It also works for Hamming distance of bit strings. For an n-bit string there are C(n,k) ways of achieving a Hamming distance of k, for example there is only 1 string with distance 0 (the original string) and 1 string with distance n (the complement of the original string), and everything in the middle goes first up and then down—following the components of Pascal's triangle.

This could also be viewed as, for example, the distance along edges from one corner of an n-dimensional hypercube to other corners. There is 1 corner where you don't move at all and 1 corner where you move in every dimension, and the largest number of corners in which you move in half of the dimensions.

I agree with your intuition that this phenomenon will apply to a whole lot of contexts and situations.

Feels to me like this is required of any 'lattice' (partial ordered set with a unique supremum and infinum for every sub-set).

Notably, given any 'cross-section' of a lattice (a set A such that given an x in A, and any x > y then y in A) has a 'surface' (the set of all points x in z such that for all y in A we NOT z > x). That cross section needs to start (1 element) at the cross section containing only the 'infinum' of the entire lattice. Similarly, the surface needs to end small (1 element again) at the cross-section that is the entire lattice.

In between, it will generally by larger. I think there might be some 'convexity' results that can be proven about the surfaces of increasing sequences of cross-sections. Given specific kinds of lattices.

It feels to me like a cartesian-lattice would always have this convexity porperty. (By convexity I mean that the surface starts of increasing, might be constant for a while, and then must continue decreasing.)

Cool subject, makes me wish I was doing a PhD in mathematics.

For reference, what you call a 'cross section' is sometimes called an ideal or a downset. I can't quite make out your definition of 'surface', but, if it is meant to read "the set of all points x in A such that for all y in A we have not y > x", then it is the set of maximal elements of A. I'm not an expert on posets, but, if I've got the terminology right, your notion of convexity sounds like you want to deal with a ranked poset with unimodal h-vector.

I think you mean the growth of that set as d goes to half circumference. The absolute size increases up to the area of the sphere.

If we take the definition to refer to points that are exactly distance d, rather than points that are within distance d, I think the original formulation is correct.

Oh, yeah, my mistake.