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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 5 16:19 EDT 2020. Contains 334852 sequences. (Running on oeis4.)