Timeline for Nested Set Permutations and their Enumeration
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Aug 27 at 3:37 | comment | added | მამუკა ჯიბლაძე | Is the following correct?$$\begin{array}{rl}S_1&=\{1\}\\S_2&=\{1,2,\{1\}\}\\S_3&=\{1,2,3,\{1\},\{1,2,\{1\}\}\}\\S_4&=\{1,2,3,4,\{1\},\{1,2,\{1\}\},\{1,2,3,\{1\},\{1,2,\{1\}\}\}\}\end{array}$$ | |
| Aug 22 at 1:55 | comment | added | Sam Hopkins | Can you do an example? Say the case $i=3$ or $i=4$? Clearly you have something specific in mind but it seems you are having trouble communicating it. | |
| Aug 22 at 1:38 | review | Close votes | |||
| Sep 13 at 3:02 | |||||
| Aug 22 at 1:15 | comment | added | Max Alekseyev | In condition 1, shouldn't it be $a,b\in\mathbb N$ rather than $a\ne S_{i-1}$ and $b\ne S_{i-1}$? Otherwise one can take, say, $a=S_{i-2}$ with no "natural ordering". And even if $a,b$ are numbers, $\sigma^{-1}(a)$ and $\sigma^{-1}(b)$ may be not numbers but some $S_i$ - how do you compare them? | |
| Aug 21 at 23:34 | history | edited | Max Alekseyev |
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| Aug 21 at 22:35 | comment | added | Riley | @SamHopkins final update hopefully | |
| Aug 21 at 22:34 | history | edited | Riley | CC BY-SA 4.0 |
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| Aug 21 at 22:17 | comment | added | Sam Hopkins | Your update makes less sense. Now each $S_i$ is a two element set, so there are only two "permutations" of it (if permutations are understood in the usual sense as bijections from a finite set to itself). | |
| Aug 21 at 22:10 | comment | added | Riley | @LSpice iodated | |
| Aug 21 at 22:10 | comment | added | Riley | @SamHopkins Updated | |
| Aug 21 at 22:08 | history | edited | Riley | CC BY-SA 4.0 |
added 39 characters in body
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| Aug 21 at 22:01 | comment | added | LSpice | I don't think I understand the definition. If $\sigma^{-1}(i) > \sigma^{-1}(x)$ for all $x \in S_{i - 1}$, then $\sigma^{-1}(i)$ is greater than $i - 1$ distinct elements of $S_i$, so it must equal $i$. Then (1) just says that $\sigma$ is order-preserving on $S_{i - 1}$. (Ah, and now I see that I read the formal definition and missed the informal description of $S_i$, which, as @SamHopkins says, doesn't match the formal definition. If you did mean the informal definition, then what's the (total?) order on $S_i$?) | |
| Aug 21 at 22:01 | comment | added | Sam Hopkins | "Thus, each $S_i$ is a finite set containing the natural numbers from $1$ to $i$, and also all preceding sets $S_j$ for $j < i$.": this does not match the way you defined $S_{i+1}$ in the preceding line; the way you defined it, $S_{i+1}=\{1,\ldots,i+1\}$. | |
| S Aug 21 at 21:57 | review | First questions | |||
| Aug 21 at 22:48 | |||||
| S Aug 21 at 21:57 | history | asked | Riley | CC BY-SA 4.0 |