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A219539 T(n,k) is the number of k-points on the left side of a crosscut of simple symmetric n-Venn diagram.

%I A219539 #26 Nov 01 2024 05:11:00
%S A219539 1,1,1,1,2,3,2,1,1,4,11,19,23,19,11,4,1,1,5,17,38,61,71,61,38,17,5,1,
%T A219539 1,7,33,107,257,471,673,757,673,471,257,107,33,7,1,1,8,43,161,451,977,
%U A219539 1675,2303,2559,2303,1675,977,451,161,43,8,1
%N A219539 T(n,k) is the number of k-points on the left side of a crosscut of simple symmetric n-Venn diagram.
%C A219539 A crosscut of a Venn diagram is defined as a segment of a curve which sequentially "cuts" (i.e., intersects) every other curve without repetition.
%C A219539 For n=2 and 3, there are 4 and 6 crosscuts respectively.
%C A219539 For n>3, there are either n crosscuts or none.
%C A219539 A k-point in a simple monotone Venn diagram is defined as being an intersection point that is incident to two k-regions.
%C A219539 The corresponding row sums are 3, 9, 93, .... (that is A007663).
%H A219539 K. Mamakani and F. Ruskey, <a href="http://arxiv.org/abs/1207.6452">A New Rose: The First Simple Symmetric 11-Venn Diagram</a>, arXiv:1207.6452 [cs.CG], 2012.
%H A219539 Andrei K. Svinin, <a href="https://arxiv.org/abs/1610.05387">On some class of sums</a>, arXiv:1610.05387 [math.CO], 2016. See p. 11.
%F A219539 For 1<=k<n-1, T(n, k) = (binomial(n-1, k) + (-1)^(k+1))/n, with n>=5 being prime.
%F A219539 T(n, k) - T(n, k-1) = (A000108(k-1) + 2*(-1)^(k+1))/n.
%e A219539 T(n, k) is defined for n>=5 being prime:
%e A219539   5:  1, 1, 1,
%e A219539   7:  1, 2, 3, 2, 1,
%e A219539   11: 1, 4, 11, 19, 23, 19, 11, 4, 1,
%e A219539   ...
%o A219539 (PARI) a(m) = {for (n=5, m, if (isprime(n), for (k=1, n-2,if (k==1, rk =1, rk = (binomial(n-1, k)+ (-1)^(k+1))/n);print1(rk, ", "););););}
%Y A219539 Cf. A000108, A007663.
%K A219539 nonn,tabf
%O A219539 5,5
%A A219539 _Michel Marcus_, Nov 22 2012