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A248929 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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

REFERENCES

Ian Anderson, Combinatorics of Finite Sets, Oxford University Press, 1987, pages 1-9.

LINKS

Table of n, a(n) for n=1..36.

FORMULA

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

EXAMPLE

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

CROSSREFS

Cf. A001206 (first column).

Sequence in context: A067410 A213947 A188403 * A109977 A201199 A318459

Adjacent sequences:  A248926 A248927 A248928 * A248930 A248931 A248932

KEYWORD

nonn,tabl,more

AUTHOR

John M. Ingram, Oct 17 2014

EXTENSIONS

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

Another row added to the data sequence by John M. Ingram, Nov 02 2014

Several new terms added to the example table by John M. Ingram, Nov 02 2014

STATUS

approved

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Last modified October 16 12:52 EDT 2019. Contains 328060 sequences. (Running on oeis4.)