Timeline for Enumerative characterisation of boolean lattices
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 30, 2016 at 14:42 | vote | accept | Sebastien Palcoux | ||
| Aug 30, 2016 at 14:21 | comment | added | Sebastien Palcoux | So that converse is true for $B_n$ with $n \le 3$ and false for $n \ge 4$. | |
| Aug 30, 2016 at 13:58 | comment | added | მამუკა ჯიბლაძე | Alternatively, your fake $B_4$ can be viewed as the (upside down) partition lattice of a 4-element set, with an extra bottom added (or you could also add an extra top instead; this is still another (non-isomorphic) example). (For a picture, see here) | |
| Aug 30, 2016 at 13:17 | comment | added | Hugh Thomas | I hope the number of vertices is now clear. Please go ahead and draw the picture of the rank 4 case if you would like. | |
| Aug 30, 2016 at 13:16 | history | edited | Hugh Thomas | CC BY-SA 3.0 |
add clarification in response to comment
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| Aug 30, 2016 at 11:46 | comment | added | Sebastien Palcoux | After your modification, your construction admits now $2^n - n + 3 $ vertices. If you can't fix that then your last comment should be your answer (if you can draw it then this would be nice, otherwise I will draw it myself). | |
| Aug 30, 2016 at 5:51 | comment | added | Hugh Thomas | To show that $B_4$ is not uniquely characterized as in the question, you can take $X$ consisting of 7 elements, and a collection of 6 of the seven triples from the Fano arrangement as the collection of subsets. The rank numbers are 1, 7, 6, 1, 1; there are 16 vertices and 32 edges. | |
| Aug 30, 2016 at 4:36 | comment | added | Hugh Thomas | It is now of length $n$. (I forgot that condition to begin with.) It's clearly not equivalent to a Boolean lattice provided $n>3$. | |
| Aug 30, 2016 at 4:36 | history | edited | Hugh Thomas | CC BY-SA 3.0 |
corrected to add in the height condition
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| Aug 30, 2016 at 4:34 | comment | added | Sebastien Palcoux | Is your construction of length $n$ and non-equivalent to a boolean lattice? | |
| Aug 30, 2016 at 4:29 | history | answered | Hugh Thomas | CC BY-SA 3.0 |
