac{x^{\bar{n}}}{(x+1)^{\bar{r-1}}}$ [duplicate]

Let's define $b_r(n,k)$ as $n$-permutations with $k$ cycles where numbers $1\dots r$ belong to one cycle. I tried to first define closed form for $b_r(n,k)$. My idea: We need to put $1 \dots r$ into ...
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### stirling numbers of second kind

i am new to combinatorics and just encountered stirling numbers of second kind the book i am using does not provide much info about it except number of ways of distributing "r" distinct objects ...
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### Number of ways to arrange $n$ numbers based on their relative values to each other

EDIT I've found a formula to solve this question, but I don't understand the reasoning behind it. Can someone explain this formula? $s(n - 1, x + y - 2) \times C(x + y - 2, x - 1)$ $s$ being ...
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### Stirling Numbers of the First Kind and Permutations [closed]

How to prove that the number of $[n]$ permutations with $k$ cycles is equal with $|s(n,k)|$?
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### Relationship between Riemann Zeta function and Prime zeta function

In his paper, Daniel Grunberg shows a relationship between the Stirling Numbers of the first kind and the Harmonic numbers via series of partitions (see Equation 3.1 on Page 5 in the link above). If ...
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### Interpretation of the unsigned Stirling number of the first kind.

Let $C_{2}, C_{3},\dots, C_{n}$ be the directed star graphs: the vertex set of $C_{j}$ is $\{1, 2, \dots, j\}$ and its edge set is $\{(j, i): 1\leq i <j\}$ . Let $c'(n,i)$ be the number of sets $X$ ...
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### Why $S(n,k)$ is the coefficient of $x^{n-k}$ in $\prod_{t=0}^{k}(1+tx+t^2x^2+\cdots+t^nx^n)$?

$S(n,k)$ is the Stirling number of the second kind. I think an algebraic proof has something to do with the generating function. But I'm more interested in combinatorial proof. Could you please give ...
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### If $S(n, n - 3) = a \binom n 4 + b \binom n 5 + c \binom n 6$, find $a, b, c$ (where $S(n, k)$ denotes a Stirling number of the second kind)

Given the identity $S(n, n - 3) = a \binom n 4 + b \binom n 5 + c \binom n 6$, find $a, b, c$. $S(n, k)$ denotes a Stirling number of the second kind, i.e., the number of ways to place $n$ labeled ...
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### Showing that $x^n=\sum_{k=1}^{n}{n\brace k}(x)(x-1)\ldots (x-k+1)$ holds for all numbers, not just positive integers

I just finished proving that this statement holds for all positive integers $r$ (through a combinatorial argument) $$r^n=\sum_{k=1}^{n}{n\brace k}(r)(r-1)\ldots (r-k+1)$$ (where the curly braces ...
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### Eulerian Numbers Generalization

Does anyone have a combinatorial proof for the following identity: $\sum_{i=j}^n S(n,n-j)(n-j)!(-1)^{n-j-1}=A(n,j)$. I have tried using ordered set partitions with inclusion/exclusion, but I have yet ...
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### Bijection for Rook placecement and Stirling number of 2nd kind

Say we have an nxn chessboard from which the squares below the diagonal are removed to obtain a new board $C_n$. The board $C_3$ is shown below. Let the number of ways to place k non-attacking ...
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### partitioning of a set with kn members into k subsets such that each subset has n members

we know that $S(n,k)$ is the number of ways we can partition a set with $n$ members to $k$ subsets ( each subset has at least one member). imagine we have a set with $k*n$ members. we want to ...
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### Understanding Stirling no of first kind

I was reading about Stirling no of the first kind, so $\left[ \frac{n}{k} \right]$ represent no of k cycles of n items, so in $\left[ \frac{4}{2} \right]$ there would be 11 such combinations, of ...
I would like to prove that: $$f_m(x) = \dfrac{x^m}{(1-x)(1-2x)...(1-mx)}$$ Where $$f_m(x) = \sum_{n=0}^{\infty} S(n,m)x^n$$ and $S(n,m)$ is stirling number of 2nd kind Multiplying the recurrence ...