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SUMMARY:Powers of 2 in high-dimensional lattice walks - Nikolai Beluhov (C
 ambridge)
DTSTART:20251106T143000Z
DTEND:20251106T153000Z
UID:TALK240550@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:Let Wd(n) be the number of 2n-step walks in ℤd which begin a
 nd end at the origin. We study the exponent of 2 in the prime factorisatio
 n of this number\; i.e.\, wd(n)=ν2(Wd(n)). We show that\, for each d\, th
 ere is a relationship between wd(n) and the number s2(n) of 1s in the bina
 ry expansion of n. For example\, wd(n)=s2(n) if d is odd and wd(n)=2s2(n) 
 if ν2(d)=1\; while wd(n)≥3s2(n) if ν2(d)=2. The pattern changes furthe
 r when ν2(d)≥3. However\, for each d\, we give the best analogous estim
 ate of wd(n) together with a description of all n where equality is attain
 ed. The methods we develop apply to a wider range of problems as well\, an
 d so might be of independent interest. 
LOCATION:MR12
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