1 | initial version |

For decent values of $n$ one can try the following:

```
N = 6
S = [1..N]
grading = {1: 1, 2:1, 3: 1, 4: 2, 5: 2, 6: 3}
# Question 0
S2 = Combinations(S,2)
S2_list = S2.list()
# Alternatively:
S2_list = [(j, k) for j in [1..N] for k in [j+1..N]]
# Question 1
g0 = 4
S2_list_in_deg_g0 = [(j, k) for j in [1..N] for k in [j+1..N] if grading[j] + grading[k] == g0]
# Question 2 :: same game
S3_list_in_deg_g0 = \
[(j, k, n) for j in [1..N] for k in [j+1..N] for n in [k+1..N]
if grading[j] + grading[k] + grading[n] == g0]
S3_list_in_deg_g0_starting_with_one = \
[(1, k, n) for k in [2..N] for n in [k+1..N]
if 1 + grading[k] + grading[n] == g0]
```

This is slow, since the python loops are slow, but should work good enough for values of $N$ up to $10000$ say. But i wonder if this lists are really needed as lists, to be stored on the HD as such. Instead, if the lists are an intermediate step, iterables may be better. (But python loops are still to slow.) To build the last as an iterable...

```
S3_iterable_in_deg_g0_starting_with_one = \
((1, k, n) for k in [2..N] for n in [k+1..N]
if 1 + grading[k] + grading[n] == g0)
```

For instance, consuming the iterable:

```
sage: S3_iterable_in_deg_g0_starting_with_one = \
....: ((1, k, n) for k in [2..N] for n in [k+1..N]
....: if 1 + grading[k] + grading[n] == g0)
....:
sage: S3_iterable_in_deg_g0_starting_with_one
<generator object <genexpr> at 0x7f5f920eba50>
sage: for triple in S3_iterable_in_deg_g0_starting_with_one:
....: print(triple)
....:
(1, 2, 4)
(1, 2, 5)
(1, 3, 4)
(1, 3, 5)
```

The above solutions are using list comprehension. If the above is slow for the needed purpose, than to improve consider the following ideas:

- I suppose that $N$ may be bigger $10^6$, but the "grading(s)" have a small range. In our case, the values of the gradings / weights dictionary, taken as a set, is {1,2,3}. Suppose this list is relatively small. (Less than $10^5$.) Then build all tuples of two weights which sum up to
`g0`

. Then define the iterable by passing through the weights first.

For instance:

```
weights = list(set(grading.values()))
weights2 = [(u, v) for u in weights for v in weights if u <= v if u + v == g0]
S2_iterable_in_deg_g0 = \
((j, k) for (u, v) in weights2 for j in [1..N-1] for k in [j+1..N]
if grading[j] == u and grading[k] == v)
# consume it via...
for tup in S2_iterable_in_deg_g0:
print(tup)
```

(The above can be done better...)

- Use
`numpy`

for vectorialization.

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.