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Results tagged with permutations
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user 46140
Questions related to permutations, bijections from a finite (or sometimes infinite) set to itself.
1
vote
Ordering and place in sets
A quick computer program throws up a counter-example for $n=2$, $k=4$: take subsets $\{1\}, \{1, 2, 3\}, \{2\}, \{3\}$. The maximum score of $\frac{115}{36}$ is obtained by $(1, 2, 3, 4)$, $(1, 3, 2, …
- 7,226
3
votes
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 suppo …
- 7,226
4
votes
Accepted
Property of some permutations of non-negative integers such that $a(n)<2^k$ iff $n<2^k$
I'm going to use $\operatorname{msb}$ (for most significant bit) as an alias of $f$.
Since $q_i$ is a permutation, the property that $q_i(n)<2^k$ iff $n < 2^k$ is equivalent to $\operatorname{msb}(q_i …
- 7,226
2
votes
Accepted
Relationship between fixed points and inversions in permutations
Looking at permutations with $j−1$ inversions we get $S_\beta=\{1432,2341,2413,3142,3214,4123\}$ and $S_\delta=\{2341,2413,3142,4123\}$, which is a counterexample to the conjecture in the question. … I've tested it numerically up to $n=28$, and by examining those data I conjecture that the expected number of inversions for a permutation uniformly selected from permutations on $n$ elements with $k …
- 7,226
3
votes
Accepted
Permutation of the natural numbers from operation related to binary expansion of $n$
$\ell(0)$ is problematic, so I will assume that you actually mean to restrict to positive integers rather than natural numbers.
We can rephrase the construction of $a(n)$ to emphasise the increment: l …
- 7,226