2
votes
2answers
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 ...
2
votes
2answers
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 ...
1
vote
2answers
195 views

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 ...
4
votes
1answer
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 ...
6
votes
4answers
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 ...
7
votes
2answers
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 ...
4
votes
1answer
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 ...
3
votes
2answers
511 views

I calculated the number of permutations with no 2-cycles in two ways but I got 2 different results

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 ...
4
votes
3answers
1k views

Combination problem with constraints

You have four containers and one pitcher of water that holds 100L. Each container has different capacities with maximums of, say...70L, 45L, 33L and 11L levels respectively. What is the formula that ...