[Retros] happy prime new year
Noam Elkies
elkies at math.harvard.edu
Sun Jan 29 14:05:45 EST 2017
andrew buchanan <andrew at anselan.com> writes:
> (1) There is a naturally occurring combinatorial way to reach 2017.
> It's the number of permutations of 7 elements which *don't* contain
> a double rise (e.g. 1342 contains a rise from 1 to 3, and then
> another rise from 3 to 4, so is excluded from the count).
> Equivalently, it's those permutations which *don't* have an embedded
> increasing run (e.g. 1423 contains 123 (although these do not appear
> adjacently) so is excluded from the count.
This equivalence is not obvious, though I have not tried to
confirm or refute it.
> Several queue problems are already built around the Euler zigzag
> function, which is not unrelated. However I can't begin to figure out
> a way to represent this new approach in chess terms.
Me neither... 2017 is also the number of permutations of 1,2,...,16
that, when arranged in a 4x4 array, are increasing by rows, columns,
and diagonals. (Without the diagonal condition that's a special case
of the hook-number formula, giving 24024.) This seems more amenable
to a chess realization, though it would still take some work even once
we remove 1,2,15,16 whose locations are forced.
NDE
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