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URL: https://oeis.org/A218384

⇱ A218384 - OEIS

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A218384
Number of nonempty subsets S of the powerset of a set of size n that have the even intersection property.
2
1, 7, 71, 3071, 1966207, 270499994623, 2342736474457787596799, 86772003564839307784895323681111305093119, 59169757600268575861444773339439520883460632949720404019392912099891777942585343
OFFSET
1,2
COMMENTS
If A is a set, an element S of P(P(A)) \ {{}} has the even intersection property (eip) if there exists a set B (necessarily nonempty) included in A with |B∩S| even for each s in S.
For instance for S={{},{1}} of A={1,2}, we can take B={2}, and then |{}∩{2}|=0 (even) and |{1}∩{2}|=0 (even), so S has eip.
LINKS
Tilman Piesk, relationship to Boolean functions (Wikiversity)
V. Scharaschkin, The Odd and Even Intersection Properties, The Electronic Journal of Combinatorics, Volume 18, Issue 1 (2011), #P185.
Steve Wright, Some enumerative combinatorics arising from a problem on quadratic nonresidues, Australas. J. Combin. 44 (2009), 301-315.
FORMULA
a(n) = 1 + 2*Sum_{i=0..n-1} (-1)^(n-i-1)*(2^(2^i-1)-1)*(Product_{j=1..i} (2^(n-j+1)-1)/(2^j-1)) * 2^binomial(n-i,2).
a(n) ~ 2^(n + 2^(n-1)). - Vaclav Kotesovec, Apr 03 2021
Apparently, a(n) = A051179(n) - A305737(n). - Tilman Piesk, Jan 26 2025
EXAMPLE
For |A|=2, A = {1,2} and P(A) = {{}, {1}, {2}, {1,2}}
S can be
{{}, {1}, {2}, {1,2}}
{{}, {1}, {2}}
{{}, {1}, {1,2}}
{{}, {2}, {1,2}}
{{1}, {2}, {1,2}}
{{}, {1}} has eip, with B={2}
{{}, {2}} has eip, with B={1}
{{}, {1,2}} has eip, with B={1,2}
{{1}, {1,2}}
{{2}, {1,2}}
{{1}, {2}}
{{}} has eip, with B={1,2}
{{1}} has eip, with B={2}
{{2}} has eip, with B={1}
{{1,2}} has eip, with B={1,2}
So we have 7 S with eip.
MAPLE
A218384:=n->1+2*add((-1)^(n-i-1)*(2^(2^i-1)-1)* product((2^(n-j+1)-1)/(2^j-1), j=1..i)*2^binomial(n-i, 2), i=0..n-1): seq(A218384(n), n=1..10); # Wesley Ivan Hurt, Dec 11 2015
MATHEMATICA
Table[1 + 2 Sum[((-1)^(n - i - 1)) (2^(2^i - 1) - 1) Product[(2^(n - j + 1) - 1)/(2^j - 1), {j, 1, i}] 2^Binomial[n - i, 2], {i, 0, n - 1}], {n, 9}] (* Michael De Vlieger, Dec 11 2015 *)
PROG
(PARI) e(m) = {for (n=1, m, v = 1+2*sum(i=0, n-1, ((-1)^(n-i-1))*(2^(2^i-1)-1)* prod(j=1, i, (2^(n-j+1)-1)/(2^j-1))*2^binomial(n-i, 2)); print1(v, ", "); ); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 27 2012
STATUS
approved