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A365447
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Number of nonempty choice functions on a set of n alternatives
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0
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OFFSET
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1,2
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COMMENTS
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Number of choice functions f:2^A\{{}}->2^A\{{}} where A is an n-element set of variants such that f(X) is a nonempty subset of any nonempty X in 2^A (here 2^A denotes the power set of A).
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REFERENCES
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F. Aleskerov, D. Bouyssou, and B. Monjardet, Utility, Maximization, Choice and Preference, Springer, 2007, pp. 28-31.
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LINKS
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FORMULA
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a(n) = Product_{k=1..n} (2^k-1)^binomial(n, k).
log_2 a(n) = n*2^(n-1) + O(2^n/sqrt(n)).
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EXAMPLE
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a(1) = 1 since 2^{1} = {{}, {1}} and there exists only one function f:2^{1}/{{}}->2^{1}/{{}} satisfying f(X) is a nonempty subset of any nonempty X in 2^A, i.e., f({1})={1}.
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MATHEMATICA
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a[n_] := Product[(2^k - 1)^Binomial[n, k], {k, 1, n}]; Array[a, 6] (* Amiram Eldar, Oct 03 2023 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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