Re: some questions on mapping from an integer to a vector and inversely

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Fri, Nov 15, 2024 at 04:03:43PM -0800, American Citizen wrote:
> Hello:
> 
> My topic is on how to develop mapping from one integer n into n-integers
> [vector] , or 1-dimensional space to n-dimensional space.
> 
> This is a mapping of 1d(n) --> nd[n] and of course, the inverse map which
> takes the n-dim vector back to the integer.
> 
> map_xy_to_n(x,y) = (y^2 + (2*x + 1)*y + x*(x + 3))/2;
> map_n_to_xy(n) =
> S=floor((sqrt(8*n+1)-1)/2);T=S*(S+1)/2;x=n-T;y=S-x;return([x,y]);
> bi_Z_to_N(Z)=if(Z<0,return(-2*Z-1),return(2*Z));
> bi_N_to_Z(N)=if(N%2==0,return(N/2),return(-(N+1)/2));

Maybe you have extra requirement, but otherwise you can just apply
map_xy_to_n recursively. For example for 3 variables:

map_xy_to_n(x,y) = (y^2 + (2*x + 1)*y + x*(x + 3))/2;
map_n_to_xy(n) = S=floor((sqrt(8*n+1)-1)/2);T=S*(S+1)/2;x=n-T;y=S-x;return([x,y]);

map_xyz_to_n(x,y,z)=map_xy_to_n(x,map_xy_to_n(y,z))

map_n_to_xyz(n)=my([x,yz]=map_n_to_xy(n),[y,z]=map_n_to_xy(yz));[x,y,z]

for(i=1,20,my([x,y,z]=map_n_to_xyz(i));print(i,":",[x,y,z],":",map_xyz_to_n(x,y,z)))
1:[0,0,1]:1
2:[1,0,0]:2
3:[0,1,0]:3
4:[1,0,1]:4
5:[2,0,0]:5
6:[0,0,2]:6
7:[1,1,0]:7
8:[2,0,1]:8
9:[3,0,0]:9
10:[0,1,1]:10
11:[1,0,2]:11
12:[2,1,0]:12
13:[3,0,1]:13
14:[4,0,0]:14
15:[0,2,0]:15
16:[1,1,1]:16
17:[2,0,2]:17
18:[3,1,0]:18
19:[4,0,1]:19
20:[5,0,0]:20

Cheers,
Bill.