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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A being a set, S belonging to 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}, let's take B={2}, then |{}∩{2}|=0 (even) and |{1}∩{2}|=0 (even), so S has eip.

LINKS

Table of n, a(n) for n=1..9.

V. Scharaschkin, The Odd and Even Intersection Properties, The Electronic Journal of Combinatorics, Volume 18, Issue 1 (2011), #P185.

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).

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

Cf. A218383.

Sequence in context: A146752 A022518 A113053 * A022503 A063861 A093632

Adjacent sequences:  A218381 A218382 A218383 * A218385 A218386 A218387

KEYWORD

nonn

AUTHOR

Michel Marcus, Oct 27 2012

STATUS

approved

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Last modified April 23 07:51 EDT 2019. Contains 322381 sequences. (Running on oeis4.)