Ok lets do this from the ground up
The T31/kiss/D600 is a crop frame camera so we need to use a 1.6x crop factor when talking about lens fields of view.
A 50mm lens has a horizontal field of view (HFOV) of 40° and a Vertical field of view (VFOV) of 27°
As you are using a crop sensor the HFOV will be 25° and the VFOV will be 16.9°
Your panorama is 3x as wide as it is high and ASSUMING you are only shooting a single row we can work out the following
The height will be 25° (we are shooting in portrait orientation) so the width will be 3x25°=75°
I use an overlap of around 1/3 but wouldn't go lower than that
So with 1/3 overlap you get all of the pixels from the first frame, the second frame you ignore its first 1/3 as its covered by the first frame, the first third of the 3rd frame is covered by the second frame etc etc... (its easier to visualize if you lay down 5 playing cards, you get all of the first card and only 2/3 of each other card)
so we need to solve the formula 16.9° + X (2/3 x 16.9°) = 75° (16.9 is the first frame, x is the number of partial frames which only add 2/3 of their width)
75-16.9 = X (11.255)
X=(75-16.9) / 11.255
X=5.16 partial frames, don't forget the full frame at the start, so total frames is 6.16
So you shoot 8 frames (an extra one at each end), overlap by 1/3 and then crop out the excess once it is stitched (always better to have too much rather than not enough)
If you wanted to do 1/2 overlap then the same thing applies
16.9° + X (1/2 x 16.9°) = 75°
X=6.8 partial frames (+1 and an extra one at each end means you shoot 10)
Damn it, this just got more complicated as we cant assume its a single row panorama.
If you set the minimum resolution for a print to 250ppi then 18Mp wont give you enough to get 24" on its long side, so its going to have to be a multi-row panorama.
18Mp = 5,184×3,456
5184/250 = 20.7 inches
So back to the start again,
If we overlap the top and bottom row by 1/3 then the height of the two merged frames is 25°+(2/3 x 25°) =41.7°
So if that's the height of the panorama then it's width is 3x that at 125.1°
Now we just run the above math again
16.9° + X (2/3 x 16.9°) = 125.1°
125.1-16.9 = X (11.255)
X=(125.1-16.9) / 11.255
X=9.6 partial frames, don't forget the full frame at the start, so total frames is 13 in each row
But.... if you are shooting a top and bottom row some stitching software out there doesn't like not having a horizon line image and gets all snooty about stitching them together. So shoot the middle frame of the panorama so that the horizon runs through it (if your stitching software has a mare and cant work out where the horizon should be add the extra frame and use that as the orientation frame [looking at you Hugin, yes I am, don't think I'm falling for that problem again]) , then shoot the top 13 and the bottom 13. Then do it all again in case something went wrong the first time.
The end result should be around 8640x31108 pixels which when you crop it will be 8640x25920 and will be around 360ppi when printed not the 216ppi if we only used one row.
Hope that helps.