A256333 Number of R&G Family matchings on n edges.

1, 3, 13, 61, 301, 1552, 8277, 45284, 252753, 1433633, 8239993, 47887467, 280927846, 1661387046, 9894376821, 59288650788, 357198545904, 2162437157263, 13147835385477, 80251977589719, 491573099486143, 3020738578507674, 18617035563669489, 115046892012376542, 712710925868858139, 4425312432316379040, 27535525144298975942, 171670784266383750322, 1072246008621559982926, 6708644077265798380125

Offset: 1

Author

Aziza Jefferson, Mar 25 2015

Keywords

• nonn

Comments

The R&G Family of matchings is the family of matchings formed by first vertex insertions into the hairpin or single edge (as long as the inserted edge does not have an outer edge connecting the first and last vertex), then edge inflations by ladders of the original single edge or hairpin.

Examples

			a(3)=13 because of the 15 matchings on 3 edges, two do not lie in the R&G Family. In canonical sequence form the missing matchings are given by 121323 and 123123. a(4)= 61 out of the 105 matchings on 4 edges, one such matching which does not lie in the R&G Family is given by 12234314.
		

Links

• Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.
• J. Reeder and R. Giegerich, Design, implementation and evaluation of a practical pseudo knot folding algorithm based on thermodynamics, BMC Bioinform. 5 (2004), Article #104.
• C. Saule, M. Régnier, J.-M. Steyaert, and A. Denise, Counting RNA pseudoknotted structures, J. Comput. Biol. 18(10), (2011), 1339-1351.

Programs

Maple

f := RootOf(x^2*_Z^4 + x*(1-x)*(_Z-x*_Z)*_Z - (1-x)^2*_Z + (1-x)^2);
series(f, x=0, 30);

Formula

G.f. f satisfies x^2f^4 + x(1-x)^2f^2 - (1-x)^2f + (1-x)^2.