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Thursday, October 8, 2020 evatutin#90

Some of my colleagues do not understand why OEIS is needed at all and why calculate numerical series that no one needs (in their opinion). I will show one of the very interesting applications of the encyclopedia in my opinion - with its help you can establish correspondences between various combinatorial objects, which at first glance may seem different, but in fact are the same, only viewed from different angles.

So, not so long ago we counted X-based diagonal fillings of the DLS diagonals, which resulted in 3 sequences in the OEIS: A309283, A337302 and A337303. So it turns out that if in A337302 (the number of X-based fillings with a fixed diagonal) we remove the starting value a(1)=1 and duplicates of the other values, then we get the sequence A000316 associated with card matchings). Or, in other words,

A337302(n) = A000316(floor(n/2)) for all n>1.

This fact was noticed by Andrew Howroyd, for which special thanks to him!!!

And for the sequence A000316, which has been known for almost half a century, there are also exact formulas (for example, through the permanent of a 2n x 2n matrix with zeros on its diagonals, and all other values are filled with ones or in the form of a recurrent dependence (2*n-3)*a(n) = 2*(n-1)*(2*n-1)^2*a(n-1) + 4*(n-1)*(2*n-3)*a(n-2) - 16*(n-2)*(n-1)*(2*n-1)*a(n-3)), and a generating function, and much more...

The bottom line is that thanks to our calculations, OEIS and Andrew Howroyd, another connection has been established between the DLS and the already known combinatorial objects and problems, which is good news! And for A337302 there are isomorphism classes expressed in terms of the A309283 series. Maybe they will find a connection with something else...

PS. Now I'm worried about the fate of the sequence A337302... Will it be removed? Wouldn't want to... Will add a short description of this fact to A000316? In general, let's see how the OEIS editors will look at the revealed pattern ...

PPS. The same trick with the sequence A309283 does not work: after deleting the starting unit and duplicates, the sequences "0, 2, 3, 20, 67, 596" or "2, 3, 20, 67, 596" are not represented in the OEIS. Perhaps (again according to Andrew Howroyd) they need to be added...