OFFSET
0,4
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. T(n,k) is the number of such crystallized diagrams built from n > 0 chords, exactly k > 0 of which are short.
LINKS
Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024.
EXAMPLE
Triangle begins:
1;
1, 1;
2, 3, 1;
6, 12, 6, 1;
24, 62, 39, 10, 1;
120, 396, 296, 95, 15, 1;
...
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), and so T(3,1) = 2. There are three diagrams with two short chords: (AB)(CF)(DE), (AD)(BC)(EF), and (AF)(BC)(DE), and so T(3,2) = 3. Finally, there is one diagram with all three chords short: (AB)(CD)(EF), and so T(3,3)=1.
MATHEMATICA
F[n_]:=Sum[Factorial2[2*i-1]*x^i, {i, 0, n}];
T[n_, k_]:=Sum[(-1)^(n-k-l)*Factorial2[2*l-1]*Binomial[2*n-k, 2*l]*Coefficient[F[n]^(k+1), x, n-k-l], {l, 0, n-k}];
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Donovan Young, Aug 18 2024
STATUS
approved