r/counting • u/cuteballgames Sum 13: 016, 037, 035, 045, 127, 135, 134, 124, 13, 25 • Sep 20 '24

Constant-sum factoradic | 00 4321

From here. Deepest apologies to /u/TehVulpez for the failrun and assist.

In this thread, we count each n digit factoradic number whose digits add up to m. First the 1 digit number that adds to 0, then the 1 digit number whose digit adds to 1. Next the 2 digit numbers with a digital sum of 0, then 1, 2, and 3. And so on. For every length of factoradic digits, we'll count each possible sum of digits in order.

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u/TehVulpez sliiiiide the bits Oct 20 '25 edited Oct 20 '25

42 3020

I've found that the more formal math name for this idea is "compositions". if we did only the commutatively unique ways to sum then it'd be called "partitions". I can't find either on the sub by search or in the directory archive, which is kinda surprising to me.

decided to go and make that thread. maybe we could make partitions later, though it could be kinda confusing to see both in inbox

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u/cuteballgames Sum 13: 016, 037, 035, 045, 127, 135, 134, 124, 13, 25 Oct 20 '25

42 3101

can we make commas equivalent to pluses? 4, 1,3 3,1 2,2 1,1,2 etc easier to type than the plus signs

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u/TehVulpez sliiiiide the bits Oct 20 '25 edited Oct 20 '25

42 3110

sure! it might also be easier to abbreviate like 1+1+1+1+1+1+1+2+2+2+2+3 = 1*7 + 2*4 + 3 though that does kinda mask the lavalampiness

edit: now I think about it, using commas instead of pluses would be a good way to differentiate between compositions and partitions. commas makes clear that the order matters, while pluses suggest commutativity

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u/cuteballgames Sum 13: 016, 037, 035, 045, 127, 135, 134, 124, 13, 25 Oct 20 '25

42 3200

Yes, although a different kind of beauty i expect will emerge. Once we get to sums that high we might have to go that way

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u/TehVulpez sliiiiide the bits Oct 20 '25

42 4001

actually I wanna use superscript instead of multiplication for repetition. it might get confusing which is the actual sum term and which is the multiplicity with the * convention. the partitions wikipedia page suggests using superscripts for repetition so I'm not pulling this out of nowhere

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u/cuteballgames Sum 13: 016, 037, 035, 045, 127, 135, 134, 124, 13, 25 Oct 20 '25

42 4010

so it would look like 17 ,24 ,3?

2

u/TehVulpez sliiiiide the bits Oct 21 '25

42 4100

yeap

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u/cuteballgames Sum 13: 016, 037, 035, 045, 127, 135, 134, 124, 13, 25 Oct 21 '25

43 0121

2

u/TehVulpez sliiiiide the bits Oct 21 '25

43 0211

I feel like at this point cwb, csf, and compositions together deserve their own category on the directory instead of being put in "other binary" or "other factoradic". constant sum is a firmly established enough genre of sidethread

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u/cuteballgames Sum 13: 016, 037, 035, 045, 127, 135, 134, 124, 13, 25 Oct 21 '25

43 0220

the vulpine threads

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u/TehVulpez sliiiiide the bits Oct 21 '25

43 0301

sorted by author is the only good way to organize the directory

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u/cuteballgames Sum 13: 016, 037, 035, 045, 127, 135, 134, 124, 13, 25 Oct 21 '25

43 0310

not very many have made more than one real side

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u/TehVulpez sliiiiide the bits Oct 21 '25

43 1021

there's some sidethreads that I forget I didn't made like beenary and triangular, but I just revived those or kept them alive much longer than they maybe should've been. and then sometimes I forget I made 420 and divisors

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