Necklaces and Slimes
Date
2020-05
Authors
Park, Jina
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Abstract
It was asked if one can find a bijective map between the following two objects:
binary necklaces with n black beads and k white beads and certain (n, k)-codes
whose weighted sum is 0 modulo n (Brauner et al.,2019 [9]; Chan, 2019 [10]).
The former object is one that has been studied for ages, whereas the latter one
was shown to be the states in a dollar game played on a cyclic graph (Corry &
Perkinson, 2018 [11]).
The case when n and k are coprime was shown in [9] and it is easily described
by using rotation. We show that in the general case, all that one needs to
construct the bijective map is to construct a rotation-invariant and weight
increasing map (riwi-map) on the codes. When n and k are coprime the simple
cyclic rotation works as a riwi-map. We show that when n or k is prime, a new
map called a slime migration works as a riwi-map and hence allows one to get a
bijective map as a result.
Description
Keywords
an (n, k)-binary necklace, a chip-firing game, a slime migration, a riwi-map,
Citation
Park, J. (2020). Necklaces and Slimes (Unpublished thesis). Texas State University, San Marcos, Texas.