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T(n,k) is the number of k-points on the left side of a crosscut of simple symmetric n-Venn diagram.
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%I #26 Nov 01 2024 05:11:00

%S 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 1,7,33,107,257,471,673,757,673,471,257,107,33,7,1,1,8,43,161,451,977,

%U 1675,2303,2559,2303,1675,977,451,161,43,8,1

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

%C 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 For n=2 and 3, there are 4 and 6 crosscuts respectively.

%C For n>3, there are either n crosscuts or none.

%C A k-point in a simple monotone Venn diagram is defined as being an intersection point that is incident to two k-regions.

%C The corresponding row sums are 3, 9, 93, .... (that is A007663).

%H 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 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 For 1<=k<n-1, T(n, k) = (binomial(n-1, k) + (-1)^(k+1))/n, with n>=5 being prime.

%F T(n, k) - T(n, k-1) = (A000108(k-1) + 2*(-1)^(k+1))/n.

%e T(n, k) is defined for n>=5 being prime:

%e 5: 1, 1, 1,

%e 7: 1, 2, 3, 2, 1,

%e 11: 1, 4, 11, 19, 23, 19, 11, 4, 1,

%e ...

%o (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 Cf. A000108, A007663.

%K nonn,tabf,changed

%O 5,5

%A _Michel Marcus_, Nov 22 2012