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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.

%I A367000 #34 Sep 02 2024 00:19:48
%S A367000 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,
%T A367000 99,82,72,0,329,2730,2310,1920,1605,1356,1155,988,848,730,658,0,3655,
%U A367000 30240,26460,22890,19845,17280,15105,13242,11634,10240,9027,7968,7310,0,47844
%N 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.
%C A367000 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.
%H A367000 Donovan Young, <a href="https://arxiv.org/abs/2311.01569">Counting Bubbles in Linear Chord Diagrams</a>, arXiv:2311.01569 [math.CO], 2023.
%H A367000 Donovan Young, <a href="https://arxiv.org/abs/2408.17232">Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams</a>, arXiv:2408.17232 [math.CO], 2024.
%F A367000 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)!)).
%e A367000 The first few rows of T(n,k) are:
%e A367000    0,   0;
%e A367000    2,   0,   0,   1;
%e A367000    8,   4,   2,   2,   0,   5;
%e A367000   42,  30,  20,  15,  12,  10,   0,  36;
%e A367000 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.
%o A367000 (PARI) N=2*n;
%o A367000 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); );
%o A367000 Tn=vector(N,x,0);
%o A367000 for(k=1,k=N,Tn[k]=polcoeff(polcoeff(G,N,y),k,w););
%Y A367000 The last entry in each row forms A278990. See also A079267.
%K A367000 nonn,tabf
%O A367000 0,3
%A A367000 _Donovan Young_, Oct 31 2023