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

5,5

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).

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

Links

Table of n, a(n) for n=5..64.

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.

Formula

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

Example

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,
  ...

Prog

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

Crossrefs

Cf. A000108, A007663.

Sequence in context: A058294 A323834 A082868 * A154556 A260228 A316657

Adjacent sequences: A219536 A219537 A219538 * A219540 A219541 A219542

Keyword

nonn,tabf

Author

Michel Marcus, Nov 22 2012

Status

approved