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A248929
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Triangle read by rows: T(n,k) = PIP(n,k) is the number of maximal families of sets from {1,2,...,n} with the property that if A and B are sets in the family, then |AB|>=k.
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0
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1, 2, 1, 4, 3, 1, 12, 7, 4, 1, 81, 25, 11, 5, 1, 2646, 216, 46, 16, 6, 1, 1422564, 12784, 477, 77, 22, 7, 1, 229809982112, 11115851, 45104, 925, 120, 29, 8, 1
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OFFSET
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1,2
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COMMENTS
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A family of sets has the k (k>=1) pairwise intersection property (PIPk) means that if A and B are sets in the family, then |AB|>=k. A family of sets with PIPk is maximal means no set can be added to the family while maintaining PIPk. (If C is a set not in the family, then there exists a set D in the family such that |CD|<=k-1.) PIP(n,k) is the number of maximal families of sets from {1,2,...,n} with PIPk.
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REFERENCES
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Ian Anderson, Combinatorics of Finite Sets, Oxford University Press, 1987, pages 1-9.
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LINKS
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FORMULA
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PIP(k,k)=1
PIP(k+1,k)=C(k+1,1)=k+1
PIP(k+2,k)=C(k+2,2)+1
PIP(k+3,k)=2*C(k+3,3)+C(k+3,1)
PIP(k+4,k)=12*C(k+4,4)+C(k+4,3)+C(k+4,2)+1
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EXAMPLE
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Triangle PIP(n,k) begins:
n\k 1 2 3 4 5 6 7...
1 1
2 2 1
3 4 3 1
4 12 7 4 1
5 81 25 11 5 1
6 2646 216 46 16 6 1
7 1422564 12784 477 77 22 7 1
8 229809982112 11115851 45104 925 120 29 8
9 129315 1633 177 37
10 320026 2686 250
11 4181
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Term PIP(6,2) (12778 should be 12784) in the data sequence and in the example table corrected by John M. Ingram, Nov 02 2014
Several new terms added to the example table by John M. Ingram, Nov 02 2014
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STATUS
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approved
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