| Abstract: |
Combinatorial mathematics is often said to be the language of computer science:
the interplay between the two works in both directions.
Enumerative combinatorics is clearly important in computing,
since it is always handy to know how many things you might encounter.
Two on-line services run by N.J.A. Sloane,
as part of the On-Line Encyclopaedia of Integer Sequence (OEIS)
- and -
help identify integer sequences you might come across in your research.
Neil Sloane, in his personal testiment, ''My Favorite Integer Sequences'',
remarks of the common appearance of a particularly challenging integer sequence in two different settings,
the theories of the four dimensional root lattice and of quasi-crystals,
''we have not yet found a direct connection ...
Nevertheless, the occurrence of the same numbers in the two problems cannot be entirely coincidental.''
Is this mathematics or more mathematical moonshine?
This talk, like Sloane's testiment, proceeds by way of examples.
What about a mathematical expression like
k ( k - 1 ) / ( n + 1 ) * 1 / ( k n + 1 ) * binomial( ( k + 1 ) n, n ), n >= 0,
that for each fixed positive integral k gives an integer sequence?
Must there be nice combinatorial objects that these sequences enumerate?
For k = 2, the sequence is indeed that conjectured by Julian West, and first proved by Doron Zeilberger,
to enumerate certain two-stack sortable permutations, and is know to appear in other settings
(which was helpful in developing further proofs of West's conjecture).
But the OEIS has been silent on these sequences for k > 2, although, as we shall see,
there is a lot of highly suggestive circumstantial evidence.
One good sequence almost always leads to another,
and consideration of the divisibility properties of this family of sequences
leads to the enumeration of light paths bouncing through piles of reflecting glass plates.
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