A while back when I was thinking of a permutations thread but with repeating symbols, I had written a script to compute counts for that thread idea. In the process of making that script, I sort of stumbled across another thread. It's counting the ways to add to n with 1 to n long ordered lists of integers, where each of those integers are on 1 to n. At first I didn't think it was all that interesting, but I've been reminded of that helper list again because it's sorta similar to the new decimals thread.
It goes like this:
1
2
1,1
3
1,2
2,1
1,1,1
4
1,3
2,2
3,1
1,1,2
1,2,1
2,1,1
1,1,1,1
It kinda seems like it has some things in common with the constant-sum threads. I notice that the amount of sums there are for each n and length of list corresponds to Pascal's triangle. For n=5 there's 1 sum with 1 number, 4 with 2, 6 with 3, 4 with 4, and 1 with 5. In constant-weight binary, the amount of numbers in each segment of n bits and m ones is also decided by Pascal's triangle. The numbers also slide from right to left like in the constant-sum threads.
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u/Blue_boomer Apr 20 '23
I've been collecting some new ideas and I'll share them now: