A256330 Number of H&S Family matchings on n edges.

1, 3, 14, 84, 592, 4659, 39699, 359004, 3399164, 33378417, 337584612, 3498553682, 37006524557, 398312230440, 4351822041763, 48169486233388, 539303075161814, 6099303431601708, 69604032964928589, 800737747350839332, 9279033826462097649, 108236883894562489628

Offset: 1

Author

Aziza Jefferson, Mar 25 2015

Keywords

• nonn

Comments

The H&S Family of matchings is the family of matchings that can be drawn in the plane without crossings.

Jay Pantone has computed the first 1500 terms and has a conjectured g.f. - N. J. A. Sloane, Oct 06 2016

Consider the graph whose vertices are the arcs of a matching, where two vertices are connected if the corresponding arcs cannot be drawn on the same side without crossing. Matchings, where the graph obtained this way is connected, are in bijection with 3-edge-connected rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle (cf. A000264). - Ludovic Schwob, Jun 17 2025

Examples

			a(5)= 592; in canonical sequence form the two 3-noncrossing matchings it does not include are 1231435425 and 1234254153.
		

Links

• Ludovic Schwob, Table of n, a(n) for n = 1..200
• Michael Albert and Mireille Bousquet-Mélou, Permutations sortable by two stacks in parallel and quarter plane walks, European Journal of Combinatorics 43 (2015): 131-164. Also arXiv:1312.4487 [math.CO], 2013-2014.
• C. Haslinger and P. F. Stadler, RNA structures with pseudo-notes: Graph-theoretical, combinatorial, and statistical properties, Bulletin of Mathematical Biology 61 (1999), 437-467.
• Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.
• Jay Pantone, Approximate Asymptotic Analysis of Combinatorial Sequences, Experimental Math Seminar, Rutgers University, Oct 06 2016.

Crossrefs

Cf. A000264.