Timeline for "Shape"/"norm" of a uniformly random set partition
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 15, 2021 at 0:01 | history | edited | Joseph Van Name | CC BY-SA 4.0 |
corrected typos and link
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| Sep 14, 2021 at 14:23 | comment | added | Joseph Van Name | I just verified these identities with code that produces the exact same sequence in 3 different ways. Here is the code in GAP. par:=[]; par[1]:=[[[1]]]; for n in [2..10] do par[n]:=[]; for a in par[n-1] do Add(par[n],Concatenation(a,[[n]])); for i in [1..Length(a)] do aa:=StructuralCopy(a); Add(aa[i],n); Add(par[n],aa); od; od; od; mak:=function(xx) return Sum(List(xx,v->Length(v)^2)); end; List(par,v->Sum(List(v,mak))); moo:=function(n) return Sum(List([1..n],k->Binomial(n,k)*Bell(n-k)*k^2)); end; List([1..10],moo); List([1..n],v->v*((v-1)*Bell(v-1)+Bell(v))); | |
| Sep 14, 2021 at 13:05 | comment | added | Marcus M | Are you sure about that? You should double check your answer; if you make that edit then you'd end up with $E [|A|^2 ] = \Theta (n^2 )$, which doesn't seem like the right order. | |
| Sep 14, 2021 at 12:00 | vote | accept | Vilhelm Agdur | ||
| Sep 13, 2021 at 15:25 | comment | added | Joseph Van Name | @MarcusM Thanks for spotting that. The correct link should be oeis.org/A175716. These are closely related sequences since A175716(n) = n*A124427(n), so I must have put the wrong sequence in the clipboard. | |
| Sep 13, 2021 at 15:14 | comment | added | Marcus M | Are you sure that OEIS sequence is the right one? It looks like you've got a $k^2$ in your identity and the linked sequence has a $k$. | |
| Sep 11, 2021 at 5:39 | history | edited | Joseph Van Name | CC BY-SA 4.0 |
added 585 characters in body
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| Sep 11, 2021 at 4:51 | history | answered | Joseph Van Name | CC BY-SA 4.0 |