All Questions
14 questions
1
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91
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Generate the nth permutation [closed]
I'm just trying to write a little algorithm. I've got nine objects, so there's 9! permutations. My question is, is there a way of turning a number from 1 to 9! into a permutation?
for example, f(1)=[1,...
- 111
1
vote
1
answer
94
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Calculating the values of a generalization of binomials to permutations
let $$\Pi\binom{n}{k}:=\mathrm{card}\left( \left\lbrace \lbrace \Pi_1^n\,\cdots\,\Pi_k^n\rbrace\,|\,0\leq \pi_{r,c}\in\sum_{i=1}^k\Pi_i^n\ni\pi_{r,c}\leq 1\right\rbrace\right)$$ be the number of sets ...
- 13.2k
3
votes
0
answers
627
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Loopless algorithm for generating permutations (Steinhaus-Johnson-Trotter)
The following is a description of the well-known Steinhaus-Johnson-Trotter algorithm to generate all permutations of an $n$-element ground set using adjacent transpositions.
In fact, it is a loopless ...
- 499
8
votes
1
answer
1k
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Is there an efficient algorithm to check whether two matrices are the same up to row and column permutations?
Define $\mathcal M_n$ as the set of all $n\times n$ matrices with each entry either 1 or $x$. Two such matrices are equivalent iff they can be obtained from each other by swapping pairs of rows and ...
- 13.4k
9
votes
0
answers
534
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Generating $S_n$ with a fundamental transposition and a big cycle
I apologize in advance if this is too amateur, this is not really my area, but I'm very curious.
We have a permutation $\pi \in S_n$ and we want to represent it as a product of $\sigma = (1\;2)$ and $...
- 231
1
vote
1
answer
196
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Algorithm for Removing Inverted Elements from a Permutation
I currently have a problem, whose solution requires to remove from a permutation of $\lbrace 1,\ \dots,\ n\rbrace$ those values that are to the left of a smaller one.
My idea was to remove the ...
- 13.2k
12
votes
2
answers
292
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Permutation search problems with no known $o(n!)$ algorithms
I am looking for problems for whose solution no known subfactorial algorithms are known. I am particularly interested in questions of isomorphism; that is, is there a permutation that converts one ...
- 333
15
votes
2
answers
512
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Does a classification of simultaneous conjugacy classes in a product of symmetric groups exist?
Let the symmetric group $S_n$ on $n$ letters act on $S_n^d=S_n\times\cdots\times S_n$ by simultaneous conjugation, i.e. $\pi\in S_n$ acts on $(\sigma_1,\ldots,\sigma_d)\in S_n^d$ by $\pi.(\sigma_1,\...
- 4,902
10
votes
1
answer
519
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Explicit algorithm for composing permutations in factorial notation
Given two permutations p1 and p2 in factorial notation, is there an algorithm which computes their composition directly, i.e. without translating to a different notation (like Cauchy's 2-line notation,...
- 11.8k
3
votes
2
answers
333
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Combinatorial design for minimization problem over binary strings
Suppose the cost of a binary string $B$ of length $k$ is the number of $1$s that occur before the last $0$. For example, $1110$ has cost 3 while $0111$ has cost 0. Now suppose you can choose $k$ ...
1
vote
2
answers
163
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Draws from multiple non-disjoint urns
Let $C = \{1,...,n\}$ be a set of $n$ colors. Let $S_1,...,S_k$ be non-empty subsets of $C$, that is, $S_i \subseteq C$ for all $i \in \{1,...,k\}$. It is helpful to think of the $S_i$ as urns with ...
- 83
1
vote
1
answer
879
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Generating fixtures for a chess league, with a twist
Hello,
I am in the process of building some software to generate fixtures for a chess league. Which has a little twist which complicates matters. I would like to introduce a constraint. Where by a ...
- 111
4
votes
2
answers
2k
views
How to compute the rook polynomial of a Ferrers board?
Given a Ferrers board of shape $(b_1,\ldots,b_m)$, we define $r_k$ as number of ways to place $k$ non-attacking rooks (as in Chess). In section 2.4 of Stanley's Enumerative Combinatorics (vol. 1) it's ...
- 1,015
19
votes
9
answers
3k
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How can I generate random permutations of [n] with k cycles, where k is much larger than log n?
I've been thinking a lot lately about random permutations. It's well-known that the mean and variance of the number of cycles of a permutation chosen uniformly at random from Sn are both ...
- 14k