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The answer depends on the enumeration of the permutations; the most common being lexicographic order, whence the factorial number system allows one to efficiently calculate permutations from their position …

let $\mathcal{ABP}$, the set of "asymmetric, balanced permutation matrices", be defined as the set of permutation matrices, that aren't equal their transpose but, for which the number $1$s below the p …

Is there an algorithm that enumerates all permutations that are "square roots" of derangements, i.e. permutations that, when applied twice, yield a derangement? … Other information about those kind of permutations is also welcome. …

Given a subset $P\subset\mathcal{P}_n$ of the permutations of $1,\dots,n$ Question: how can a maximal subset $p\subseteq\lbrace1,\dots,n\rbrace$ be determined, whose elements appear in the same relative …

This should be seen as a comment: I was tempted into combinatorial thinking because of the permutations and thus didn't see the solution, while it is actually embarrassingly simple: take $\lbrace1,\dots … i,i+1)$ for which $(i+1,1)$ is an element of some other Hamilton path resembled by some $p\in P$ python code: from numpy import random n = 10 # number of elements to be permuted k = 3 # number of permutations …

Is it true that the number of carries, when calculating the sum of a finite set of finite positive integers, is constant (i.e. independent of their permutation and the order in which the additions ar …

Has the following kind of problem been investigated previously and, where can I find information about it: Given the set $\mathbb{P}_{n_0}$ of all permutations of $n_0$ elements a predicate $P: p\in … I could find information about the statistics of random permutations (e.g here, but nothing that relates to invariant sub-permutations as specified above, so any pointers to literature or online resources …

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 complem …

If $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ is the cost-matrix of an assignment problem, then the usual statement of the problem of finding an optimal assignment is to identify $n$ elements $a_{i,\,\ …

It contains code for determining the $k$th permutation and also discusses successive refinement of a set of permutations. … More generally, you could look for Gray codes for permutations like e.g. in this paper …

The Quadratic Assignment Problem is another example of notoriously hard Problem.

I just found the paper "In-Place Transposition of Rectangular Matrices" by Fred G. Gustavson and Tadeusz Swirszcz, which provides a solution for the problem. An online version of the paper can be f …

let the entries of a rectangular matrix $A\in\mathbb{C}^{m\times n}; m,n\in\mathbb{N}$ be stored in row-order in a linear vector $v$, i.e. $A_{i,j}=v_{i*m+j}$ Question: How can the first element an …

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 …

I am especially interested in $K_5$ and $K_{3,3}$, mainly, because due to Kuratowski's characterization of planar graphs, one can be sure that permutations of $5$ (resp. $6$) cities, i.e. those permutations … Letting $p$ denote the maximum number of simple polygonal tours and $h$ the number of Hamiltonian tours, $h-p$ permutations of a subset of $m$ cities can be excluded from the optimal tour through all $ …