New answers tagged permutations
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Prime numbers made of permutations of digits of consecutive positive integers
We knows that no prime of the form $1\_2\_...\_n$ has been found yet... at least for $n$ up to $10^6$ (see https://mathworld.wolfram.com/SmarandachePrime.html).
Now, a partial answer to your question ...
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Prime numbers made of permutations of digits of consecutive positive integers
My impression is that the prime numbers in $S(3k+1)$ are exactly in the proportion they should be. Note that numbers in $S(3k+1)$ are smaller than $10^{D(3k+1)-1}$ because they have $D(3k+1)$ digits.
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Vandermonde $V_n$ mod $n$
OP asked me to fill in the details of my comment, and in attempting to do so I realised that I claimed too much. However, a very similar argument proves a weaker result which is strong enough to ...
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Vandermonde $V_n$ mod $n$
Per comments above, for a counterexample we have with necessity $\pi_n=n$ and prime $n$. The case $n=2$ is trivial, so I assume that $n$ is an odd prime.
The elements $U:=\{ 1,2,\dots,n-1\}$ form the ...
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5
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Accepted
Given $\pi$ permutation on $\{1,\dotsc,n\}$, what is the sign of a permutation of $\{2,\dotsc,\hat\jmath,\dotsc,n\}$?
$\DeclareMathOperator\sgn{sgn}$This is almost the same as your previous question, just with the order of the operations switched—whether you think of $\pi$ as ordering or disordering is just a matter ...
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4
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Accepted
Sign of the permutation which brings a subsequence back to its original form
Imagine a bubble sort where you bring each element to its original position. $x_1$ would have taken $\pi^{-1}(1) - 1$ transpositions to bring it back to its original position. Let's perform those ...
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Integer eigenvalues of a class of matrices inspired by Prof. Zhi-Wei Sun's conjecture
Fourier transform does it.
Denote by $u_j$ $(j=0,1,\ldots,n-1$) the column-vector with coordinates $(x_{ji})_{1\leqslant i\leqslant n-1}$. Note that $u_0+u_1+\ldots+u_{n-1}=0$ and any $n-1$ vectors $...
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Permuting subgroups with the same finite index
The main result of
Lubotzky, Alexander; Mann, Avinoam; Segal, Dan, Finitely generated groups of polynomial subgroup growth, Isr. J. Math. 82, No. 1-3, 363-371 (1993). ZBL0811.20027.
states that a ...
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A determinant involving the cotangent function
The conjecture has been proved by my graduate student Han Wang and me via the Eigenvector-eigenvalue Identity, for our joint paper see http://arxiv.org/abs/2206.02589 .
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