How To Filter Elements Of Cartesian Product Following Specific Ordering Conditions

I have to generate multiple reactions with different variables. They have 3 elements. Let's call them B, S and H. And they all start with B1. S can be appended to the element if there is at least one B. So it can be B1S1 or B2S2 or B2S1 etc... but not B1S2. The same goes for H. B1S1H1 or B2S2H1 or B4S1H1 but never B2S2H3. The final variation would be B5S5H5. I tried with itertools.product. But I don't know how to get rid of the elements that don't match my condition and how to add the next element. Here is my code:

import itertools

a = list(itertools.product([1, 2, 3, 4], repeat=4))

#print (a)

met = open('random_dat.dat', 'w')
met.write('Reactions')
met.write('\n')
for i in range(1,256): 
    met.write('\n')
    met.write('%s: B%sS%sH%s -> B%sS%sH%s' %(i, a[i][3], a[i][2], a[i][1], a[i][3], a[i][2], a[i][1]))
    met.write('\n')
met.close()

Simple for loops will do what you want:

bsh = []
for b in range(1,6):
    for s in range(1,b+1):
        for h in range(1,b+1):
            bsh.append( f"B{b}S{s}H{h}" )

print(bsh)                

Output:

['B1S1H1', 'B2S1H1', 'B2S1H2', 'B2S2H1', 'B2S2H2', 'B3S1H1', 'B3S1H2', 'B3S1H3', 
 'B3S2H1', 'B3S2H2', 'B3S2H3', 'B3S3H1', 'B3S3H2', 'B3S3H3', 'B4S1H1', 'B4S1H2', 
 'B4S1H3', 'B4S1H4', 'B4S2H1', 'B4S2H2', 'B4S2H3', 'B4S2H4', 'B4S3H1', 'B4S3H2', 
 'B4S3H3', 'B4S3H4', 'B4S4H1', 'B4S4H2', 'B4S4H3', 'B4S4H4', 'B5S1H1', 'B5S1H2', 
 'B5S1H3', 'B5S1H4', 'B5S1H5', 'B5S2H1', 'B5S2H2', 'B5S2H3', 'B5S2H4', 'B5S2H5', 
 'B5S3H1', 'B5S3H2', 'B5S3H3', 'B5S3H4', 'B5S3H5', 'B5S4H1', 'B5S4H2', 'B5S4H3', 
 'B5S4H4', 'B5S4H5', 'B5S5H1', 'B5S5H2', 'B5S5H3', 'B5S5H4', 'B5S5H5']

Thanks to @mikuszefski for pointing out improvements.