An unusual series that produces pi was discovered by Jonas Castillo Toloza in 2007; the series consists of the reciprocals of the triangular numbers and, as such, could be detected in Pascal’s triangle.

http://www.cut-the-knot.org/arithmetic/algebra/TriPiInPascal.shtml ]]>

Unfortunately, though, that doesn’t help us obtain a good approximation for the product of all the numbers in each row of Pascal’s triangle, which is the aim of the calculations in this blog post. For n=1,..,10, the ratio between the actual product and the product of the numbers you get from the Gaussian approximation takes the values: 1.06747, 1.28577, 1.36777, 1.3446, 1.237, 1.07058, 0.87397, 0.674019, 0.49155, 0.339207. The limit of this ratio for large n turns out to be zero.

So, alas, this is one problem where the approximation you’ve described is not applicable.

]]>Bernoulli 0.5, De Moivre 0.798 | Bernoulli 0.5, De Moivre 0.564 | Bernoulli 0.375, De Moivre 0.461 | Bernoulli 0.375, De Moivre 0.399 | Bern 0.312, De M 0.357 | Bern 0.312, De M 0.326, | Bern 0.273, De M 0.302 | Bern 0.273, De M 0.282 | Bern 0.246, De M 0.266 | Bern 0.246, De M 0.252 |

The way I do it the approximation improves with increasing n. My goal is to set a lower bound on the number of genes having an influence on any given biological manifestation. So, for example, if the bell curve for adult height solves to a 12 element binomial (13 columns) I think that implies a minimum of 12 genes with an influence on cell production rate.

]]>I’ve been using

to mean, not the binomial coefficient, but a certain set whose size is that binomial coefficient: the set of -element subsets of But this set corresponds to a basis of the th exterior power of , usually called In quantum physics, this is the Hilbert space for identical fermions each having degrees of freedom.

I’ve been using

to mean, not the number of balls in a -dimensional pyramid with edges of length but a certain set whose size is that number: the set of -element multisubsets of But this set corresponds to a basis of the th symmetric power of usually called This is the Hilbert space for bosons each having degrees of freedom.

So, when we get our hands on a specific bijection

when we also get an isomorphism of Hilbert spaces

relating bosons and fermions. Is this good for something? I don’t know. But this is the kind of thing one can do.

]]>I believe this is because the approximation works very well around the mean, but not so well at the tails, and while that doesn’t matter much if you’re summing the terms (which are small at the tails), when you’re taking their product it leads to large discrepancies.

]]>where:

will yield non-negative values for , and a dot product with of .

So what the bijection is doing is taking the -dimensional pyramid, changing the coordinates to embed it in the appropriate affine hyperplane, and then treating the new coordinates of each object in the pyramid as counts for the various values in the coordinates of each object of the -dimensional pyramid.

]]>The other specifies the *number of elements* of each possible value, giving a -tuple:

The bijection between triangular numbers:

gives the “count representation” of one triangular number from a change of basis and origin of the “coordinate representation” of the other triangular number:

where and .

]]>If , then the number of coordinates of that are equal to a certain value, , is:

where we define and in addition to the actual coordinates , .

]]>If and , the coordinates of are given by:

where is the count of the number of coordinates of that are equal to .

Or to put this another way, is one more than the number of coordinates of that are less than or equal to .

]]>Actually, you can’t quite just flip the coordinates, if you want to start both paths at (1,0). But that’s just matter of tweaking the choice of coordinates; geometrically, the paths are flipped.

I don’t have time to draw a diagram of this right now, but it’s very easy to draw these paths by hand, and doing that for the first dimension-changing example:

… makes it easy to see what’s going on.

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