All Questions
Tagged with combinatorial-identities permutations
9 questions
2
votes
1
answer
507
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Integer eigenvalues of a class of matrices inspired by Prof. Zhi-Wei Sun's conjecture
Theorem: Let $n>1$ be an odd number and $\zeta$ a primitive $n$-th root of unity. Then
\begin{eqnarray}
&&\sum_{\tau\in D(n-1)}\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1}{1-\zeta^{j-\tau(j)...
10
votes
1
answer
565
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Identities involving derangements and roots of unity
For a positive integer $n$, a derangement of $\{1,\ldots,n\}$ is a permutation $\tau$ of $\{1,\ldots,n\}$ with $\tau(j)\not=j$ for all $j=1,\ldots,n$. For convenience, we let $D(n)$ denote the set of ...
- 15.6k
6
votes
2
answers
217
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A convolution-type identity for the "major index"
For a permutation $\pi\in\frak{S}_n$, define the number of descents of $\pi$ as $$\text{des}(\pi)=\vert\{i: \pi(i)>\pi(i+1)\}\vert.$$
The following is a well-known (and interesting) identity:
$$\...
- 43.2k
5
votes
0
answers
350
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Sum over permutations involving sign
The problem is to evaluate the following sum over all permutations $\sigma\in S_{d}$ of $\{1,2,...,d\}$:
$\displaystyle\sum_{\sigma\in S_{d}}\text{sgn}(\sigma)\displaystyle\frac{1}{\prod_{i=1}^{d}(\...
- 1,538
7
votes
1
answer
1k
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Identity involving sum over permutations
In some work on QFT the following identity has come up:
$$
\sum_{\sigma \in S_n}\sum_{j=1}^n \left(\sum_{l=1}^j w_{\sigma_l}\right)\prod_{i=1,i\neq j}^{n}\frac{1}{\sum_{l=1}^j z_{\sigma_l}-\sum_{l=1}^...
- 237
4
votes
3
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509
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How to calculate: $\sum\limits_{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}$
How to calculate:
$$\sum _{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}.$$
- 327
10
votes
2
answers
1k
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A cancellation property for permutations?
Let $S_n$ be the group of $n$-permutations. Denote the number of inversions of $\sigma\in S_n$ by $\ell(\sigma)$.
QUESTION. Assume $n>2$. Does this cancellation property hold true?
$$\sum_{\...
- 43.2k
28
votes
3
answers
3k
views
Sum over permutations is 1
This might be easy, but let's see.
Question 1. If $\mathfrak{S}_n$ is the group of permutations on $[n]$, then is the following true?
$$\sum_{\pi\in\mathfrak{S}_n}\prod_{j=1}^n\frac{j}{\pi(1)+\pi(...
- 43.2k
5
votes
1
answer
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Number of Permutations with k-inversions and with a single clamped value
This question is cross-posted from math.stackexchange because it might be too technical.
Let $S_n$ be the symmetric group. Recall that the number of inversions of a permutation $\sigma\in S_n$ is the ...
- 4,952