A183757 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a circle.

1, 1, 2, 6, 17, 74, 324, 1558, 7640, 38245

Offset: 0

Author

David Scambler, Jan 06 2011

Keywords

• nonn
• more

Comments

Interpret the super-Catalan sequence (A001003) as follows: Ways to insert parentheses in a string of n+1 symbols. The parentheses must be balanced but there is no restriction on the number of pairs of parentheses. The number of letters inside a pair of parentheses must be at least 2. Parentheses enclosing the whole string are ignored. Now picture the x's and parentheses as equally spaced unlabeled points on a circle with chords joining paired parentheses and x's having no chord. Circles thus produced may have n+1, n+3, ..., 3*n-1 points and up to n-1 chords. The circle may be rotated. a(n) is the count of unique patterns.

Examples

			super-Catalan(3) = 11: (xx)xx, x(xx)x, xx(xx), (xx)(xx), (xxx)x, x(xxx),((xx)x)x, (x(xx))x, x((xx)x), x(x(xx)), xxxx. This sequence counts unique patterns up to rotation so  a(3) = 6: (xx)xx, (xx)(xx), (xxx)x, ((xx)x)x, x(x(xx)), xxxx.
		

Crossrefs

Cf. A001003 super-Catalan numbers. A183758 similar but with reflections discounted. A183759 this sequence decomposed by number of chords in the circles.