Thread by @pgeerkens

@nklym143 @GWOMaths Then I'd note:

- Generating Function f(x) for a single standard die is
x.(1+x+x²+x³+x⁴+x⁵)
= x+x²+x³+x⁴+x⁵+x⁶
= Σ xⁿ . C(n,1)

1/

@nklym143 @GWOMaths - For two Std. die f(x)² is
= 1.x²+1.x³+1.x⁴+ 1.x⁵+ 1.x⁶+ 1.x⁷
+ 1.x³+1.x⁴+1.x⁵+ 1.x⁶+ 1.x⁷+ 1.x⁸
+ 1.x⁴+1.x⁵+1.x⁶+ 1.x⁷+ 1.x⁸+ 1.x⁹
+ 1.x⁵+1.x⁶+1.x⁷+ 1.x⁸+ 1.x⁹+1.x¹⁰
+ 1.x⁶+1.x⁷+1.x⁸+ 1.x⁹+1.x¹⁰+1.x¹¹
+ 1.x⁷+1.x⁸+1.x⁹+1.x¹⁰+1.x¹¹+1.x¹²

2/

@nklym143 @GWOMaths = 1.x²+2.x³+3.x⁴+4.x⁵+5.x⁶+6.x⁷+5.x⁸+4.x⁹+3.x¹⁰+2.x¹¹+1.x¹²

@nklym143 @GWOMaths - For three die f(x)³
= 1.x³+1.x⁴+1.x⁵+1.x⁶+1.x⁷+ 1.x⁸
...
+ 5.x⁷+5.x⁸+5.x⁹+5.x¹⁰+5.x¹¹+5.x¹²
+ 6.x⁸+6.x⁹+6.x¹⁰+6.x¹¹+6.x¹²+6.x¹³
+ 5.x⁹+5.x¹⁰+5.x¹¹+5.x¹²+5.x¹³+5.x¹⁴
...
+1.x¹³+1.x¹⁴+1.x¹⁵+1.x¹⁶+1.x¹⁷+1.x¹⁸

@nklym143 @GWOMaths = 1.x³+3.x⁴+6.x⁵+10.x⁶+15.x⁷+21.x⁸
+25.x⁹
+27.x¹⁰
+27.x¹¹
+25.x¹²
+21.x¹³+15.x¹⁴+10.x¹⁵+6.x¹⁶+3.x¹⁷+1.x¹⁸

@nklym143 @GWOMaths where the 1, 3, 6, 10, 15, 21 are just C(n,2) for n {1..6) and:

- 25 = 2+3+4+5+6+5
- 27 = 3+4+5+6+5+4.

@nklym143 @GWOMaths Or, back to the diagram alone:

# | n | 1 2 3 4 5 6
---+--+---------------------
2 | 1 | 3 4 5 6 7 8
3 | 2 | 4 5 6 7 8 9
4 | 3 | 5 6 7 8 9 10
5 | 4 | 6 7 8 9 10 11
6 | 5 | 7 8 9 10 11 12

@nklym143 @GWOMaths ...
7 | 6 | 8 9 10 11 12 13
8 | 5 | 9 10 11 12 13 14
9 | 4 | 10 11 12 13 14 15
10 | 3 | 11 12 13 14 15 16
11 | 2 | 12 13 14 15 16 17
12 | 1 | 13 14 15 16 17 18

@nklym143 @GWOMaths The diagram is constructed with the results of the previous generation (here for two std. dice) as the pair of columns Value (#) and Count (n) and the new die as columns 1 .. 6.

@nklym143 @GWOMaths @threadreaderapp unroll

@nklym143 @GWOMaths In the simpler diagram for just two dice one simply counted appearances of the result being analyzed.

Here, one instead sums the 'n' values for each row in which the result being analyzed appears.