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Simplicial sequences generalize the familiar "linear", "triangular", and "tetrahedral" number sequences. (A line segment is a 1-simplex, a triangle is a 2-simplex, a tetrahedron is a 3-simplex, and so on, so I'm calling these simplicial sequences.)

In general, the n-th d-simplicial number is the binomial coefficient C(n+d-1,d).

The pictures below graphically show the n-th element (modulo 512) of 512 simplicial sequences side by side. The white dots in a given frame have coordinates (d,C(n+d-1,d) mod 512), where d runs from 0 through to 511, and the value of n is displayed at the bottom of the diagram.

Also, if you are using a "modern browser" (i.e. not IE6,7,8), you can see an animation running through all values of n by following this link. The animation allows you to start and stop, and even save individual frames.

If you watch the animation, you'll see that binary visual patterns emerge, in ways that are threaded through the image and increase and decrease in strength. The images below for n=5 and n=256 indicate the extremes .. for n=5 the points look randomly distributed, for n=256 they are strongly patterned.

So my question is whether there are any explanations for the emergence and disappearance of visual binary patterns as we iterate through n.

alt text alt text

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I felt like entering the matrix at 500s. – Pedro Tamaroff Jun 19 '12 at 3:47
@Peter, for anybody interested in the "trippiness" aspect, there is an animation at that uses the patterns to generate sounds as well. Just set "Formula" to "Simplicial Numbers" and turn "Volume" to something non-zero. – brainjam Jun 19 '12 at 4:00
WOW. But how do I get that one? – Pedro Tamaroff Jun 19 '12 at 4:01
@Peter, I don't understand your question. – brainjam Jun 19 '12 at 4:18
up vote 5 down vote accepted

Theorem (Kummer): The greatest power of $p$ dividing ${m+n \choose n}$ is the number of carries it takes to add $m$ and $n$ in base $p$.

In particular the greatest power of $2$ dividing ${n+d-1 \choose d}$ is the number of carries it takes to add $n-1$ and $d$ in base $2$, and so depends entirely on the binary digits of both $n-1$ and $d$. (When $n = 256$, for example, there are likely to be a lot of carries, which is why the position of the dots is so restricted in that case. More generally the same happens when $n-1$ has lots of $1$s in its binary representation.)

The above theorem places a strong restriction on where the white dots can be, which I think will be clearer visually if you plot things in the shape of Pascal's triangle instead of whatever it is you're doing now (for example $\bmod 2$ you will get the Sierpinski triangle. More generally I think you should try plotting the entries of Pascal's triangle $\bmod 512$ where the modulus determines the color of a block. This won't be an animation but will allow you to consolidate all of the patterns you're seeing into one image which I think will ultimately be clearer).

See also Lucas' theorem for a partial explanation of patterns in the exact remainder $\bmod 512$.

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+1: I was just about to write about this (Kummer's theorem)! – Aryabhata Sep 22 '10 at 20:52
@brainjam: the Wikipedia article on Lucas' theorem links to a paper by Andrew Granville ( that you might be interested in. For searchability purposes I suggest you retitle your question as being about binomial coefficients; this "simplicial sequence" thing is a distraction. – Qiaochu Yuan Sep 22 '10 at 20:54

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