Necklaces and Slimes




Park, Jina

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<p>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]).</p> <p>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.</p>



An (n, k)-binary necklace, A chip-firing game, A slime migration, A riwi-map,


Park, J. (2020). <i>Necklaces and Slimes</i> (Unpublished thesis). Texas State University, San Marcos, Texas.


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