[seqfan] Re: Some number triangles

[Dale Gerdemann]

Hello Olivier, Hello SeqFans,

I do have a general story about what all of these triangles are counting.
It's a little easier to explain t(n,m) because it's more symmetric than
T(n,k). So for t(n,m), think of tiling a 1x(n+m) board with two kinds of
squares. They could be black and white squares, but for reasons that I'll
explain, I like to call them "up" and "down" squares (or tiles). Now t(n,m)
counts the sum of the "weights" of all n-up, m-down tilings of this board,
where the "weight" of a tiling of the board is the product of the "weights"
of the individual tiles. So it remains to explain how the weight of an
individual tile is calculated. The weight of an up tile depends on the
number of down tiles to the left and the weight of a down tile depends on
the number of up tiles to the left. This is where the side function f(x)
comes in. If u is the number of leftward up tiles and d is the number of
leftward down tiles, then the weight of an up tile is f(d) and the weight
of a down tile is f(u).

For some kinds of triangles, a kind of asymmetry is introduced with the
"up" tiles shifted upward and the "down" tiles shifted downward (hence the
names). So if the shift is 1, then an up tile get weight f(d+1) and a down
tile gets weight f(u-1). The Fibonomial triangle, for example, works this
way. For Pascal's triangle, on the other hand, the side function is f(x) =
1, so shifting plays no role.

I realize that counting "weights" is rather abstract. Nevertheless, this
tiling approach should be usable to prove some rather abstract theorems
across all the triangles. For example, I'm pretty sure that some
generalization of the hockey stick identity is valid for all.

What I'd like to understand is why this kind of tiling works for the
various variants of the Eulerian triangle. I can understand in a general
way, that rises and falls are like my up and down squares. The
possibilities for a fall are dependent on the number of preceding rises,
and vice versa. But I've never studied these Eulerian triangles, so I don't
know anything in detail.

What other counting problems work this way with two kinds of objects, each
dependent on the previous number of the other kind of object?

Dale

On Thu, Apr 9, 2015 at 7:31 PM, Olivier Gerard <olivier.gerard at gmail.com>
wrote:

> On Thu, Apr 9, 2015 at 3:35 PM, Dale Gerdemann <dale.gerdemann at gmail.com>
> wrote:
>
> > Dear SeqFans,
> >
> >
> > I've collected here some number triangles whose row sums are OEIS
> > sequences. They all follow a basic parameterized formula. Most of these
> > triangles are not in OEIS, and those that are in OEIS are usually defined
> > with rather different formulae.
> >
> >
> Dear Dale,
>
> you have yourself noticed that the simplest cases are already in the OEIS,
> for good reasons, as generalizations of the binomial and eulerian
> triangles.
>
> Half of your triangles have simple and growing GCDs that should
> probably be simplified, factored away, and the sums have simple
> definitions.  If you submit them as Neil suggested, please come
> up with an idea of what they are counting (the binomial triangle
> decomposes subsets by number of elements = binary words by number
> of bits equal to 0 or 1, the eulerian triangle decomposes permutations
> by, among other things, rises). It will give some motivations to these
> entries.
>
> You sent the link to your videos. Next time please do it for your
> (proposed or not) sequences too instead of including them in a message.
> It was much too big for the standards of that mailing list.
>
> Regards,
>
> Olivier
>
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