A375505 Number of crystallized linear chord diagrams on n chords.

1, 2, 6, 25, 136, 923, 7557, 72767, 807896, 10180274, 143741731, 2250285510, 38715864581, 726596076239, 14780041925011, 324070919795226, 7622475922806634, 191515981769983447, 5120787153821434468, 145222986971201544125, 4355043425181710241819, 137728970544635824065325

Offset: 1

Author

Donovan Young, Aug 23 2024

Keywords

• nonn

Comments

In a linear chord diagram a "bubble" is defined as a set of consecutive vertices such that no two adjacent vertices are joined by a chord, i.e., "short" chords are not allowed. A bubble is therefore bounded externally either by short chords, or by the ends of the diagram. In a crystallized diagram, all chords are either short, or "bridge" two distinct bubbles, i.e., they have one vertex in one bubble, and the other vertex in a separate bubble. a(n) is the total number of such diagrams built from n chords.

Examples

			For n = 3, let the vertices of the linear chord diagram be A,B,C,D,E,F. There are two diagrams with a single short chord: (AF)(BE)(CD) and (AE)(BF)(CD). There are three diagrams with two short chords: (AB)(CF)(DE), (AD)(BC)(EF), and (AF)(BC)(DE). Finally, there is one diagram with all three chords short: (AB)(CD)(EF). In total, there is therefore a(3) = 6 crystallized diagrams.
		

Links

• Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024. See p. 18.

Crossrefs

Row sums of triangle A375504.