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

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, 1, 7, 33, 107, 257, 471, 673, 757, 673, 471, 257, 107, 33, 7, 1, 1, 8, 43, 161, 451, 977, 1675, 2303, 2559, 2303, 1675, 977, 451, 161, 43, 8, 1

Offset: 5

Author

Michel Marcus, Nov 22 2012

Keywords

• nonn
• tabf

Comments

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.

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

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

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

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

Examples

			T(n, k) is defined for n>=5 being prime:
  5:  1, 1, 1,
  7:  1, 2, 3, 2, 1,
  11: 1, 4, 11, 19, 23, 19, 11, 4, 1,
  ...
		

Links

• K. Mamakani and F. Ruskey, A New Rose: The First Simple Symmetric 11-Venn Diagram, arXiv:1207.6452 [cs.CG], 2012.
• Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016. See p. 11.

Crossrefs

Cf. A000108, A007663.

Programs

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, ", "););););}

Formula

For 1<=k=5 being prime.

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