MathDB

Let

n=2^{2018}

and let

S=\{1,2,\ldots,n\}

. For subsets

S_1,S_2,\ldots,S_n\subseteq S

, we call an ordered pair

(i,j)

murine if and only if

\{i,j\}

is a subset of at least one of

S_i, S_j

. Then, a sequence of subsets

(S_1,\ldots, S_n)

S

is called tasty if and only if:
1) For all

i

i\in S_i

. 2) For all

i

\displaystyle\bigcup_{j\in S_i} S_j=S_i

. 3) There do not exist pairwise distinct integers

a_1,a_2,\ldots,a_k

with

k\ge 3

such that for each

i

(a_i, a_{i+1})

is murine, where indices are taken modulo

k

. 4)

n

divides

1+|S_1|+|S_2|+\ldots+|S_n|

.
Find the largest integer

x

such that

2^x

divides the number of tasty sequences

(S_1,\ldots, S_n)

.
Proposed by Vincent Huang and Brandon Wang

Problem Statement