# Questions tagged [permutations]

Questions related to permutations, bijections from a finite (or sometimes infinite) set to itself.

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### Can Matsumoto's theorem for the symmetric group be proved using a monovariant?

This is a question that can be asked for any Coxeter group, but for the sake of simplicity I will restrict myself to symmetric groups. Recall the main definitions: Let $n$ be a nonnegative integer. ...
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### Distribution of peaks in permutations, after a sorting operation

Let $S_n$ be the set of permutations on $\{1,2,\dotsc,n\}$. A peak-value of $\pi$ is some $\pi_i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$, where $1<i<n$. Let $PV(\pi)$ denote the set of ...
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### How to construct such a series of “partial” symmetric polynomials?

As we know, a symmetric polynomial is a polynomial P(X1, X2, …, Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric ...
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### A conjecture on circular permutations of n elements in an abelian group of odd order

In 2013 I formulated the following conjecture in additive combinatorics. Conjecture. Let $G$ be an additive abelian group of odd order, and let $A$ be a subset of $G$ with $|A|=n>2$. Then, there is ...
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### Number of permutations with longest increasing subsequences of length at most $n$

Is there a known expression for, or a nontrivial upper bound on, the number of permutations in $S_k$ with longest increasing subsequence of length at most $n$? Let $l(\sigma)$ denote the length of the ...
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Consider a partition $f:S_n\rightarrow [n]$ of $S_n$ into $n$ parts. Denote the permutations that map $j$ to $k$ by $s(j,k)$. Set $S(f):=\Sigma_{1\leq i,j\leq n}max_{1\leq k\leq n}|f^{-1}(i)\cap s(j,k)... 0answers 85 views ### Is there a characterization for the finite sequences of natural numbers which are the shifts of a permutation? Given a natural number$k<n$and a permutation$\sigma\in S_n$, I will call$k$a shift of$\sigma$if there is some$m$with$\sigma(m)-m\equiv k \mod n$. If one is given a sequence$s=(x_1,...,...
Given a reductive group $G$ over a finite field and a parabolic subgroup $P$ , I wonder what are the orbits in $G$ under the adjoint action of $P$. This should be standard, but I can only find results ...
I would like to understand commutation classes of reduced expressions of the longest element in $S_5$ a little better. For this, it makes sense to look at the permutohedron of order 5. Since I am only ...