Skip to main content
MathOverflow

Questions tagged [permutations]

Ask Question

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

591 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
2 votes
2 answers
360 views

Recently, I have been wondering about how to stack a deck in my favor using minimal moves for Poker. Concretely, I want to know if any deck can be stacked in my favor in 2 or 3 card moves. I have been ...
Tomodovodoo's user avatar
  • 31
13 votes
1 answer
532 views

For the set $\omega$ of non-negative integers, we let $\newcommand{\oo}{[\omega]^\omega}\oo$ be the collection of infinite subsets of $\omega$. If $U\in \oo$, there is a unique order-preserving ...
Dominic van der Zypen's user avatar
  • 54.4k
0 votes
0 answers
28 views

Question: what, besides publishing, should I do with a new interpretation of how to formulate hard problems for optimal permutations with constraints on the cycle structure? Currently I have ...
Manfred Weis's user avatar
  • 14k
6 votes
3 answers
493 views

The Fisher-Yates shuffle is the standard implementation for randomly permuting a finite list of $n$ elements. The algorithm has several incorrect implementations, one being that in each step permuting ...
Markus Klyver's user avatar
  • 241
13 votes
0 answers
320 views

Let $n \in \mathbb N$ and let $\sigma,\tau \in {\rm Sym}(n)$. I am looking for a permutation $x \in {\rm Sym}(n)$ that minimizes the Hamming distance between $x^2 \sigma$ and $\tau x$. Here, the ...
Andreas Thom's user avatar
  • 26.2k
1 vote
0 answers
101 views

Consider the symmetric group $S_{2n}$ and $[2n]:=\lbrace 1,..,2n\rbrace$. All notations regarding the symmetric group come from its action on the set $[2n]$. We define discrete torus braids of size $k$...
Jens Fischer's user avatar
  • 377
15 votes
0 answers
308 views

Let $w$ be a permutation of $\{1,2,\dots,n\}$ chosen uniformly at random. You have to determine $w$ by successively guessing permutations $v_1, v_2, \dots$. After each guess $v_j$ you are told where $...
Richard Stanley's user avatar
  • 53.7k
0 votes
0 answers
102 views

Let $T(n,k)$ be A375837, i.e., triangle read by rows: $T(n,k)$ is the number of rooted chains starting with the cycle $(1)(2)(3)\dotsc(n)$ of length $k$ of permutation poset of $n$ letters. $a(n)$ be ...
user avatar
4 votes
1 answer
199 views

Consider the Young diagram of an integer partition $\lambda \vdash n$. I can fill the boxes of the Young diagram with the integers $1,2,\ldots,n$ in row-major order (i.e., in increasing order row-by-...
Christopher Drupieski's user avatar
  • 2,202
10 votes
1 answer
365 views

Consider a uniform random permutation of $\{1,\dots, n\}$, and let $D_n$ be its number of descents (indices $i$ such that $\sigma(i)>\sigma(i+1)$). There is a nice result by Tanny where they show ...
dori's user avatar
  • 103
4 votes
0 answers
166 views

Let $s = (s_1, s_2, \ldots, s_N) \in \mathbb{R}^N$ be a fixed vector with distinct elements. We define the label $l(s) = (l_1, \ldots, l_N) \in \{1, \ldots, N\}^N$ as the argsort of $s$, i.e., the ...
ABB's user avatar
  • 4,150
0 votes
1 answer
268 views

We call a finite subset $S\subseteq \mathbb{N}$ arithmetical if there are $n, k\in\mathbb{N}$ with $k>1$ such that $S = \{n+j: 0 \leq j\leq k\}$. Given an integer $\ell>0$ and a bijection $\...
Dominic van der Zypen's user avatar
  • 54.4k
2 votes
0 answers
214 views

Let $M$ be an $n \times m$ matrix, with $1, \dots, m$ in the first row, $m+1, \dots, 2m$ in the second row, etc. $$M = \left[ \begin{array}{c} 1 & 2 & \dots & m \\ m+1 & m+2 & \...
So Ya's user avatar
  • 51
2 votes
0 answers
109 views

Consider the Bruhat decomposition of a simple linear algebraic group $G$: $$G = \bigsqcup_{w\in W} B w B.$$ There are rules for multiplying two elements $g_1 \in B w_1 B$, $g_2\in B w_2 B$, in the ...
H A Helfgott's user avatar
  • 22k
8 votes
1 answer
399 views

I recently became aware of the paper, The Geometry and Combinatorics of Some Hessenberg Varieties Related to the Permutohedral Variety, by Jan-Li Lin. In it, the author defines prepermutohedral ...
Timothy Chow's user avatar
  • 88.5k
4 votes
0 answers
217 views

Let $A_n$ be the number of permutations $\pi$ of $[n]=\{1,2,\ldots,n\}$ such that $\pi(i)\neq i$ for all $i \in [n]$ and $|\pi(i+1) - \pi(i)| > 1$ for all $i \in [n-1]$. So $A_n$ counts the ...
user967210's user avatar
  • 557
14 votes
1 answer
1k views

