A367000 Triangle read by rows: T(n,k) is the total number of bubbles of size k found in linear chord diagrams on 2n vertices.

0, 0, 2, 0, 0, 1, 8, 4, 2, 2, 0, 5, 42, 30, 20, 15, 12, 10, 0, 36, 300, 240, 186, 147, 120, 99, 82, 72, 0, 329, 2730, 2310, 1920, 1605, 1356, 1155, 988, 848, 730, 658, 0, 3655, 30240, 26460, 22890, 19845, 17280, 15105, 13242, 11634, 10240, 9027, 7968, 7310, 0, 47844

Offset: 0

Author

Donovan Young, Oct 31 2023

Keywords

• nonn
• tabf

Comments

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. T(n,k) counts the total number of bubbles consisting of k > 0 vertices, counted across all linear chord diagrams on 2n > 0 vertices.

Examples

			The first few rows of T(n,k) are:
   0,   0;
   2,   0,   0,   1;
   8,   4,   2,   2,   0,   5;
  42,  30,  20,  15,  12,  10,   0,  36;
For n = 2, let the four vertices be A, B, C, D. The diagram consisting of the chords (A,B) and (C,D) has no bubbles. The diagram consisting of the chords (A,D) and (B,C) has two bubbles of size 1: The vertex A is one bubble and the vertex D is the other. The diagram consisting of the chords (A,C) and (B,D) is itself a bubble of size 4. Hence T(2,1) = 2 and T(2,4) = 1.
		

Links

• Donovan Young, Counting Bubbles in Linear Chord Diagrams, arXiv:2311.01569 [math.CO], 2023.
• Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024.

Crossrefs

The last entry in each row forms A278990. See also A079267.

Programs

PARI

N=2*n;
G=0; for(j=0,j=N/2, G=G+taylor((1/((1 + w*(-1 + w*y^2))^2))*((((w^2*y^2)/(2*(1 + w^2*y^2)^2))^j*(2*j)!/j!* (-1 + w)^2*(-1 + w*y^2)^2)/(1 + w^2*y^2) - ((y^2)/2)^j/j!*w*y^2*((-2 + 2*w + (3 -4*w)*w*y^2 + (w + 2*(-1 + w)*w^2)*y^4 + w^3*y^6 )*(2*j)!+(-y^4 + w*y^4+ w*y^6 - 2*w^2*y^6 + w^3*y^8 )*(2*j+2)!)),y,N+1); );
Tn=vector(N,x,0);
for(k=1,k=N,Tn[k]=polcoeff(polcoeff(G,N,y),k,w););

Formula

G.f.: Sum_{j=0..n} (1/(1 + w*(-1 + w*y^2))^2)*((((w^2*y^2)/(2*(1 + w^2*y^2)^2))^j*((2*j)!/j!)* (-1 + w)^2*(-1 + w*y^2)^2)/(1 + w^2*y^2) - ((y^2)/2)^j*(w*y^2/j!)*((-2 + 2*w + (3 - 4*w)*w*y^2 + (w + 2*(-1 + w)*w^2)*y^4 + w^3*y^6)*(2*j)! + (-y^4 + w*y^4 + w*y^6 - 2*w^2*y^6 + w^3*y^8)*(2*j+2)!)).