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December 30

Real unit disc

The complex unit disc $|z|<1$ is very important in analysis, and similarly the real unit interval (0,1). What about the real unit disc (-1,1)? Is that as ubiquitous? Does it have a name other than (-1,1) or real unit disc? Thanks. I I want to parametrize something to $t\in (-1,1)$ for expository purposes and am wondering if that seems a little weird compared to the more conventional (0,1). 2601:648:8202:350:0:0:0:9435 (talk) 10:31, 30 December 2021 (UTC)[reply]

It is the (open) unit 1-ball. I can't think of a snappier term. --Lambiam 11:00, 30 December 2021 (UTC)[reply]

certain restricted permutations

Is there a name for the set of permutations of { 1, 2, .., n } such that the difference between adjacent items in the sequence is no more than k?

If we call the count S,

S(n,0) = 0 (no solution)
S(n,1) = 1 (the identity permutation)
S(n,k≥n-1) = n!

but 1<k<n-1 may be more interesting. —Tamfang (talk) 18:50, 30 December 2021 (UTC)[reply]

I don't know if this concept has a name. For n ≥ 2, S(n, 1) = 2 (there is also the reverse permutation). S(n, n-2) = (n-2)(n-1)! (subtract off the ones where 1 and n touch). Aside from that I suspect there will not be nice formulas. Danstronger (talk) 19:42, 30 December 2021 (UTC)[reply]

Thanks for the correction. So S(n,k) is always even. —Tamfang (talk) 22:46, 30 December 2021 (UTC)[reply]

S(n,2) is OEIS sequence A003274. Assuming the g.f. has been proven (it's not entirely clear), there is a linear recursive formula. --RDBury (talk) 01:44, 31 December 2021 (UTC)[reply]

PS. S(n,3) is OEIS sequence A174700. It lists an "empirical" g.f. which, if true, would imply it also has a linear recursive formula. It appears S(n,4), and S(n,k) with fixed k>4, are not in the OEIS. It's perhaps not too much of a leap to conjecture that S(n,k) has a linear recursive formula for any fixed k, though the formulas increase in complexity as k increases. Anyone need a topic for a Master's Thesis? --RDBury (talk) 02:29, 31 December 2021 (UTC)[reply]