Cross-posted from MSE where the question didn't get much attention. The question is related to and has a similar motivation as this MO question. Let$\newcommand{\from}{\colon}\newcommand{\sgn}{\...
Jakob Werner's user avatar
  • 1,648
16 votes
0 answers
374 views

Let $w=a_1 a_2\cdots a_n\in S_n$, the symmetric group of all permutations of $1,2,\dots,n$. The descent set $D(w)$ is defined by $D(w)=\{1\leq i\leq n-1\,:\, a_i>a_{i+1}\}$. Let $f(n)$ be the ...
Richard Stanley's user avatar
  • 53.7k
3 votes
1 answer
255 views

Theorem 1 of On abelian quotients of primitive groups states that $$ |G/G'| \leq n $$ for any primitive permutation group $ G $ of degree $ n $. In other words, for primitive permutation groups the ...
Ian Gershon Teixeira's user avatar
  • 4,577
1 vote
1 answer
86 views

Let $\Pi=\{\pi_1,\pi_2,\dots,\pi_n\}$ be the rows of an $n\times n$ Latin square (the order of the rows does not matter). Each row $\pi_i$ induces an order $\prec_i$ on the elements of $[1,n]$, where $...
Arnaud Casteigts's user avatar
  • 235
2 votes
0 answers
144 views

Given a matrix $n \times m$, I want to find the submatrices $a \times m$ by selecting $a$ columns such that their rank is minimal. Can this problem be solved efficiently?
Alm's user avatar
  • 1,217
1 vote
1 answer
137 views

Fix $n\geq 1$ and let $p_k(x) := x^k(x-1)^{n-k}$. Suppose $\pi$ is a permutation on $\{0,1,\dotsc,n\}$, such that $$ \sum_{k=0}^n (-1)^k \binom{n}{k} p_{\pi(k)}(x) \text{ is a constant}. $$ Must it be ...
Per Alexandersson's user avatar
  • 16.1k
3 votes
0 answers
116 views

This question was asked several months ago on Math.SE, but remains unsolved. For any collection of permutations of $\{1,2,\dots,n\}$, we say that it realizes a directed multigraph with $1,2,\dots,n$ ...
Karo's user avatar
  • 277
2 votes
1 answer
141 views

Consider the following refinement of permutation statistics. For $π ∈ S_n$, let: $\mathrm{fix}(π) = |\{i : π(i) = i\}|$ (number of fixed points) $\mathrm{exc}(π) = |\{i : π(i) > i\}|$ (number of ...
Peter Thomas's user avatar
  • 21
10 votes
0 answers
393 views

Let $a_n$ be a sequence such that $a_1=1$ and for each $n \geq 1$ $a_{n+1}$ is the smallest positive integer distinct from $a_1,a_2,...,a_n$ such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$....
jack's user avatar
  • 3,143
19 votes
4 answers
1k views

Motivation. Suppose we are given $6$ boxes, arranged in the following manner: $$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]$$ Two of these boxes contain a ...
Dominic van der Zypen's user avatar
  • 54.4k
4 votes
0 answers
201 views

Let $S_n$ be the set of all permutations of $\{1, \ldots, n\}$, thereafter treated as integer sequences. Let $A_n$ be the set of all such permutations $\sigma \in S_n$ that we can choose two ...
Mikhail Tikhomirov's user avatar
  • 5,812
4 votes
0 answers
156 views

Vexillary permutations are an important family of permutations in Schubert calculus. There are several definitions, for example that they avoid the pattern 2143. Recall the Lehmer code of a ...
Zach H's user avatar
  • 2,009
1 vote
0 answers
151 views

Imagine a circular permutation of n points on a circle, if we draw a line connecting any pair of points, the rest of the points are divided into two sets that are on the same side. We can partition a ...
puzzler's user avatar
  • 11
2 votes
0 answers
103 views

The question itself is formulated in the title, so below I specify the matrices and expressions mentioned there. In case if this is something known or can be easily deduced from something known, this ...
Nikita Safonkin's user avatar
  • 351
0 votes
0 answers
139 views

Let $(S_i)_{i \in \mathbb{N}}$ be a sequence of sets defined recursively as follows: $S_1 = \{1\}$ $S_{i+1} = S_i \cup \{S_i, i+1\} \quad \forall i \in \mathbb{N}$ A permutation $\sigma$ of $S_i$ is ...
Riley's user avatar
  • 1
4 votes
1 answer
263 views

Let $S\subset S_{\infty}$ be a set of permutations of $\mathbb{N}$. A real series $\sum_{n\geq0}u_{n}$ will be called $S$-conditionally convergent if it is absolutely divergent and if, for all $\sigma\...
abeaumont's user avatar
  • 117
2 votes
1 answer
333 views

A visible permutation $\sigma$ of $[1,2, ...,n]$ has a permutation matrix such that all "1" locations are visible from the origin $(0,0)$. Two "1" locations are visible if the two ...
Mohammad Al-Turkistany's user avatar
  • 3,035
6 votes
0 answers
312 views

