# Tagged Questions

125 views

### Coefficient of $x^n$

What is the coefficient of $x^n$ of this following expression: $$(x+x^2+x^3+ \dots + x^r)^n$$ I tried it but failed. I know that it can be solved by combinations but ...
287 views

### Exponential generating function for permutations with descent set whose least element is even

Let $E(n)$ be the number of permutations $w\in S_n$ such that the least element of the set $Des(w)\cup \{n\}$ is even, where $Des(w)$ is the descent set of $w$. I need to find the exponential ...
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### Combinatorics: counting sums with conditions

Hi guys, Here's a combinatorial nut to crack. I've been struggling with this one: Count the number of ways summing a set of $n$ non negative integers $i_1, \cdots, i_n \in \{ 0, \cdots , n-1\}$ so ...
327 views

### Number of Permutations with $k$-inversions and with a single clamped value

Let $S_n$ be the symmetric group. Recall that the number of inversions of a permutation $\sigma\in S_n$ is the number of ordered pairs $i<j$ with $\sigma(i)>\sigma(j)$. Now, call the number of ...
2k views

### Exponential Generating Functions For Derangements

I have been introduced to the concept of exponential generating functions a few days ago. However, my understanding of them are still quite limited, and I would like to see some examples. Earlier this ...
294 views

### Generating function for permutations in $S_n$ with $k$ cycles.

I was reading a little bit about Galois theory, and read that some computer algebra software try to compute Galois groups by finding cycle types. Anyway, this led me to a curious question. If I fix ...
266 views

### The generating function for permutations indexed by number of inversions

For $\sigma\in S_n$ an inversion is a pair $(\sigma_i,\sigma_j)$ such that $i<j$ and $\sigma_i>\sigma_j$. Could you help me to prove that the generating function of $S_n$ by number of ...
I calculated the number of permutations in $S_n$ with no 2-cycles in two ways but I got 2 different results. The first time I used the principle of inclusion-exclusion and I got \$\sum_{k=0}^n ...