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The 2-ordinal polytopes associated with the symmetric groups are the permutohedra. The number of codimension $k$ faces of the $n$th permutohedron is given in sequence A019538.

The second diagonal has some nice properties. For example, Alexander Povolotsky observed that these numbers $P_{i}$ arise as the right hand coefficients for the following sequence of expressions, indexed by $i$.

$n(n+1)[n + (n+1)] = 6(1 + 4 + 9 + \cdots + n^2 )$

$n(n+1)(n+2)[n + (n+1) + (n+2)] = 36(1 + (1+4) + (1+4+9) + \cdots + (1 + 4 + \cdots + n^2 ))$

This brings to mind the Leech sequence

$1 + 2^2 + 3^2 + \cdots + 24^{2} = 70^{2}$

for $n=24$, for which the first element of the list is expressed

$\frac{1}{3}(\sum_{i=1}^{n} i ) (2n + 1) = n^{2}$

If the squares of integers up to $n > 24$ cannot be summed to a square, it follows that the left hand side can never be a square.

The second diagonal has some nice properties. For example, Alexander Povolotsky observed that these numbers $P_{i}$ arise as the right hand coefficients for the following sequence of expressions, indexed by $i$.

$n(n+1)[n + (n+1)] = 6(1 + 4 + 9 + \cdots + n^2 )$

$n(n+1)(n+2)[n + (n+1) + (n+2)] = 36(1 + (1+4) + (1+4+9) + \cdots + (1 + 4 + \cdots + n^2 ))$

This brings to mind the Leech sequence

$1 + 2^2 + 3^2 + \cdots + 24^{2} = 70^{2}$

for $n=24$, for which the first element of the list is expressed

$\frac{1}{3}(\sum_{i=1}^{n} i ) (2n + 1) = n^{2}$

If the squares of integers up to $n > 24$ cannot be summed to a square, it follows that the left hand side can never be a square.

## 3 Comments:

12 16 07

Nice post Kea. I have learned about associahedra from your posts and now permutohedra, what other hedras are you thinking about? hehehehe Anyway, you inspired me to do one of my latest posts about the quantum numbers in Chemistry generated from the solutions to the Schroedinger eqn. Anyhow, there is a very interesting structure to the numbers and if we came up with just one other parameter, we'd have a set of 5 quantum numbers closed under the Reals...Check it out here. Happy Holidays BTW.

Hi Mahndisa! Good to hear from you. As usual, I am interested in all higher operad polytopes, as described by Batanin, Loday et al. I'll have a look at your post right now...

Dear Marni/Kea,

I hope that you will be not too much disappointed to find out that your =

link went to the wrong (but obviously more famous than me ;-) )

Alexander M. Povolotsky.

"The second diagonal has some nice properties. For example, Alexander Po=

volotsky observed that these numbers $P_{i}$ arise as the right hand coe=

fficients for the following sequence of expressions, indexed by $i$."

http://www.research.att.com/~njas/sequences/A000330

http://www.pme-math.org/journal/Proble

msF2006.pdf.

"1147 Proposed by Alexander Povolotsky, Goodrich Optical Systems, Chelmsford, MA"

There is also the proof (proved amongst others by the "submitter" :-))

http://www.math.fau.edu/web/PiMuEpsilon/pmespring2007.pdf.

Best Regards,

Alexander R. Povolotsky

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