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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.

%I A248929 #12 Nov 02 2014 11:58:35
%S A248929 1,2,1,4,3,1,12,7,4,1,81,25,11,5,1,2646,216,46,16,6,1,1422564,12784,
%T A248929 477,77,22,7,1,229809982112,11115851,45104,925,120,29,8,1
%N 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.
%C A248929 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.
%D A248929 Ian Anderson, Combinatorics of Finite Sets, Oxford University Press, 1987, pages 1-9.
%F A248929 PIP(k,k)=1
%F A248929 PIP(k+1,k)=C(k+1,1)=k+1
%F A248929 PIP(k+2,k)=C(k+2,2)+1
%F A248929 PIP(k+3,k)=2*C(k+3,3)+C(k+3,1)
%F A248929 PIP(k+4,k)=12*C(k+4,4)+C(k+4,3)+C(k+4,2)+1
%e A248929 Triangle PIP(n,k) begins:
%e A248929 n\k 1             2         3      4       5       6     7...
%e A248929 1   1
%e A248929 2   2             1
%e A248929 3   4             3         1
%e A248929 4   12            7         4      1
%e A248929 5   81            25        11     5       1
%e A248929 6   2646          216       46     16      6       1
%e A248929 7   1422564       12784     477    77      22      7     1
%e A248929 8   229809982112  11115851  45104  925     120     29    8
%e A248929 9                                  129315  1633    177   37
%e A248929 10                                         320026  2686  250
%e A248929 11                                                       4181
%Y A248929 Cf. A001206 (first column).
%K A248929 nonn,tabl,more
%O A248929 1,2
%A A248929 _John M. Ingram_, Oct 17 2014
%E A248929 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
%E A248929 Another row added to the data sequence by _John M. Ingram_, Nov 02 2014
%E A248929 Several new terms added to the example table by _John M. Ingram_, Nov 02 2014