We say that a family ${\cal S}\subseteq{\cal P}(\mathbb{N})$ is bijection-dodging if there is a bijection $\varphi:\mathbb{N}\to\mathbb{N}$ with $\varphi(T)\notin {\cal S}$ for all $T\in{\cal S}$. ...
Dominic van der Zypen's user avatar
  • 54.4k
45 votes
2 answers
3k views

This is a follow-up to this question by Dominic van der Zypen. For each bijection $f:\mathbb{N}\to\mathbb{N}$, let $$\operatorname{rc}(f) := \liminf_{N\to\infty} \frac{\left|\left\{(m,n)\in\{1,\dots,N\...
Saúl RM's user avatar
  • 13.1k
9 votes
1 answer
559 views

For any set $X$, let $\newcommand{\N}{\mathbb{N}}[X]^2 = \big\{\{x,y\}:x\neq y \in X\big\}$ and set $[n]^2 = [\{0,\dotsc,n-1\}]^2$ for any positive integer $n$. For $A\subseteq [\N]^2$ we set $$\...
Dominic van der Zypen's user avatar
  • 54.4k
11 votes
2 answers
658 views

Let $\mathbb{N}$ denote the set of non-negative integers. We say $A\subseteq \mathbb{N}$ is a finite arithmetic progression if there are $a, n, d\in\mathbb{N}$ with $d \geq 1$ and $n \geq 2$ such that ...
Dominic van der Zypen's user avatar
  • 54.4k
1 vote
1 answer
265 views

Let $w=\sum_{I} a_{I}dx_{I}$ be a $k$-form in $\Bbb R ^n$. Let us consider the Hodge operator in a combinatorial form, i.e. as an $(n-k)$ form such that $$\star(dx_{i_{1}} \wedge \dotsb \wedge dx_{i_{...
Wrlord's user avatar
  • 261
1 vote
0 answers
84 views

This question has relation with this previous one, although the two cases are not likely solved with the same method. Let us consider a function $P:\{0,1\}^*\to\{0,1\}^*$ that can be calculated in ...
Doriano Brogioli's user avatar
  • 489
5 votes
1 answer
284 views

Given $n\in \mathbb{N}$, we have a bijection $P:\{0,1\}^n\to\{0,1\}^n$, i.e. $P$ is a permutation of $2^n$ symbols, $P\in S_{2^n}$. The permutation $P$ can be calculated efficiently, i.e. by a ...
Doriano Brogioli's user avatar
  • 489
-4 votes
1 answer
265 views

I claim, for $\operatorname{artanh}(\rho) = \frac{1}{2} \ln\left(\frac{1+\rho}{1-\rho}\right)$, i.e., the inverse hyperbolic tangent function, the following holds approximately under assumptions given ...
virtuolie's user avatar
  • 191
11 votes
1 answer
1k views

Motivation. My eldest son thinks the following procedure is a "perfect shuffle" for a deck of cards: Take the first card, put the second on top of it, put the 3rd below cards 2 and 1, put ...
Dominic van der Zypen's user avatar
  • 54.4k
5 votes
1 answer
260 views

Let $a_1a_2\ldots a_{2n+1}$ ($n\geq 2$) be a given permutation of the numbers from $1$ to $2n+1$ and let $\alpha_i=\{i,i+1,i+2\},~1\leq i\leq 2n-1$ $\alpha_{2n}=\{2n,2n+1,1\}$ $\alpha_{2n+1}=\{2n+1,1,...
W. Paul Liu's user avatar
  • 567
2 votes
1 answer
382 views

Motivation. I am trying to make an interesting infinite version out of this fascinating problem from the Russian mathematical olympiad: There are $c$ flavours of cookies, we are given $n$ cookies of ...
Dominic van der Zypen's user avatar
  • 54.4k
0 votes
1 answer
343 views

I have been studying the following situation, and I have a claim I believe to be true, but am unsure on how to approach it. I would appreciate any references I could look into where others have ...
NathanLiitt's user avatar
  • 3
1 vote
1 answer
206 views

Let $S_n$ be the collection of bijections $\varphi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. In an earlier question, the insert-and-shift graph structure was introduced on $S_n$ and the resulting graph is ...
Dominic van der Zypen's user avatar
  • 54.4k
1 vote
1 answer
231 views

Motivation. I am working with a database software that allows you to sort the fields of any given table in the following peculiar way. Suppose your fields are numbered $1,\ldots, 18$. Next to every ...
Dominic van der Zypen's user avatar
  • 54.4k
1 vote
0 answers
98 views

Suppose we want to build a primality testing algorithm for the numbers limited to the set $A =\{1, ..., 2^n\}$ and $n$ is reasonably large. The prime-number theorem tells us that there are ...
user1747134's user avatar
  • 283
2 votes
2 answers
116 views

[Repost of same question math stack exchange which got no answers] I'm looking for literature on the following family of graphs: Call a regular graph $G=(V,E)$ (of regularity degree $d$) nice if there ...
jojo's user avatar
  • 21
4 votes
0 answers
260 views

This is motivated by Aaronson's post, Probability of generating a desired permutation by random swaps. I am interested in a related problem where the swaps are given in the input. We're given as input ...
Mohammad Al-Turkistany's user avatar
  • 3,035

1
2 3 4 5
…
12